Glossary

A single alphabetical list of every term defined across the book. Each term is defined in full in its home chapter; this page gathers them for quick reference and search.

A

Term	Meaning
A/B test (online eval)	Serve two models to live user halves and compare a business metric — the real verdict.
Ablation	A test that removes one part of a method to see what the part was worth (§4).
Ablation study	An experiment that removes one component of a model at a time to see what each contributes (§8.6, Step 0).
Activation function	The fixed, simple nonlinear bend a layer applies to its weighted sums, e.g. the sigmoid or ReLU (§3).
Activation function \(\phi\)	Nonlinear function applied to \(z\) (sigmoid, tanh, ReLU); its output \(a\) is the neuron’s activation.
Activations	The numbers a layer outputs; each unit’s is computed from the previous layer’s (§5.2).
Adam	An optimizer that divides each parameter’s step by the recent size of its own gradient (§5; Kingma and Ba, 2015).
Adaptor	A small trained network that maps the LLM’s vector into the recommender’s space and shrinks it, as in KAR (§4).
Adjacency matrix \(A\)	The whole graph’s connections as a square grid of 0/1.
Advantage	How much better an output scored than the group’s mean reward, divided by the group’s standard deviation in GRPO (§4).
Agentic recommender	An LLM that plans, remembers, and calls tools over multi-turn conversation (§2.4).
Alignment	A positive pair is pulled together: two views of one node, or in DirectAU an interacting user and item (§3.3).
Alignment & uniformity	Pull positives together / spread embeddings on the sphere (DirectAU); its popularity-weighted uniformity is said to help reduce exposure bias (items the system showed more got more clicks; a claim of the DirectAU paper, not tested here).
Alignment loss (RLMRec)	Ties the vectors of LLM-written user/item profiles to the collaborative (CF) embeddings — contrastively (RLMRec-Con) or by rebuilding masked profiles (RLMRec-Gen).
ALS / iALS	Alternating least squares: fix \(Q\) and solve for \(P\) in closed form, then swap. iALS is the implicit-feedback version, weighting each cell by a confidence (§5).
Anchor	The node \(i\) whose first view a contrastive loss compares with all second views (§3.2).
Angle (of an arrow)	Its direction, measured from the romance axis and positive toward action (§8).
ANN	Approximate nearest-neighbor search — fast, slightly-inexact vector retrieval; the candidate-generation engine.
ANN (approximate nearest neighbor)	Fast similarity search that makes dot-product retrieval scale to millions of items.
ANN (approximate nearest-neighbor)	Fast vector search that returns almost the closest vectors to a query, trading a little accuracy for large speed-ups; how embedding retrieval is served (Training and Serving a Recommender §7).
ANN (Approximate Nearest-Neighbor) search	Search that finds almost-the-best items by consulting a precomputed index instead of scoring every item, trading exactness for a large speed-up.
AP / MAP	Average Precision (Precision@\(k\) summed over hit positions, divided by $
API (application programming interface) model	One you cannot download: your program sends each request to the maker’s servers over the internet and pays per use.
Approximate nearest neighbor (ANN)	Finding items whose vectors are close to a query vector by skipping most of the catalogue, at the price of sometimes missing the very best one (§15).
Arm	One choosable option (an item); “pulling” it = showing it. From the slot-machine metaphor.
Arm (tower, stream)	One of the two parallel sub-networks whose outputs meet in one logistic head; the papers also say tower or stream. Not the retrieval chapters’ tower, an encoder of only the user or only the item.
Attention	\(\mathrm{softmax}(QK^\top/\sqrt{d_k})V\): each position takes a blend of the value vectors of the positions it may see, weighted by how well its query matches their keys (§4.1).
Attention mask	Which tokens each token may look at: all of them (bidirectional — BERT, BERT4Rec) or only itself and earlier ones (causal — GPT, SASRec; §5.1).
Attention weight	The softmax of the scaled query-key scores: how much one position attends to another; positive, summing to \(1\).
Attributed / featureless	Whether nodes carry input feature vectors; recsys nodes are featureless (IDs only).
AUC	Area under the ROC curve: the fraction of (positive, negative) pairs ordered correctly; BPR is its smooth stand-in (§9).
AUC / ROC	Area Under the Receiver Operating Characteristic (ROC) Curve = P(score of a random relevant > random irrelevant) = the fraction of correctly ordered (relevant, irrelevant) pairs.
Augmentation	A small change to the input that should not change what it is (a flipped photo is the same cat); the recipe comes from image learning (SimCLR) and gives GCL its views (§3.1).
Autoencoder	A network trained to reproduce its own input through a narrow middle layer (§6.2).
Autoregressive decoding	Generate one token, append it, run the model again, repeat — the GPT generation loop (SASRec’s next-item prediction is one step of it; §5.4).
Autoregressive generation	Writing an output one token at a time, each token chosen given the ones before it (§2.2).
Auxiliary task	The extra, label-free objective added alongside the main (BPR) loss.

B

Term	Meaning
Back-propagation	Computing all weight-gradients by the chain rule, backward through the net.
Back-propagation (backprop)	The chain rule run backward from the loss, to find how the loss changes with every weight (§3).
Backbone	The base model an extra loss is added to; here LightGCN (§1, §7).
Backbone, lift	The base model a method is added to, and the gain over it (§4, RLMRec).
Bag-of-words vector	A paper described by which vocabulary words occur in it: one entry per word; in the standard Cora files (\(1{,}433\) words) each entry is \(1\) if present and \(0\) if absent; counts in our toy (§6).
Bandit	An MDP in which actions never change the next state (one state, or a fresh context each round); such actions are called one-shot.
Bandit feedback	You observe the reward only for the action you took, never the alternatives.
Base-rate effect	A rare condition (low prior) makes even an accurate test’s positives mostly false.
Baseline	A simple model to compare others with, such as plain matrix factorization with BPR (§15).
Basis	A set of vectors from which every vector can be built in exactly one way as a weighted sum.
Batch / Layer Norm	Re-center/re-scale activations for training stability (LayerNorm is the normalization the Transformer uses, §7).
Batch normalization / layer normalization	Re-center and re-scale activations across a batch / within one example: \([2,4,6]\to[-1.22,0,1.22]\) (§5.2).
Bayes’ rule	posterior \(\propto\) likelihood \(\times\) prior.
Bayesian Personalized Ranking (BPR)	A ranking loss for implicit data: train so that an item the user interacted with scores above a randomly sampled unobserved one (§1; Loss Functions & Regularizers §9).
Beam search	A search that keeps only the best few partial lists at each step (§4).
Beam search (decoding)	Keep the \(W\) best partial ID prefixes per step; the completed tuples, ranked, are the top-\(K\) recommendations (§2.2).
Beam width \(W\)	How many partial candidates beam search carries forward at each step; the surviving completed tuples, ranked by probability, are the top-\(K\) (§2.2).
Benchmark framework	RecBole / Cornac / Elliot / Microsoft Recommenders / LensKit — unified datasets + models + protocol.
Benchmaxxing	Training on (or near) benchmark data so a high score can reflect memorization, not ability — why one leaderboard number is untrustworthy.
Bernoulli	One yes/no with parameter \(p\); \(-\log\) → BCE.
Bernoulli trial	One try with two outcomes, such as click / no click, with unknown chance \(\mu\).
BERT	Bidirectional Encoder Representations from Transformers: trained by masked language modeling (pick about 15% of the tokens, mostly replaced by [MASK], and predict them), originally with a next-sentence objective that was later often dropped; an encoder for understanding text → embeddings (Devlin 2019; §5.1).
BERT4Rec	A bidirectional Transformer trained with the cloze (masked-item) objective.
Beta	Distribution over a probability; conjugate prior for Bernoulli.
Beta posterior	\(\text{Beta}(1+s,1+f)\) for \(s\) clicks, \(f\) non-clicks; conjugate to the Bernoulli click.
Bias–variance trade-off	Error splits into systematic mis-fit (bias) and sensitivity to the sample (variance); model complexity trades one against the other (§4).
Biases \(\mu\), \(b_u\), \(b_i\)	The global mean rating, a user’s generosity or harshness, and an item’s broad appeal; added before the dot product in biased MF (§5).
BIGRec	Bi-step grounding: an instruction-tuned LLM writes the next item by name, and the written text is matched to the nearest catalog item by embedding (§5).
Binary entropy	The entropy of a two-outcome variable with probabilities \(\theta\) and \(1-\theta\): \(-\theta\log_2\theta-(1-\theta)\log_2(1-\theta)\), which peaks at \(1\) bit for \(\theta=\tfrac12\) (§7).
Bipartite graph	Nodes in two groups; edges only cross between groups (users ↔︎ films).
Bit	The information in one fair coin flip; an event of probability \(\tfrac12\) carries \(1\) bit, and \(-\log_2 p\) bits in general.
Bits / nats	Units of entropy: \(\log_2\) gives bits, \(\ln\) gives nats; \(1\) bit \(=\ln 2\approx0.693\) nats (§7).
Blockbuster	A film that very many people watch: a node of huge degree (§13).
BM25 / hybrid retrieval	“Best Matching 25”, the standard keyword-scoring refinement of TF-IDF / blending its (sparse) scores with embedding (dense) scores in one query.
BM3	A multimodal recommender that trains without negative samples by comparing an embedding with a dropout copy of itself (§5).
Bonferroni / Holm	Multiple-comparison fixes for \(m\) tests: Bonferroni compares each \(p\) to \(\alpha/m\); Holm is a stepwise version that is never more conservative: it keeps every result Bonferroni keeps, and sometimes more (for \(p=0.012,\,0.020,\,0.024\) Bonferroni keeps one, Holm all three; §8.6 Step 3).
Bootstrap	Estimating how much a score wobbles by re-drawing the test cases at random with replacement (a case may be drawn more than once) many times and recomputing the score each time; it needs no bell-shape assumption (Evaluation Metrics §11d; named in §8.7).
BPR	Bayesian Personalized Ranking: a loss that teaches a recommender to rank an item a user chose above one they did not; it applies a sigmoid to the score gap (§3).
BPR loss	Pairwise ranking loss: push observed pairs above sampled negatives; Ann’s triple (Titanic over Matrix) costs \(0.027\) (§15).

C

Term	Meaning
Calibration	A model is calibrated when the numbers mean what they say: of impressions scored \(0.3\), about \(30\%\) are clicked (Evaluation Metrics §11).
Calibration / ECE	A model is calibrated if items it scores \(0.8\) are clicked about \(80\%\) of the time; the Expected Calibration Error averages the per-bin absolute gap between predicted and observed rates, weighting each bin by its share of predictions.
Capacity	The range of functions a model can express; a weight matrix on already-trained vectors adds none (§14).
Categorical	One of \(K\) classes; \(-\log\) → cross-entropy; paired with softmax.
Causal mask	Lower-triangular mask: position \(t\) may attend only to positions \(\le t\) (no peeking at the future).
CDF \(F(x)\)	Cumulative distribution function \(F(x)=\Pr(X\le x)\) — the running total of probability; a different function describing the same distribution.
Cell state	The LSTM’s protected memory line, edited only by a forget multiply and a write add (§3.3).
Censored reward	A reward not yet known: no click so far, but one may still come (§7).
Central Limit Theorem	The average of many independent noisy pieces is approximately bell-shaped around the truth; with enough pairs it keeps the \(t\)-test a fair approximation when the differences are not exactly normal, but not with a handful (§8.3, Fact 2).
Chain rule	Derivative of \(f(g(x))\) = \(f'(g)\cdot g'\); the basis of back-propagation.
Characteristic equation / polynomial	\(\det(A-\lambda I)=0\); its left side is the characteristic polynomial, and its roots are the eigenvalues.
Chebyshev polynomials	The family \(T_0=1\), \(T_1=\lambda\), \(T_{j+1}=2\lambda T_j-T_{j-1}\): the same filters a plain polynomial can express, the basis ChebyCF interpolates in; interpolating at Chebyshev points spreads the error nearly evenly over \([-1,1]\).
CIN (compressed interaction network)	xDeepFM’s explicit-interaction arm: crosses between whole field vectors, mixed into a few new vectors by learned weights.
Citation network	A graph whose nodes are papers and whose edges are citations; the GCN paper treats them as undirected (§6).
Client / server	Devices holding private data / the coordinator that aggregates their updates.
Closed-form / training-free CF	Recommenders that apply a fixed filter with no gradient training (GF-CF, PSGE, by their authors’ account).
Cloze objective	Hide some of a user’s items and predict them from the rest (BERT4Rec; §10).
Codebook / codeword	A small fixed menu of prototype vectors (the codebook); each entry is a codeword. Quantizing a vector means replacing it by its nearest codeword’s symbol (§2.2).
Cold item	A film with almost no clicks yet (§4 box).
Cold start	No history yet for a new user/item; CF cannot act.
Cold start (serving)	Having no learned vector for a brand-new item to put in the index, or for a brand-new user to query it with (§8).
Cold-start	A new user or item with no history, so no learned row to look up (§3).
Collaborative filtering (CF)	Recommend using interaction patterns across users (no content).
Collaborative signal	What can be learned from who interacted with what (From Graphs to LightGCN, Self-Supervised & Contrastive Learning, The Spectral / Graph-Filter View), independent of item content.
Column space / image	Every output a matrix can produce — all weighted sums of its columns.
Complement \(\neg B\)	“Not \(B\)”; \(\Pr(\neg B)=1-\Pr(B)\) (§1).
Component / entry	One number inside a vector.
Computational graph	One node per quantity, one arrow per “is computed from” — the picture back-propagation walks backward (§5).
Concat (early fusion)	Append the LLM feature vector to the ID embedding — the quick first try, at the cost of a fatter serving vector. A frozen vector adds a fixed content score; only an adaptor or an FM or MLP scorer lets the model re-weigh it (§4).
Concatenation	Gluing vectors side by side: three layers of \(2\) numbers give \(6\) numbers per node (§14).
Condition number	The largest singular value divided by the smallest, \(\sigma_1/\sigma_{\min}\) (§6); a huge value means a solve or inverse with that matrix is numerically unreliable.
Conditional probability \(\Pr(A\mid B)\)	Probability of \(A\) given \(B\) occurred \(=\Pr(A\cap B)/\Pr(B)\).
Confidence \(c_{ui}\)	How much to trust an implicit signal: \(c_{ui}=1+\alpha\,n_{ui}\), growing with the play count \(n_{ui}\) (§5).
Confidence interval (CI)	\(\bar d \pm t^{*}s_d/\sqrt n\); the effect size with its uncertainty. A \(95\%\) CI excludes \(0\) exactly when the two-sided test at \(\alpha=0.05\) is significant (a \(99\%\) CI goes with \(\alpha=0.01\); §8.5, §8.6 Step 4).
Conjugate prior	A prior whose posterior is the same family (Beta↔︎Bernoulli).
Consistent policy	One whose regret grows more slowly than any power \(T^{c}\) on every bandit of the class; none can beat \(\ln T\) growth (Lai and Robbins, §2).
Constrained decoding	Restricting what the model may generate to a valid vocabulary (here, real item identifiers), so an invalid answer cannot be produced at all (§2.2).
Content-based filtering	Recommend items similar (in features) to what the user liked.
Context (context window)	The longest input a model reads at once, in tokens: “8K context” \(\approx\) 8,000 tokens (word pieces). Longer text is cut off.
Context vector	The weighted average of the values under the attention weights; the summary that scores the next item (§4).
Contextual bandit	The reward depends on a context (user/item features); LinUCB models it linearly.
Contour (level set)	A curve along which \(f\) keeps one constant value, like a height line on a map; the gradient is perpendicular to it (§4).
Contrastive (InfoNCE) term	Added to the recommender’s loss: it pulls a node’s collaborative and projected semantic vectors together (the numerator) and pushes its collaborative vector away from other nodes’ semantic vectors (the denominator).
Contrastive learning	SSL pretext: pull a node’s two views together, push other nodes apart; needs negatives (DirectAU: no, §3.3); discriminative mechanism (this chapter’s own reading, not a standard label; §6.3).
Contrastive learning / views	Learning embeddings by comparison: two views of the same thing are pulled together, other things pushed apart (§2.4).
Convex	Curves up everywhere: for a function with second derivatives, one variable \(f''\ge0\), many variables a Hessian that is PSD at every point. A bowl, a trough, or a slope with no bottom (like \(e^x\)) — no bad local minima; any minimum is global.
Cosine similarity	\(\dfrac{\mathbf{u}\cdot\mathbf{v}}{\lVert\mathbf{u}\rVert\lVert\mathbf{v}\rVert}\in[-1,1]\) for nonzero vectors; direction agreement, ignoring length.
Cost / objective / criterion	Near-synonyms for loss (cost = total; objective = neutral; criterion = the code object).
Coverage	The fraction of the catalog that ever appears in anyone’s top-\(K\) list (§6).
CPU / GPU / RAM / VRAM	The general-purpose processor / the chip that does thousands of multiplications at once / the computer’s fast working memory / the GPU’s own memory, which must hold the whole model.
Credit assignment	Deciding which earlier action deserves credit for a reward that arrives later.
Critical / stationary point	Where \(f'(x)=0\) (many variables: \(\nabla f=\mathbf 0\)): peak, valley, plateau or saddle.
Critical value	The cutoff that pure luck exceeds only \(\alpha\) of the time (\(2.776\) for df \(=4\), \(\alpha=0.05\), two-sided) (§8.3).
Croissant	MLCommons machine-readable dataset-metadata standard (2024); NeurIPS encouraged a record in 2024, worded it “should use” in 2025 and “must use” in 2026, with the Responsible Artificial Intelligence (RAI) fields only requested in the 2026 call (the organizers’ post of 4 May 2026 requires them, §6).
Cross network	Stacked layers that raise interaction degree by one each, explicitly and at linear cost (\(2d\) parameters per layer). DCN = a cross network beside a deep MLP.
Cross-encoder	A model that reads a (user, item) pair together in one forward pass, so its score cannot be precomputed. A specialist reranker is usually one, and it is not an LLM (§2.1).
Cross-entropy / BCE / log loss	Classification loss; average code length (nats with \(\ln\), bits with \(\log_2\)) to encode the truth \(p\) with a code built for the prediction \(q\); BCE = −log-likelihood of a Bernoulli (categorical cross-entropy: of a categorical).
Cross-entropy \(H(p,q)\)	\(-\sum p\log q\); surprise of \(p\)’s outcomes scored by model \(q\) — the standard classification loss.
Cross-view alignment	Pulling a node’s two embeddings (collaborative \(\mathbf{e}_i\) and semantic \(\mathbf{s}_i\)) together with InfoNCE (§3).
CTR (click-through rate)	\(P(\text{click}\mid \text{user, item, context})\); the score the ranking stage predicts.
CTR / impression	Click-through rate \(=\) clicks \(/\) impressions, where an impression is one showing of an item to a user.
Curvature	How a curve bends: \(f''\) in one variable; in many, the second derivative along a chosen direction (§6).
Cutoff \(K\)	The list length in Recall@\(K\) (not the layer count \(K\) of §3.4): only a user’s top \(K\) films count (§7).

D

Term	Meaning
\(D^{-1/2}\)	The degree matrix with each diagonal degree replaced by \(1/\sqrt{\deg}\) — an entry-by-entry shortcut that works only because \(D\) is diagonal (§7).
Data augmentation	Train on randomly perturbed copies of inputs; one history \([A,B,C,D]\) gives four views (§5.2).
Data leakage	Test-period information seeping into training; silently inflates every metric.
DCG / IDCG / NDCG	Discounted Cumulative Gain (gains \(2^{\text{rel}}-1\), each divided by \(\log_2(i+1)\) at rank \(i\)) / its ideal / the normalized ratio in \([0,1]\).
DCNv2	The cross network with a full weight matrix per layer instead of a vector; its authors report it better than DCN, a report §9 says to test (low-rank form and a mixture of experts keep it affordable).
Debiasing	Methods that correct for users rating mostly what they chose to see (§7).
Decision tree / information gain	A flowchart of yes/no questions that sorts examples into groups / how much a question reduces the entropy of the labels; the tree asks the question with the largest gain first (§7).
Decoder	A network that turns an embedding back into the input it came from; the generative end of a network (§6.3).
Decoy items, hybrid filling	Randomly sampled unrated items whose gradients are uploaded with virtual ratings filled in (the user’s average rating or a predicted one), so the real ones do not stand out; \(\rho\) decoys per rated item (§3).
DeepFM	Wide & Deep with the wide arm replaced by an FM, sharing one embedding layer; no hand-crossing.
Degree	How many neighbors a node has.
Degree matrix \(D\)	Diagonal matrix of degrees; used for normalization.
Degrees of freedom	The number of values still free to vary once their mean is fixed: \(n-1\) for \(n\) runs (\(4\) for \(5\) seeds).
Degrees of freedom (df)	\(n-1\) for the paired test; sets which Student-\(t\) curve gives the \(p\)-value.
Demographic parity	Equal selection / exposure rate across groups.
Density	Interactions \(\div\) (users \(\times\) items) — the share of the user–item grid that is filled. The graph-CF trio sits near \(0.1\%\) (§2), but MovieLens-1M is about \(4.5\%\) (\(1{,}000{,}209/(6{,}040\times3{,}706)\)) and Netflix about \(1.2\%\) (\(100{,}480{,}507/(480{,}189\times17{,}770)\)), so check.
Derivative \(f'(x)\)	Instantaneous rate of change = slope of the curve at a point.
Determinant \(\det A\)	Area/volume-scaling factor (\(ad-bc\) for \(2\times2\)); equals the product of the eigenvalues (shown here for \(2\times2\)); \(\det=0\) ⇒ collapses a dimension / non-invertible.
Diagonal matrix	A matrix whose only possibly-nonzero entries lie on its diagonal; it just scales each axis (its singular values are the absolute values of its diagonal entries, put in order of size).
Difference quotient	\(\frac{f(x+h)-f(x)}{h}\) — rise over run before taking \(h\to0\).
Differential privacy	Add calibrated noise to updates so that how much any single user’s data can be inferred is bounded — by a privacy budget \(\varepsilon\) (smaller means more noise and more privacy; not the exploration rate \(\varepsilon\) of the Bandits chapter), not reduced to zero; §3’s numbers are not calibrated to any \(\varepsilon\) (Dwork et al., 2006; Abadi et al., 2016).
Differentiate	Compute a derivative (limit of differences).
Dimension	How many components a vector has (\(\mathbb{R}^d\) = all \(d\)-vectors).
Direct preference optimization (DPO) / S-DPO	A loss that learns from pairs of a preferred and a rejected output with no separate reward model; Softmax-DPO (S-DPO) widens the pair to one preferred item against several negatives (§4).
DirectAU	A negative-free loss (Wang et al., KDD 2022) that optimizes alignment and uniformity directly, instead of via a softmax; its positive pair is a user and an item she interacted with, not two views (§3.3).
Directed / undirected	Whether an edge has a direction (A→B \(\ne\) B→A) or not.
Directional derivative	Rate of change of \(f\) along a unit direction \(\hat{\mathbf d}\): \(\nabla f\cdot\hat{\mathbf d}\).
Discount \(\gamma\)	How much a later reward is worth vs. now; \(\gamma\to0\) is myopic (cares only about the immediate reward, like a bandit’s objective), \(\gamma\to1\) far-sighted.
Discriminative vs. generative	Axis 1 — whether a model learns \(p(y\mid x)\) (a boundary; can’t generate) or \(p(x)\)/\(p(x,y)\) (the data; can sample). BPR/LightGCN is discriminative; VAEs/LLMs are generative.
Distributional hypothesis	“You shall know a word by the company it keeps” (Firth 1957; Harris 1954) — words in similar contexts have similar meanings; word2vec turns this into similar vectors (§2.1).
Diversity	Average dissimilarity within a recommended list.
DNN (deep neural network)	DeepFM’s name for the MLP arm — the same object as Wide & Deep’s deep arm.
Dot / inner / scalar product \(\mathbf{u}\cdot\mathbf{v}\)	\(\sum_i u_iv_i\); one number measuring agreement.
Dot product	Multiply-and-sum two embeddings → a similarity / ranking score.
DPO (direct preference optimization)	Fine-tuning a model on pairs of preferred and rejected outputs so it favors the preferred kind; the “alignment” of OneRec’s training, not the embedding alignment of §3 (§2.2).
Dropout	Randomly zero units during training; an ensemble-like regularizer; “edge dropout” on graphs.
Dynamic negative sampling	Keeping hard negatives hard: draw a few un-clicked candidates and keep the one the current model scores highest (§3).

E

Term	Meaning
Early stopping	Stop when validation stops improving; implicit regularizer.
EasyRec	A language model tuned with contrastive learning so that vectors from text line up with collaborative behavior: the feature-encoding cell of the enhancer (§2.3).
Edge	A connection (user liked film, rated \(4\) or \(5\)).
Effect size	The magnitude of the difference (\(\bar d\)); report it alongside \(p\) (significance \(\ne\) size).
Eigenbasis	A basis made of eigenvectors; a real symmetric matrix always has one of mutually perpendicular unit eigenvectors (the spectral theorem, §5, stated without proof).
Eigenvalue	One of the two extreme curvatures of a symmetric \(2\times2\) Hessian; their product is \(D\) (§6 optional box; Linear-Algebra primer §5).
Eigenvalue \(\lambda\)	The stretch factor of an eigenvector.
Eigenvector	A direction a matrix only stretches, not rotates: \(A\mathbf{v}=\lambda\mathbf{v}\).
Eigenvector \(\mathbf{v}\)	A special direction a matrix only stretches (not rotates): \(\tilde{A}\mathbf{v}=\lambda \mathbf{v}\). The graph’s eigenvectors are its “frequency patterns.”
Elastic net	L1 + L2 combined; “stretchable net” keeping correlated features.
ELBO	Evidence lower bound: a computable lower bound on the data’s log-likelihood that a VAE (variational autoencoder) maximizes; Mult-VAE minimizes reconstruction loss + β·KL, which is exactly −ELBO when β = 1 (§11).
Element-wise product \(\odot\)	Multiply two vectors number by number: \([2,-2]\odot[2,0]=[4,0]\); the dot product is the sum of the result (§14).
Elo	The chess rating system, reused for chatbot arenas: beating a strong opponent raises your rating more than beating a weak one.
Embedding	A learned vector representing a user/item as a point in space.
Embedding (layer)	The table that replaces each field’s active category by its learned dense vector \(\mathbf e_f\); one copy, read by every arm that needs it.
Embedding layer	A learnable lookup table; id → its row (= one-hot × matrix); the input side for discrete data.
Embedding matrix \(E\)	All embeddings stacked, one row per node: \(N\times d\); reading row \(k\) is the lookup of Neural Networks & Back-propagation §3 (§4).
Empirical mean	An arm’s clicks divided by its pulls so far, written \(\hat\mu_i\); \(2\) clicks in \(3\) pulls give \(0.667\) (§3).
Encoder	Whatever turns raw inputs into embeddings; here, LightGCN’s propagation over the ID table (§3.1).
Encoder / decoder	An encoder reads a sequence into context-aware vectors, every token seeing every other (BERT); a decoder writes text one token at a time, each token seeing only itself and earlier ones (GPT; §5.1).
Enhancer / reranker (LLM role)	Recsys LLM roles: generate item/user text features / re-order a candidate list — not chat.
Enhancer pipeline	Using an LLM offline to manufacture features for a fast collaborative backbone that serves.
Enhancer-style	An LLM enriches the inputs of an existing recommender instead of replacing it (§17).
Entropy bonus / exploring	In reinforcement learning (RL, learning by trial and reward), a reward proportional to the entropy of a model’s choices / what the bonus buys: trying actions other than the one that looks best so far (§7).
Entropy \(H(p)\)	\(-\sum p\log p\): average surprise / uncertainty (maximal at uniform). In bits with \(\log_2\) (the average yes/no questions, or bits to store one sample, as in §7’s examples), in nats with \(\ln\).
Epoch	One pass over all the training triples (§13; Neural Networks & Back-propagation §6).
Epoch / batch	One full pass over the training data / the small group of examples used for one update (§5.2).
Equal opportunity	Equal true-positive rate across groups (the truly-relevant are surfaced equally).
ERM	Empirical Risk Minimization — “fit the training data” (usually + a regularizer).
Euclidean distance \(\lVert\mathbf{u}-\mathbf{v}\rVert\)	Straight-line gap between two arrow-tips; feels magnitude (unlike cosine).
Evidence \(p(D)\)	Normalizer; data probability averaged over \(\theta\), each \(\theta\) weighted by its prior (a.k.a. marginal likelihood).
Expectation \(\mathbb{E}[X]\) / mean \(\mu\)	Probability-weighted average value.
Explicit / implicit feedback	Stated ratings (likes and dislikes) vs. behavioral positives (\(1\) = clicked / played / bought; blanks are unknown, not \(0\) — no true negatives).
Explicit vs implicit interaction	Explicit = built by a formula you can point at, with a known degree and a readable strength; implicit = mixed inside an MLP’s layers, where no single weight names a cross.
Explicit vs. implicit feedback	Star ratings vs. clicks/plays (positive-only, unlabeled negatives).
Explore vs. exploit	Try an uncertain option to learn (explore) vs. take the best-known one now (exploit).

F

Term	Meaning
F1@K	Harmonic mean of precision and recall.
Factor vector	A feature’s short learned vector \(\mathbf v_j\); the dot product of two is their interaction strength.
Factorization machine (FM)	MF generalized to any features: each feature gets a vector, each feature pair interacts via a dot product (Rendle, 2010). MF is the ids-only special case.
False negative	A “negative” the user would in fact have liked — pushed away by mistake (§3.4). Not the hypothesis-testing sense of the term.
FCF	Federated collaborative filtering: item factors are global; each user’s factor stays on-device (its uploads still carry the factor’s line, §3).
Feature	A given number about a node, measured outside the model (a paper’s word counts); training never changes it (§6).
Feature interaction / cross	A pair (or, higher-order, a larger group) of features whose combination matters (user \(\times\) genre); FM scores a pair as \(\langle\mathbf v_j,\mathbf v_{j'}\rangle\).
Feature matrix \(X\)	One row of features per node (\(N\times F\)); data, not learned (§6).
Feature store	The table of generated id → (profile, vector) the serving path reads; the LLM is never on the serving path.
FedAvg	Aggregate by a data-size-weighted average of clients’ local models (McMahan et al., 2017).
Federated learning	Train a shared model across devices that keep their raw data local; only updates are shared.
FedRec	Federated recommendation with explicit feedback (Lin et al., 2021): uploads gradients for decoy items as well as rated ones (§3); FedRec++ (Liang et al., 2021) removes the noise the decoys add.
Feed-forward (FFN)	Per-token MLP \(\max(0,\mathbf{x}W_1{+}\mathbf{b}_1)W_2{+}\mathbf{b}_2\) inside each block; transforms each token (about two thirds of each block’s weights when its hidden layer is 4× wider; §4.3).
Feedback loop	Recommend → click → log → retrain on skewed data → more skew; the “rich get richer” spiral that compounds bias.
Field	One attribute of the row (user, item, genre, device, daypart); each field contributes exactly one active category, and one embedding vector \(\mathbf e_f\).
Filter response \(h(\lambda)\)	The function deciding how much each frequency is kept; defines the filter.
FinalMLP, feature gate	A two-stream MLP model; its feature gate is a learned weight per input feature that decides how much of it a stream sees (§9).
Fine-tune / distil / gold & silver labels	Train a released model further on your data / train a small model to copy a big one’s outputs / hand-checked reference answers vs cheap machine-written ones.
Fine-tuning / instruction tuning	Continuing to train a pretrained model on task data. Instruction tuning uses prompt–answer pairs (“User liked A, B; will they like C? → Yes”), which is how TALLRec and BIGRec are trained (§2.1, §5).
Fine-tuning vs in-context prompting	Train the model’s weights on the task, or keep them fixed and put worked examples in the prompt (§5.2).
Focal loss	Cross-entropy times \((1-p_t)^\gamma\), with \(p_t\) the probability of the true class, to down-weight easy examples; for class imbalance.
Forward pass	Running an input through the net to a prediction + loss.
FPMC	Factorizing Personalized Markov Chains: MF term \(+\) a learned (factorized) transition term.
Frequency (on a graph)	An eigenvector pattern; on a path, the number of sign changes: low = smooth over neighbors, high = jagged.
Frobenius norm \(\lVert A\rVert_F\)	The square root of the sum of the squares of all a matrix’s entries — its L2 length, flattened.
Frozen split	A train/test partition released as files and reused unchanged by later papers of one lineage (the trio’s defining feature, §2), which makes their numbers mutually quotable; not every graph-CF paper reuses it (§4).
Frozen text encoder	Run text once through a fixed pretrained model and keep the output vector — the semantic \(\mathbf{s}_i\) RecSys aligns/consumes (§5.2).
Full ranking vs. sampled negatives	Score the held-out item against every item the user has not seen, or against a handful (99–100) drawn at random (negative sampling; a negative is an item the user did not interact with, treated as non-relevant). The two give different numbers for the same model (§4).
Full vs. sampled ranking	Rank the true item against the whole catalog vs. a few sampled negatives; the latter is inconsistent across papers and optimistic (with one held-out item per user, as in leave-one-out, fewer competitors mean the item can only rank higher).
FuxiCTR, BARS-CTR	An open-source library for configuring and reproducing CTR models, and the benchmark built with it that re-ran the lineage under equal tuning (§9).

G

Term	Meaning
Gap \(\Delta_i\)	\(\mu^{*}-\mu_i\): how much less an arm pays per pull than the best arm; each pull of it adds this to the regret (§2).
Gate / GRU	A gate is a number between \(0\) and \(1\) the cell computes at each step to decide how much of the new candidate to write in (near \(0\) keeps the old summary, near \(1\) overwrites it); the gated recurrent unit (GRU) is the recurrent cell built on gates (§3).
Gaussian (Normal)	Bell curve \(\mathcal N(\mu,\sigma^2)\); \(-\log\) → squared error (MSE) when \(\sigma\) is the same fixed number for every point (§4).
GCL	Graph Contrastive Learning — contrastive SSL on a graph recommender.
GCN	The model built by stacking graph-convolution layers (plus a loss, a prediction step that turns two embeddings into a score, and — in vanilla GCN — \(W\) and \(\sigma\)). Operation vs. architecture.
Generative learning	SSL pretext: reconstruct the masked/corrupted input; no negatives (for a plain reconstruction error; RLMRec-Gen is the exception, §7); generative / reconstructive mechanism.
Generative recommendation	Recommend by generating an item’s Semantic ID token by token (§2.2).
Gini coefficient	Concentration of recommendations on few items: \(0\) = perfectly even; the maximum, all on one of \(n\) items, is \((n-1)/n\) — near \(1\) only for a large catalog.
Global temporal split	One cut-off date for everybody: train on everything before it, test on everything after (Evaluation Metrics).
GNN	Graph neural network: a network that updates each node’s vector from its neighbors’, as LightGCN does (§7).
GPT	Generative Pre-trained Transformer (next-token, causal); a decoder for generating text (the model of Radford 2018; the name came later; §5.1).
Gradient	The slope (or slopes) a model follows downhill when it trains (§3; Calculus primer).
Gradient \(\nabla f\)	Vector of all partials; points in the direction of steepest increase.
Gradient clipping	Capping the gradient’s length at a ceiling so it cannot explode; the direction is kept (§5).
Gradient descent	\(\theta\leftarrow\theta-\eta\nabla f\): step against the gradient to minimize.
Gram matrix	A matrix’s transpose times itself, \(R^{\top}R\); entry \((i,j)\) counts the users who liked both films \(i\) and \(j\).
Graph	Nodes + edges.
Graph convolution	The operation (in LightGCN’s form): replace each node’s embedding with the degree-weighted sum of its neighbors’ (one matrix multiply by the normalized adjacency).
Graph Fourier Transform (GFT)	Projecting a graph signal onto the eigenvectors of the (normalized) adjacency.
Graph Laplacian	A matrix built from a graph, a close relative of its adjacency matrix; the standard forms are built like an \(A^{\top}A\) and so are PSD (taken as given in §6).
Graph signal	One value per node (e.g. a user’s interaction vector over films).
Graph Signal Processing (GSP)	Treating one-number-per-node data as a “signal” and processing it with graph frequencies.
Graph-CF trio	Gowalla / Yelp2018 / Amazon-Book — the frozen-split benchmark of graph-CF papers.
Graphics processing unit (GPU)	The chip that trains a model fast but holds little memory (§15).
GraphMAE	Graph masked autoencoder (Hou et al., KDD 2022): hides some nodes’ input features and rebuilds them with a scaled cosine error. A node- and graph-classification method; this chapter borrows its mask-and-rebuild idea as a recommender example (§6.2).
GraphSAGE	A model (Hamilton et al., 2017) that builds a node’s vector from a sample of its neighbors, so a new node gets a vector at once (§14).
ε-greedy	Exploit the best estimate with prob \(1-\varepsilon\); pull a uniform-random arm with prob \(\varepsilon\).
Greedy walk	HNSW’s search: from some item, step to the linked item that scores highest against the user vector, and stop when no link beats where you stand (§7).
GRPO	Group relative policy optimization: sample a group of outputs for one prompt and score each against the group’s mean and spread, with no value model (§4).
GRU	Gated Recurrent Unit: a lighter 2-gate cousin of the LSTM, with no separate cell state (Cho 2014; §3.4); GRU4Rec is its session-based recommender.
GRU4Rec	A gated RNN for session-based recommendation; session-parallel batches + a ranking loss.

H

Term	Meaning
Hadamard product (\(\odot\))	Entry-by-entry product of two vectors of the same length: \([1,2]\odot[3,4]=[3,8]\) (§7, §8).
Hallucination	An LLM confidently producing an item that is not in the catalog; reduced by constrained decoding, candidate-list forcing or post-hoc filtering (defined in §1, fixes in §2.2).
Hard negative	A negative that sits close to the anchor, so it dominates the loss; dangerous when it is closer than the positive (§3.3).
Harmonic mean	The reciprocal of the average of the reciprocals; for two numbers \(P,R\) it is \(2PR/(P+R)\), dragged down by the smaller one (\(0.6\) and \(0.75\) give \(0.667\)).
Harness / orchestration	The engineering around the LLM call — templates, validation, batching, retries, caching, cost control.
He / Kaiming init	Draw initial weights with variance \(2/\text{fan\_in}\) (fan_in = inputs per neuron) — the scale that keeps ReLU activations from shrinking or exploding layer by layer (§6).
Head	The few most popular items, at the left of the popularity curve (§6).
Heat kernel	The blur \(e^{-t(1-\lambda)}\): signal spreading over the graph like heat for a time \(t\) (the idea behind BSPM’s blur).
Hessian \(H\)	Matrix of second partial derivatives \(f_{xx},f_{xy},f_{yx},f_{yy}\) (multivariable curvature).
Hidden layer	A layer that is neither input nor output.
Hinge loss	The loss of the support vector machine (SVM), a classic classifier that separates two classes with the widest possible gap between them; flat once correct-with-margin (\(m=y\hat y\ge1\)), then a linear ramp — shaped like a hinge.
Hit Rate (HR@K)	Fraction of users with \(\ge 1\) relevant item in the top-\(K\).
Hit-Rate@K (HR@K) / NDCG@K	Was the held-out next item in the top \(K\)? / how high was it — normalized discounted cumulative gain (Evaluation Metrics).
HNSW (Hierarchical Navigable Small World)	An ANN index that links each item to its nearest neighbors in layered graphs and answers a query by a greedy walk of about \(\log_2 M\) steps, each scoring a few dozen linked items (a simple model of the walk, not a measurement; §7).
HNSW / IVF-PQ / DiskANN / CAGRA	ANN index types: graph (in-RAM default) / inverted-list + compression (memory-thrifty) / on-disk (billion-scale) / GPU graph (batched throughput).
Homogeneous / heterogeneous	One node type, vs. several (users and items).
Hop	One step along an edge (distance in the graph); fixed by the graph, not chosen. \(K\) layers mix the starting vectors of nodes a walk of \(K\) steps away (§5).
HR@\(K\), MRR, AUC	Hit rate: was the held-out item in the top \(K\); mean reciprocal rank: the average of \(1/\text{rank}\); area under the receiver-operating-characteristic (ROC) curve: the chance a clicked impression outscores an unclicked one. Labels here; definitions in Evaluation Metrics (§4).
HSTU	Hierarchical Sequential Transduction Units: Meta’s generative recommender that reads a user’s sequence of actions and writes the next one (§2.2).
Hub	A node joined to many others (Vendetta in the film path).
Huber loss	Quadratic near 0, linear far out; robust compromise (after P. Huber).
Hybrid	Combine content-based + collaborative, e.g. a weighted blend of their predictions, or LLM features + a CF backbone (§7).
Hypergraph	A graph whose edges may join many nodes at once; HCCF contrasts one with the ordinary graph (§2, §4).
Hyperparameter	A setting fixed before training, such as the number of layers \(K\) (§5).

I

Term	Meaning
ID embedding	An embedding that exists only to tell one node from another, with nothing measured about the node; LightGCN’s \(E^{(0)}\) (§6).
ID table	The one freely trainable vector per user and per item, before any neighbor-averaging (§3.4).
Ideal low-pass	A hard cutoff: keep the top few patterns whole and zero the rest, as GF-CF’s ideal part and a truncated SVD do (§4, §6).
Idempotency	Re-running the pipeline on unchanged input does no new work (hash → cache hit), so a profile is regenerated only when its input or the prompt version changes.
Identity \(I\)	The “do-nothing” matrix; \(I\mathbf{x}=\mathbf{x}\).
Identity matrix \(I\)	A \(1\) on the diagonal, \(0\) elsewhere; as \(X=I\) it gives each node a one-hot ID, and \(XW\) then picks out the rows of \(W\) (§6).
Impression	One logged showing of one item to one user in one slot; the unit of a CTR row, labeled \(1\) (clicked) or \(0\) (not).
In-batch negative	A negative taken from the other nodes in the current mini-batch, rather than from the whole catalog (§3.4).
Indefinite	Curvature up in some directions and down in others (\(D<0\) for a \(2\times2\) Hessian; eigenvalues of both signs) — the Hessian at a saddle.
Independence	\(\Pr(A\cap B)=\Pr(A)\Pr(B)\); one event tells you nothing about the other.
Index	A lookup structure built once, offline, over the item vectors, that finds the best-scoring items without scoring them all (§7).
Index recall (recall@\(K\))	The share of the exact top-\(K\) items that the approximate index also returns: \(4\) of \(5\) is \(0.80\) (§7).
Inductive	A model that learns a rule building a vector from what is visible about any node (features, neighbors), so a new node needs no retraining (§14).
Inference / serving	Using the trained embeddings to return a user’s top-\(K\) list.
Inflection point	Where \(f''\) changes sign (the curve switches between bending up and down).
InfoNCE	A contrastive training loss: a softmax that must pick a matching pair out of mismatched ones (§3; §7 for its link to mutual information).
InfoNCE / NT-Xent	Contrastive loss: \(-\log\) of the positive’s softmax share, the denominator running over the positive and the negatives; Info = mutual-info bound, NCE = Noise-Contrastive Estimation; temperature \(\tau\).
Integral \(\int\)	Area under a curve; the reverse of differentiation; gives probabilities/expectations.
Interaction matrix \(R\)	Users-×-films table of 0/1 (who liked what).
Inverted file	A partition index: the catalog is split into buckets, each with a center point (the average of its vectors); a query scores the centers, then only the items in the best few buckets (§7).
Invertible / singular	Invertible = has an inverse \(A^{-1}\) that undoes it, \(A^{-1}A=AA^{-1}=I\) (\(\det\neq0\), full rank); singular = no inverse (\(\det=0\)).
IPS-weighted recall	Recall with each relevant item weighted by \(1/p\) and divided by the total weight: \(0.60\) against the raw \(0.75\) on the running list (§12.4).
Isotonic regression (PAV)	Calibrates with a fitted non-decreasing staircase; the pool-adjacent-violators algorithm averages neighbors that break the order, which can create ties.
Item-kNN	Recommend the items most similar, by who interacted with both, to those the user already liked; a strong simple baseline (§5).
item2vec	word2vec applied to users’ histories (“item = word, history = sentence”; item2vec itself ignores their order); a collaborative item embedding (§2.4).

J

Term	Meaning
Jaccard similarity	\(\lvert A\cap B\rvert/\lvert A\cup B\rvert\) on liked-sets; ignores rating values.
Jacobian	The matrix of all partial derivatives of a vector function; for \(\mathbf z=W\mathbf h\) it is \(W\), and back-prop multiplies by its transpose (§5).
Jacobian \(J\)	Matrix of partials of a vector-valued function (rows = outputs, columns = inputs).
Joint / marginal	Joint \(p(x,y)=\Pr(X{=}x\text{ and }Y{=}y)\), the chance of each pair of values; marginal \(p(x)=\sum_y p(x,y)\) (sum the joint over the other variable — the row totals in the margin of the grid).
JSON (JavaScript Object Notation)	A plain-text format for records, e.g. {"brand": "Acme", "fit": "slim"} — how an extractor returns structured fields.

K

Term	Meaning
@K	Evaluated over the top \(K\) recommended positions only.
k-core filter	Iteratively drop users/items with \(<k\) interactions until none remain; stabilizes the data. The guarantee holds before the split, not after it (§3).
\(k\)-fold	Cut the data into \(k\) parts; test on each in turn, train on the rest, average (§3). A random scheme.
\(k\)-NN / neighborhood	The \(k\) most-similar users/items used for a prediction.
KAR	Knowledge Augmented Recommendation: prompt an LLM for user reasoning and item facts offline, then feed them to any backbone through a mixture-of-experts adapter (§2.3).
KL divergence	Asymmetric “distance” between distributions (Kullback–Leibler); a divergence, not a metric.
KL divergence \(D_{\mathrm{KL}}(p\Vert q)\)	\(H(p,q)-H(p)\ge0\); extra surprise from using \(q\) not \(p\); asymmetric (a divergence, not a distance).
Knob (hyperparameter)	A value set by hand before training (\(\delta\), \(\gamma\), \(\tau\), \(\eta\), \(\lambda\)), not learned from the data like the parameters \(\theta\) (§2.1).
Knowledge distillation	Training a small model to match a large model’s output probabilities; the KL divergence measures the mismatch (§7).
KV-cache	Key–value cache: store past tokens’ keys and values so each decode step pushes only the new token through the model — \(O(T)\) token-passes instead of \(O(T^2)\) (§5.4).

L

Term	Meaning
L1 / lasso	$
L2 / ridge / Tikhonov / weight decay	\(\sum\theta_k^2\) penalty; shrinks weights smoothly toward 0; = Gaussian prior centered at \(0\). Weight decay is the same thing under plain (stochastic) gradient descent, not under Adam (hence AdamW, §5.1).
L2 norm / L1 norm \(\lVert\boldsymbol\theta\rVert_2,\lVert\boldsymbol\theta\rVert_1\)	The length of the weight list as the root of the sum of squares / as the sum of absolute values; their penalties come from the Gaussian and Laplace priors (§6).
L2 penalty	The regularizer \(\lambda\lVert E^{(0)}\rVert^2\): \(\lambda\) times the sum of the squares of every entry of the starting vectors (\(0.01\times26.25=0.2625\) in the toy); LightGCN penalizes only the base embeddings (§15).
L2-normalize / ANN	Divide a vector by its length so a dot product becomes a cosine; approximate nearest-neighbor search finds the closest catalog vectors without scoring all (§4.4).
Lambda \(\lambda\) (regularization strength)	How much we weight the regularizer vs. the loss.
LambdaRank / LambdaMART	Listwise methods that write no loss, only a gradient weighted by each pair’s effect on NDCG (the per-pair gradients are Burges’s \(\lambda_{ij}\)).
Language model	A model that predicts a missing or next token from its context; an LLM is a very large Transformer one (§2.4, §5).
Laplace	Peaked/heavy-tailed \((\mu,b)\); \(-\log\) → absolute error (MAE) when \(b\) is the same fixed number for every point.
Laplacian \(L=I-\tilde A\)	The same patterns as \(\tilde A\) with labels \(\mu=1-\lambda\): smooth at \(0\), jagged at \(2\) (§1).
Large language model (LLM)	A model trained on large amounts of text, which can write and read descriptions of users and items (§13, §17).
Latency	How long the serving path takes, in milliseconds (ms); p50 is the median, the time half of the requests beat (§4.1).
Latency (and p99)	The time between a request arriving and the answer going back; a p99 budget is one that \(99\) of every \(100\) requests must meet (§8).
Latent dimension \(d\) (also called the rank)	Width of the factor vectors; the dial between under- and over-fitting (§5.2). Not the “rank” of ranking a list (§6).
Latent factor	A hidden, learned dimension of taste/content.
Latent vector	The short list of \(k\) numbers an FM learns for each category; its dot product with another category’s vector is the pair’s strength.
Law of Large Numbers	The sample average converges to the expectation as samples accumulate.
Law of total probability	\(\Pr(A)=\sum_j\Pr(A\mid B_j)\Pr(B_j)\) over cases \(B_j\) that do not overlap and together cover every outcome (e.g. \(B\) and \(\neg B\)); the engine of Bayes’ evidence term.
Layer	One application of the graph-convolution rule to all nodes (a computation step in the model); the number of layers \(K\) is a hyperparameter you choose.
Layer / width / depth	A bank of neurons / neurons-per-layer / number of layers.
Layer combination	Averaging the embeddings from all layers, \(\mathbf{e}_u=\sum_{k}\alpha_k\mathbf{e}_u^{(k)}\); for Ann’s pairs in this toy it slows over-smoothing rather than curing it, and Bob and Cara gain little from it (§12).
Layer superscript \((\ell)\)	A layer number, not a power: \(W^{(\ell)}\), \(\mathbf h^{(\ell)}\) belong to layer \(\ell\); subscripts pick entries.
Layer weights \(\alpha_k\)	The coefficients of that combination; usually all equal, \(\alpha_k = 1/(K+1)\) (§12).
Layer-combined embedding	A node’s final vector: the weighted sum of its vectors after 0, 1, …, \(K\) rounds of neighbor-averaging, as LightGCN produces it (§3.4).
LayerNorm / Pre-LN	Standardize a token’s vector to mean 0 and standard deviation 1; Pre-LN does it before each sub-layer, Post-LN after (§4.3).
Leaky-ReLU / GELU	ReLU variants that give negative inputs a small output and slope, so a unit at a negative \(z\) still passes some gradient (§2).
Learning rate	The size of each training step (§15).
Learning rate \(\eta\)	Step size in gradient descent.
Learning-rate schedule / warm-up / decay	The rate \(\eta_t\) chosen for each step \(t\) (§6) / a short rise from a small rate at the start / a fall as training ends, in jumps (step decay) or along half a cosine wave to \(0\) (cosine decay).
Leave-one-out (LOO)	Hold out one interaction per user — the chronologically last unless stated otherwise, often the second-last for validation. Leak-free within a user, not across users (§3). Some papers use the name for a random held-out interaction: check.
Leave-one-out (LOO) split	Hold out each user’s chronologically last interaction for test, the second-last for validation (a per-user cut).
License families / OSI	Permissive (Apache-2.0, MIT, BSD) / copyleft (GPL, AGPL) / non-commercial (CC-BY-NC) / custom (Llama, Gemma \(\le 3\)). Only the first two are open source as the Open Source Initiative (OSI) defines it. §1 rule 2.
LightGCN	(He et al., 2020) NGCF with \(W\), \(\sigma\), the product term and the self-loop removed, and the concatenation replaced by a weighted sum of layers; a commonly used graph-CF backbone (§14).
Likelihood / prior / posterior	\(p(\text{data}\mid\theta)\) / \(p(\theta)\) / \(p(\theta\mid\text{data})\).
Likelihood \(L(\theta)\)	\(p(\text{data}\mid\theta)\) read as a function of \(\theta\) (data fixed).
Limit	The value a quantity approaches (here as the run \(h\to0\)).
Linear (transformation)	A rule that respects adding and scaling: \(A(\mathbf{x}+\mathbf{y})=A\mathbf{x}+A\mathbf{y}\), \(A(c\,\mathbf{x})=c\,A\mathbf{x}\); every matrix is one.
Linearly independent	A set of vectors none of which is a weighted sum of the others.
LinUCB	Ridge-regression reward estimate \(\theta^{\top}\mathbf x\) plus a feature-space confidence bonus \(\alpha\sqrt{\mathbf x^{\top}A^{-1}\mathbf x}\) (§6).
Lipschitz-smooth	The gradient changes at most in proportion to the step, \(\lVert\nabla f(\theta_1)-\nabla f(\theta_2)\rVert\le K\lVert\theta_1-\theta_2\rVert\); then one fixed learning rate, any \(\eta\) between \(0\) and \(2/K\), lowers \(f\) at every step where the gradient is not zero (§7).
LlamaRec	Two-stage recommendation: a small sequential model retrieves a shortlist, a fine-tuned LLM scores it in one pass from the probabilities of the candidates’ labels (§2.1).
LLM	Large language model: a Transformer language model with (today often) billions of weights, pretrained on a huge corpus; here, demystified as this chapter’s lineage at scale (§5).
LLM-as-enhancer	The LLM produces semantic features or profiles offline that feed or align with a classical model (§2.3).
LLM-as-recommender	The LLM directly ranks or selects items from a prompt of the user’s history (§2.1).
LLM-as-reranker	The recommender role in its common production form: the LLM re-scores a short top-\(K\) shortlist that a cheap retriever already fetched, instead of ranking the whole catalog. Reranking is the funnel’s ranking stage under a second name (§2.1).
Local / global minimum	Local = lowest within a neighborhood; global = lowest anywhere.
Log-likelihood	\(\log L(\theta)\); products become sums.
Log-odds (logit)	The raw score \(z=\ln\frac{\hat p}{1-\hat p}\) the sigmoid squashes; weights add up here, not in the probability.
log-\(Q\) correction	Subtract \(\log\) of an item’s sampling probability from its score, so negatives drawn in proportion to popularity (e.g. in-batch negatives) do not over-penalize popular items; under uniform sampling it removes no popularity skew and only rescales the negatives’ total (§10).
log-sum-exp / softplus	\(\log\sum_i e^{z_i}\), a smooth stand-in for the largest \(z_i\) that normalizes a softmax (\(1.313\) for logits \(1\) and \(0\), against a largest logit of \(1\)); \(\text{softplus}(z)=\log(e^0+e^{z})\) is its two-term case with one logit at \(0\) (§2.4).
Logistic loss	BCE written on the margin \(m=y\hat y\) with labels \(\pm1\): \(\ln(1+e^{-m})\) (§2.2).
Logistic regression	Linear score \(\to\) sigmoid \(\to\) probability; the CTR baseline and every model’s output head.
Logit	The raw score an output layer gives a label before the softmax turns the scores into probabilities (§2.1).
Logit (log-odds)	\(\ln\!\big(p/(1-p)\big)\); maps \((0,1)\to(-\infty,\infty)\).
LogLoss	Log loss, or binary cross-entropy: the average of \(-\ln\)(the probability given to what happened); \(0.9\) on a click and \(0.2\) on a non-click give \(0.164\).
Long tail	The many low-degree (niche) films, each with few watchers; each is named by few training triples (§13).
Long tail / popularity bias	A few blockbusters get most interactions; metrics can be gamed by pushing them.
Lorenz curve	The share of recommendations held by the least-recommended fraction of the catalog; Gini \(=A/(A+B)\), the gap below the diagonal over the whole triangle.
Loss	A single number scoring how badly a model’s predictions fit the data; training adjusts the model to make it small (§2).
Loss \(\mathcal{L}\)	A single number scoring how wrong a model’s predictions are; training makes it as small as possible (the chapter opener and §4; its letter \(\mathcal{L}\), §3; written \(f\) in §5 to §7).
Loss function	A single number measuring how wrong the model is; training minimizes it.
Low-pass filter	Keep low frequencies (smooth signal), suppress high (noise).
LSTM	Long short-term memory: an RNN with a cell state + forget/input/output gates; the cell line, edited only by a near-\(1\) multiply and an addition, keeps gradients from vanishing quickly (§3.3).

M

Term	Meaning
MAE / absolute loss	Mean of absolute errors; from Laplace noise; outlier-robust; targets the median.
MAE / MSE / RMSE	Mean Absolute / Mean Squared / Root-Mean-Squared rating error.
Main embedding / main view	The embeddings of the un-augmented graph, the ones BPR scores with; LightGCL uses them as one of its two views (§3.4, §5).
MAP	Maximum A Posteriori; the \(\theta\) at the peak of the posterior — it maximizes likelihood \(\times\) prior, i.e. minimizes loss \(+\) regularizer.
MAP (maximum a posteriori) estimate	Maximize posterior ∝ likelihood × prior; ⟹ minimize (−log-lik) + (−log-prior) = loss + regularizer.
Markov chain (first-order)	Next item depends only on the last one; transitions estimated by counting.
[mask] token	A placeholder that replaces a hidden item in BERT4Rec’s input, so the answer is not in what the model reads (§5).
Matrix	A rectangular grid of numbers; also a linear transformation of vectors.
Matrix Factorization (MF)	\(\hat r_{ui}=\mathbf{p}_u^\top \mathbf{q}_i\); trained with regularized MSE.
Matrix–vector product \(A\mathbf{x}\)	Dot each row of \(A\) with \(\mathbf{x}\); “\(n\) in, \(m\) out.”
Matryoshka (MRL)	Truncatable embeddings — a shorter prefix of the vector still works, so you index at a smaller dim to save memory/latency.
Maximum-entropy model	Among all distributions that match what you know, the most uncertain one: knowing nothing about a die, \(\tfrac16\) for each face (§7).
MDP	Markov Decision Process: states, actions, transition, reward and discount — the problem RL solves; the policy is its solution.
Mean pooling / [CLS]	Two ways to get one vector per text: average the token vectors, or read BERT’s summary token (§5.2).
Median	The middle value once the data are sorted; the single value that minimizes the total absolute error (§2).
Memory-based CF	Predict from similar rows/columns of \(R\) at query time (\(k\)-NN).
Message passing	Another name for one graph-convolution step (also called propagation): every node sends its vector along its edges, and replaces its own with the weighted sum it receives.
Mini-batch	A small group of training triples used for one update (§15; Neural Networks & Back-propagation §6).
Minibatch	A handful of training examples drawn at random; SGD estimates the gradient on it (§5).
MIPS (Maximum Inner-Product Search)	Finding the item vectors with the largest dot product against a query: what top-\(K\) scoring is. It coincides with nearest-neighbor search for every query when all the item vectors have the same length.
Mixture of experts	Several small sub-models whose outputs a learned gate weights input by input.
ML-20M	MovieLens 20M, a standard film-rating data set: 138,493 users, 27,278 films, 20,000,263 ratings, 0.53% of the cells filled (§1).
MLE	Maximum Likelihood Estimate; \(\theta\) that best explains the data.
MLP	Multi-layer perceptron — a feedforward stack of fully-connected layers.
MNAR data	Missing-Not-At-Random: whether an interaction is observed depends on what the system chose to show — recommender logs are the textbook case (the opposite, missing completely at random, is pure coin-flip missingness).
Mode	The most plausible value — where a distribution peaks (for a continuous variable the height of the curve at the mode is a density, not a probability: §1); for Beta\((\alpha,\beta)\) with \(\alpha,\beta>1\) it is \((\alpha-1)/(\alpha+\beta-2)\) (§2, §6).
Model card	The documentation page a publisher posts beside a released model: size, license, training data, scores. The primary source for every license claim in this appendix.
Model-agnostic	Added on top of an existing recommender, whichever one it is; RLMRec is described so by its authors (§17).
Model-based CF	Learn a compact model first (e.g. matrix factorization).
MoE (mixture of experts)	Several specialist sub-networks plus a trained gate that weights them and sums their outputs; KAR’s adapter for turning LLM knowledge into features (§2.3).
MoE (mixture-of-experts)	A model that holds many parameters but activates only a few per token — compute per token is below what the total size suggests, but every expert must still sit in memory.
Momentum	Gradient descent that steps on a running total of recent gradients, \(v\leftarrow0.9\,v+g\) (§5).
Momentum / Adam	SGD upgrades: a velocity term / a per-parameter adaptive step (§6).
Monotonic	Here, increasing (a bigger input always gives a bigger output), like \(\log\) and \(e^{x}\), so it keeps the ranking of any list (§3).
MRR	Mean Reciprocal Rank — each user’s reciprocal rank (RR), \(1/\)rank of the first relevant item (\(0\) if none in the top-\(K\)), averaged over users.
MSE / squared loss	Mean of squared errors; from Gaussian noise of the same spread for every point; outlier-sensitive.
MTEB / ann-benchmarks / LMArena / CoNLL	The standard benchmarks and leaderboards for embeddings / vector indexes / chat-LLMs / NER. Priors, not verdicts.
Mult-VAE	A variational autoencoder for recommendation with a multinomial likelihood; loss = reconstruction + \(\beta\cdot\)KL (§11).
Multi-head	Several attentions (“heads”) in parallel with different learned projections; their outputs are concatenated and mixed back to length \(d\) by \(W_O\) (§4.3).
Multi-head self-attention	Each field’s new vector is a weighted mix of all the fields’ vectors, the weights computed from dot-product scores and a softmax; several heads do it independently (AutoInt, §8; built in Representation Learning & the Transformer).
Multi-label	Several independent yes/no labels at once, such as a film that is both a comedy and a drama; each gets a sigmoid (§3).
Multiple comparisons	Running several tests at once: each carries its own \(\alpha\) chance of a false positive, so the chance that at least one fires by luck grows (\(1-0.95^{3}\approx0.14\) for three tests; §8.5, §8.6 Step 3).
Mutual information	How much knowing one variable tells you about another: the KL divergence between their joint distribution and the product of their separate ones; zero exactly when they are independent (§7).
Mutual information (MI)	How much knowing one of two quantities tells you about the other. For a batch of \(N\) nodes, \(\log N-\mathcal L_{\text{cl}}\), with the loss averaged over the batch, is a lower bound on the MI between the two views (§7; Probability & Statistics for Machine Learning §7). RLMRec frames its alignment as maximizing it.

N

Term	Meaning
Nabla / del (\(\nabla\))	The symbol for the gradient operator.
NCF / NeuMF	Neural collaborative filtering: feed user and item embeddings through a small network instead of a dot product; trained with BCE (§10).
NCL	Neighborhood-enriched Contrastive Learning — makes a node’s graph neighbors and cluster prototype its positives (§4).
NDCG	Normalized discounted cumulative gain: each relevant item earns \(1/\log_2(\text{rank}+1)\), divided by the best possible total; \(1\) = perfect list (§10; Evaluation Metrics §7).
NDCG@10	Normalized Discounted Cumulative Gain at \(10\): a ranking score that rewards putting the items a user wanted near the top of a list of \(10\); bigger is better (§8.4).
NDCG@20	Normalized discounted cumulative gain: a score that also rewards ranking a user’s held-out films high in the top-20 list (§4).
Negative (sample)	An item the user did not choose next, used as a wrong answer that the true next item must outscore.
Negative sampling	Drawing un-interacted items to act as negatives — uniformly by default (mostly easy, obscure items; each candidate equally likely, so a popular item, being a positive for many users and never their negative, is drawn less often overall, §2.4), or in proportion to popularity / by hardness, which over-samples popular items (§2.4, §10).
NER (named-entity recognition)	Finding the names of things in text and labeling each span with a type (PERSON, BRAND, GENRE …). The list of types you want is your schema. §6.
Neuron / unit	A weighted sum of inputs + bias, passed through an activation.
Next-item prediction	The core task: given \((i_1,\dots,i_{t-1})\), score every item for being \(i_t\).
NGCF	Neural graph collaborative filtering (X. Wang et al., 2019): a GCN ported to recommendation, keeping \(W\) and \(\sigma\), adding a user–item product term (the element-wise product of the two vectors), and concatenating all layers (§14).
NLL	Negative log-likelihood \(=-\log L\); minimizing it = MLE; this is a loss.
Node	A thing (a user, a film).
Noise	A random error added on top of a true value (a bathroom scale’s reading is your true weight plus a small random wobble) (§2).
Non-IID data (statistical heterogeneity)	Each client’s data comes from a different distribution (one user all horror, another all cartoons), so local updates pull the shared model in different directions — a core difficulty of federation.
Non-stationarity	The arms’ true rates drift over time, so old evidence goes stale and a policy must keep exploring (§1).
Norm	The length of an arrow, \(\lVert\mathbf e\rVert=\sqrt{x^2+y^2}\) for two numbers (§8).
Norm / length \(\lVert\mathbf{u}\rVert\)	The arrow’s length; default L2 \(\sqrt{\sum_i u_i^2}\). L1 \(=\sum_i\lvert u_i\rvert\) (used in lasso).
Normalize (cosine)	Divide a vector by its length (\(\ell_2\)-normalize); the dot product of unit vectors is the cosine similarity, but it answers the cosine question, not the raw dot-product one (§7).
Normalized interaction matrix \(\tilde R\)	\(D_U^{-1/2}RD_I^{-1/2}\): each interaction divided by the square root of the user’s and the film’s degrees; the matrix GF-CF and LightGCL work on (§6).
Normalizing constant	The factor that makes a density’s area (or a list of probabilities) equal \(1\); Beta\((2,2)\) has \(6\) (§2).
Novelty	Non-obviousness, often \(-\log_2(\text{popularity})\).
Null hypothesis \(H_0\)	The skeptical default “no real difference”; a test asks whether the data are strong enough to reject it.
Null space	Every input a matrix sends to the zero vector \(\mathbf{0}\).

O

Term	Meaning
Odd cycle	A loop of an odd number of edges, such as a triangle; a connected graph with one has no \(-1\) mode.
Odds	\(p/(1-p)\).
Off-policy	Learning from data collected by a different (older) policy than the one being trained.
Off-policy, IPS, doubly-robust	Scoring a policy from another policy’s log; IPS reweights logged data by \(1/p(\text{action})\); doubly-robust adds a reward model that, when roughly right, cuts the variance, the run-to-run jitter of the estimate (§7).
Offline / online	Offline = computed in advance, in bulk, while nobody is waiting; online = computed live, while the user waits (§7).
Offline replay	Unbiased evaluation of a bandit policy on a log of actions served uniformly at random, independent of context (Li et al. 2011).
Omnibus / post-hoc test	An omnibus test (such as the Friedman test) asks one question across all the models and datasets at once; post-hoc pairwise tests, planned in advance, run only after it finds some difference (§8.5). This planned use is not the after-the-fact choice that §8.2 warns against.
One- vs two-sided test	Two-sided asks “A \(\ne\) B” (the default); one-sided asks “A \(>\) B”: it halves \(p\) when the result goes the pre-registered way and gives \(1-p/2\) (near \(1\) for a clear result) when it goes the other way; valid only if the direction was pre-registered (§8.6, Step 1).
One-hot	A list with a single \(1\) and \(0\)s elsewhere, like \([1,0,0]\) (§3).
One-hot vector	A length-\(V\) vector \(\mathbf{o}_k\), \(1\) in slot \(k\), \(0\) elsewhere; encodes identity but no similarity (all pairs orthogonal; §1.1).
One-hot vector \(\mathbf o_k\)	All zeros except a \(1\) in slot \(k\); \(\mathbf o_k^\top E\) picks row \(k\) of \(E\).
OneRec	Kuaishou’s end-to-end generative recommender, which replaces the retrieval-and-ranking cascade with one model tuned by iterative DPO (§2.2).
Open weights / open source / open data	Released weights only / released under an OSI-approved license (§1 vocabulary table) / training data published too. They are different and often confused.
Optimizer	The procedure that updates the parameters from the gradient; gradient descent is the basic one (§4, §5).
Order / degree	How many features an interaction multiplies together: second-order is a pair, third-order three at once.
Order of magnitude	A factor of 10; three orders of magnitude is a factor of 1,000 (§4.1).
Orthant	The region of space where the signs of all coordinates are fixed, the high-dimensional analogue of a quadrant; SimGCL keeps its noise in the orthant of the vector it nudges (§4).
Orthogonal	Perpendicular; dot product \(0\).
Orthonormal basis	A full set of axes that are mutually orthogonal and each unit length; a coordinate is then just a dot product (\(U,V\) in §6).
Outer product \(\mathbf{u}\mathbf{v}^{\top}\)	A column times a row = a whole rank-1 matrix, entry \((r,c)=u_rv_c\).
Over-smoothing	Too many layers make all embeddings point in (almost) the same direction, so users can no longer be told apart (§12). Their lengths still differ — it is a collapse of direction, not a collapse to a single point.
Overfitting	Memorizing training noise; great on train, poor on new data.

P

Term	Meaning
\(p\)-value	Probability, if \(H_0\) were true, of a result at least this extreme (for the \(t\)-test: a ratio \(\lvert t\rvert\) at least this large; either direction for a two-sided test); small = significant. NOT \(\Pr(H_0\text{ true})\).
P5	One text-to-text model for five families of recommendation task: “Pretrain, Personalized Prompt & Predict Paradigm”, five P’s (§2.1).
Paired \(t\)-test	Tests whether the true mean of the per-pair differences is \(0\), using their sample mean \(\bar d\): \(t=\bar d/(s_d/\sqrt n)\).
Parameter / free	A number the model stores and trains; a number is free when training may set it to any value, the opposite of a given number such as a feature (§6).
Parameters	A model’s adjustable numbers (e.g. weights, embeddings) — in a neural network mostly its weights, in a recommender also its embeddings (§5); training nudges them, step by step, until the loss is small.
Parameters, “7B”	The learned weights of a network, counted with M = million and B = billion; a 7B model has 7,000,000,000 of them. Sets memory and roughly sets speed.
Pareto frontier	The set of best-possible (recall, speed) points an index can reach by turning its accuracy/speed knob; off the frontier you can do better on both, on it you trade one for the other.
Partial derivative \(\partial f/\partial x\)	Derivative with respect to one variable, the others held fixed.
Pearson correlation	A similarity measure (centered cosine) for ratings — removes each user’s rating bias.
Percentile	The \(q\)th percentile is the value below which \(q\%\) of the values fall; the bootstrap reads its interval from the \(2.5\)th and \(97.5\)th.
PinSage	A GraphSAGE-like model (Ying et al., 2018) that picks neighbors by short random walks and ran at Pinterest on \(3\) billion nodes (§15).
Plain average	The matrix \(D^{-1}A\): each neighbor weighted \(1/\deg\) of the receiver; the weights in a row sum to \(1\) (§7).
Platt scaling	Calibrates a raw score \(s\) with a fitted S-curve \(\sigma(as+b)\); keeps the ranking exactly when the slope \(a>0\).
PMF / PDF	Probability mass (discrete) / density (continuous) function; sums/integrates to 1.
Pointwise	Scoring each candidate with its own separate model call, as against scoring a whole list at once (§4.1).
Pointwise / pairwise / listwise	Score each (user, item) pair alone / the order of two items / the whole list at once (softmax: does the positive top it; LambdaRank: pairs weighted by the metric).
Policy	The rule that picks the arm each round, using everything seen so far (§2).
Policy \(\pi(a\mid s)\)	The learned rule mapping a state to an action (\(\pi\) names the policy, not \(3.14159\)); what RL optimizes. A language model’s output rule in reinforcement learning from human feedback (RLHF) is also called a policy (Probability & Statistics for Machine Learning, glossary); §4 applies the idea to LLM-based recommenders.
Policy gradient	Optimize the policy directly by gradient ascent on expected return (REINFORCE is the basic algorithm; \(\theta\) are the policy’s parameters).
Polynomial filter	A filter that is a polynomial in \(\tilde{A}\) (what LightGCN layers compute).
Popularity baseline	A simple recommender that shows everyone the same most-popular items.
Popularity bias	Tendency to over-recommend high-degree (blockbuster) items (§13).
Popularity bias (in eval)	Popular items dominate the test set, so pushing them inflates accuracy metrics.
Popularity bias / long tail	A recommender over-suggesting already-popular items at the expense of the long tail of rarely seen ones (§2.4).
Position bias	Higher-ranked slots draw more clicks regardless of true relevance.
Positional embedding	A learned per-position vector added to item embeddings so attention can see order.
Positional encoding	A per-position signal added to embeddings so order-blind attention can see order (sinusoidal or learned; many recent LLMs instead rotate queries and keys by a position-dependent angle — RoPE; §4.3).
Positive / negative pair	Positive = two views of the same node (in DirectAU, a user and an item she interacted with); negative = views of different nodes (§3.1, §3.3).
Positive definite	Curvature strictly \(>0\) in every direction (for a \(2\times2\) Hessian: \(f_{xx}>0\) and \(D>0\); all eigenvalues \(>0\)). Positive definite everywhere ⇒ at most one minimum (§6).
Positive semidefinite (PSD)	Curvature \(\ge0\) in every direction (for a \(2\times2\) Hessian: \(f_{xx}\ge0\), \(f_{yy}\ge0\), \(D\ge0\); equivalently all eigenvalues \(\ge0\)). A function whose Hessian is PSD at every point is convex.
Positive-semidefinite (PSD)	A symmetric matrix with all eigenvalues \(\ge0\) (equivalently, for a symmetric matrix, \(\mathbf{x}^{\top}A\mathbf{x}\ge0\) always); every \(A^{\top}A\) is PSD since \(\mathbf{x}^{\top}A^{\top}A\mathbf{x}=\lVert A\mathbf{x}\rVert^{2}\).
Post-activation \(a\) / layer output \(\mathbf h\)	\(a=\phi(z)\), one neuron’s output; \(\mathbf h^{(\ell)}=\phi(\mathbf z^{(\ell)})\) collects layer \(\ell\)’s outputs into one vector.
Posterior collapse	A VAE’s code pushed all the way to the prior, so it carries nothing about the user (§13).
Posterior \(p(\theta\mid D)\)	Updated belief after data.
Power	The probability that a test detects a real difference when there is one; having more pairs gives more power, and pairing the models on the same seeds helps when their scores rise and fall together (§8.1; §8.2, §8.3).
Pre-activation \(z\) (logit, for a classifier’s last layer)	The raw score \(\mathbf{w}\cdot\mathbf{x}+b\) before the activation; for layer \(\ell\), the vector \(\mathbf z^{(\ell)}=W^{(\ell)}\mathbf h^{(\ell-1)}+\mathbf b^{(\ell)}\), starting from \(\mathbf h^{(0)}=\mathbf x\).
Pre-registered	Written down before running anything (a directional hypothesis, the tests to be run), so it cannot be chosen after seeing the data (§8.5).
Precision@K	Of the \(K\) shown, the fraction relevant.
Predicted score \(\hat y\)	The dot product of a user’s and an item’s embeddings (§4).
Predictive learning	SSL pretext: predict a property derived from the data (e.g. a masked item/attribute); one correct answer, no negatives; discriminative mechanism (this chapter’s own reading, not a standard label; §6.3).
Prefix	The beginning of a list, \((i_1,\dots,i_{t-1})\): the items seen so far, in order.
Pretext task	The self-supervised literature’s name for the auxiliary task; the two words mean the same thing (§6.2).
Pretraining	Train a big Transformer on a huge corpus with a self-supervised objective, then reuse it (§5).
Prior and posterior	Belief about a click chance before data (prior) and after data (posterior).
Prior \(p(\theta)\)	Belief about \(\theta\) before data.
Product rule	\((fg)' = f'g + fg'\) — the derivative of a product is not the product of the derivatives (§2).
Production	The live system that answers real users’ requests, as opposed to an experiment on saved data.
Profile	A short LLM-written description of a user’s taste or an item’s character, used as input to an embedder.
Projection \(\mathrm{proj}_{\mathbf{v}}\mathbf{u}\)	The shadow of \(\mathbf{u}\) on \(\mathbf{v}\)’s line, \(\frac{\mathbf{u}\cdot\mathbf{v}}{\lVert\mathbf{v}\rVert^{2}}\mathbf{v}\); in an orthonormal basis a coordinate is just a dot product.
Projection \(g\)	The small network that maps the semantic vector into the collaborative space so the two can be compared (§4).
Projector \(\mathbf v\mathbf v^{\top}\)	The matrix that keeps the part of a signal along the unit vector \(\mathbf v\) and discards the rest.
Prompt	The text handed to an LLM: the question plus whatever it should know (§2).
Propensity	The probability \(p\) that the logging policy took the action it did; a film shown with \(p=0.1\) counts \(10\) times per click (§7).
Propensity / IPS	Propensity = probability an item was shown; inverse propensity scoring (IPS) re-weights each interaction by \(1/\)propensity to undo exposure bias.
Prototype / EM	NCL’s prototype is the center of a cluster of look-alike nodes; EM (expectation-maximization) alternates guessing the clusters and improving the embeddings against them (§4).
Provider vs. consumer fairness	Comparable exposure across item creators (provider) vs. comparable quality across user groups (consumer).
Pseudo-count	The heads and tails a Beta prior behaves as if already seen: \(\alpha-1\) and \(\beta-1\) (§2).

Q

Term	Meaning
Q-learning	Learn \(Q(s,a)\) by moving it toward reward plus \(\gamma\times\) the best next \(Q\) (temporal-difference learning), then take the highest-\(Q\) (greedy) action; it must score every action, so slate methods avoid it. Deep Q-learning uses a neural network for \(Q\); DRN applies it to news.
QPS (queries per second)	How many requests a system answers each second: its throughput, as distinct from any one request’s latency.
Quadratic form	The number \(\mathbf{x}^{\top}A\mathbf{x}=\mathbf{x}\cdot(A\mathbf{x})\).
Quantization	Storing weights/vectors in fewer bits (4/8-bit, int8/binary) to shrink memory and speed inference, at a small quality cost.
Query / Key / Value	Soft database lookup: a query is matched (dot product) against keys to retrieve a blend of values (§4.1).
Query, key, value	In attention, each position’s query asks, each key says what that position is about, and each value is what it hands over when matched (§4).

R

Term	Meaning
RAG (retrieval-augmented generation)	Retrieving relevant evidence and inserting it into the LLM’s prompt so it reasons over real data instead of pretrained memory (§5).
Random seed	The number that fixes a training run’s random choices; another seed gives a slightly different, equally valid result (§8).
Random split	Hold out a random fraction of interactions, per user or globally (say which) — leaks the future for time-ordered data.
Random variable	A quantity whose value is uncertain (coin, die, temperature).
Random walk	A path that steps along randomly chosen edges; one of SGL’s ways to damage the graph (§4).
Randomized SVD	A fast approximate way to find only the top-\(q\) singular vectors, using random projections (§5).
Rank	The matrix’s number of independent directions: independent rows/columns (§2) = nonzero singular values (§6). An \(n\times n\) matrix of rank \(n\) is full rank; below \(n\), rank-deficient.
Ranking	The expensive, high-precision second stage that reorders the shortlist with a feature-rich model; search and LLM writing call the same stage reranking.
Ranking scope	How many candidate items the held-out one is ranked against: all of them (full ranking) or a sample. Printed in Table B.5 as its own column (§4).
Recall vs QPS	The ANN trade-off: fraction of true neighbors found vs queries-per-second. Always compare at a fixed recall.
Recall@20	Of the films a user liked that were held back from training, the share found in the model’s top \(20\) (§12, §14; Evaluation Metrics).
Recall@K	Of all relevant items, the fraction in the top-\(K\).
Recall@K / Precision@K / NDCG@K	Top-\(K\) ranking quality: the share of the user’s relevant items found / the share of the list that is relevant / a score that rewards putting relevant items high (§6).
Reciprocal-rank fusion	How two ranked lists are merged: each item scores \(\sum 1/(60+\text{rank})\) over the lists it appears in, so steady agreement beats one top spot. §3.
Regime	A research community’s shared package of datasets + models + one evaluation protocol. Numbers are comparable only inside a regime, and only when split, preprocessing and ranking scope match as well (§4).
Regression	Predicting a real number, such as a price (§2).
Regret	\(\sum_t(\mu^{*}-\mu_{a_t})\): expected reward lost versus always pulling the best arm.
Regularization	Making an ill-posed solution “regular” (well-behaved); from Hadamard/Tikhonov.
Regularization strength \(\lambda\)	The weight on the penalty in “loss \(+\,\lambda\times\) penalty”; a narrower prior is a larger \(\lambda\) (\(1/(2s^2)\) for L2, \(1/b\) for L1, against a summed negative log-likelihood; rescaled for an average loss) (§6).
Regularizer	A penalty added to the loss that keeps the parameters well-behaved; under MAP it is \(-\log p(\boldsymbol\theta)\) — a Gaussian prior gives L2 (weight decay), a Laplace prior gives L1 (lasso) (§6).
Reinforcement learning (RL)	Learning to choose actions to maximize long-term reward when actions can change the next state.
Relevant item	A ground-truth item the user actually likes.
Reliability diagram	Predicted vs. observed rate, one point per bin; calibrated points lie on the diagonal.
ReLU	\(\max(0,z)\) — Rectified Linear Unit; in practice the most used hidden-layer activation (a claim from practice, not tested here).
ReLU (rectified linear unit)	\(\max(0,z)\); derivative \(1\) for \(z>0\) and \(0\) for \(z<0\) (§3).
Representation collapse	The failure where the encoder maps every node to the same vector; InfoNCE then sits at its collapse value \(\log N\) (§3.3).
Representation learning	The network learning useful features by itself.
Residual	\(r=\hat y-y\), the miss on one point (§2.1).
Residual (skip) connection	A shortcut that carries an earlier stage’s output past the later ones; in LightGCN the layer-\(0\) term of the layer combination (§12).
Residual connection	A sub-layer computes \(\mathbf{x}+f(\mathbf{x})\); the \(+1\) in its derivative is a gradient highway for deep stacks (kept clean in the Pre-LN form, §4.3).
Residual quantization	Quantize a vector, subtract the codeword, quantize what is left over against a second codebook, and so on. The chosen codewords add back up to the vector, and each level usually shrinks the error, though with an arbitrary codebook it can grow (§2.2).
Retention	Whether the user comes back (next week); a long-horizon objective.
Retrieval / candidate generation	The cheap, high-recall first stage that cuts millions of items to a few hundred.
Retrieval → ranking	The production funnel: a cheap, high-recall retrieval stage (candidate generation: MF/\(k\)-NN with ANN search) then a precise ranking stage, also called reranking (§7).
Return \(G\)	Cumulative discounted reward \(\sum_t \gamma^t r_t\) — the long-term objective.
Reward model	A model trained to score outputs: RLHF trains it on preferred and rejected outputs, OneRec on the logged watch-time, view, follow and like outcomes of whole sessions; Rec-R1 uses none (§4).
Reward \(r\) / action \(a\)	The immediate payoff (a click, a watch-minute, a return visit) / what the recommender does (show an item or slate).
Ridge / L2 penalty	A penalty on large coefficients added to least squares; in LinUCB it is the identity start \(A=I\).
Risk / empirical risk	Expected loss over the true distribution / its average on the training sample.
RLHF / policy	Reinforcement learning from human feedback: tuning a language model toward the answers people prefer / the rule a model uses to choose its outputs; a KL penalty keeps the tuned policy near the original (§7).
RLMRec	Representation Learning with Large Language Models for Recommendation (WWW 2024): aligns a graph backbone’s embeddings with LLM-written profile embeddings (§7).
RLMRec-Con / -Gen	RLMRec’s contrastive variant (pull a node’s embedding toward its mapped profile vector against other users, an InfoNCE-form loss) and generative variant (hide a node’s input, rebuild its profile vector; cosine inside the same ratio): the paper’s Eqs. 16 to 18 (§7).
RMSE / MAE	Root-mean-square error / mean absolute error: how far predicted ratings miss the true ones; RMSE squares the misses first, so big misses count more (§6).
RNN	Recurrent neural network: one shared cell that updates a hidden-state “memory” \(\mathbf{h}_t=\tanh(W\mathbf{h}_{t-1}+U\mathbf{x}_t+\mathbf{b})\) along a sequence (§3.1).
RNN / hidden state \(\mathbf h_t\)	A net that folds each item into a running fixed-length summary of the session.
Robustness	Resistance to a few wildly-off values: the Laplace / MAE / median are robust, the Gaussian / MSE / mean are not (§2).
RQ-VAE	The residual-quantized variational autoencoder that learns the Semantic-ID codebooks by reconstructing the content embedding (§2.2).

S

Term	Meaning
Saddle point	Critical point that is a min in some directions, a max in others (indefinite Hessian).
Sample space / event	The set of possible outcomes / a subset of them.
Sampled softmax	Training with a softmax over a sampled handful of items rather than the whole catalog; for in-batch negatives, over the batch (§7).
Sampled softmax (SSM)	Listwise cross-entropy over a positive + many sampled negatives; structurally = InfoNCE.
Sampling bias	In-batch sampling meets an item in proportion to how often it is a positive, so popular items are over-penalized as negatives; the log-\(Q\) correction repairs it (§3).
SASRec	Self-Attentive Sequential Recommendation: a causal Transformer over the item sequence.
SASRec / BERT4Rec	Sequential recommenders: a causal (left-to-right) / bidirectional (masked) Transformer over the item history (§4.4).
Scalar	A single number (as opposed to a vector); multiplying by one scales a vector.
Scaled cosine error	GraphMAE’s loss: the mean over hidden nodes of \((1-\cos(\mathbf x,\mathbf z))^{\gamma}\) with \(\gamma\ge1\); at \(\gamma=1\) it is \(1-\cos\) (§6.2, §6.4).
Second derivative \(f''\)	Rate of change of the slope = curvature.
Second-derivative test	At a critical point of \(f(x,y)\), \(D=f_{xx}f_{yy}-f_{xy}^2\): \(D>0,\ f_{xx}>0\) bowl (min); \(D>0,\ f_{xx}<0\) cap (max); \(D<0\) saddle; \(D=0\) silent (§6).
Secure aggregation	The server sees only the sum of client updates, never any individual one (Bonawitz et al., 2017).
Selection / exposure bias	Only items the system showed can get feedback; un-shown items look “disliked” by silence.
Self-attention	Attention of a sequence to itself (Q, K, V all from the same tokens; §4.1).
Self-loop	An artificial edge from a node to itself, so that a node counts itself among its neighbors; present in GCN and NGCF, absent in LightGCN (§6, §14).
Semantic embedding	A fixed-length vector encoding the meaning of the profile text (from a frozen sentence encoder), distinct from the CF latent space.
Semantic ID	A short sequence of content-derived codewords identifying an item (a quantized text embedding) (§2.2).
Semi-supervised node classification	Predicting the label of unlabeled nodes when a few nodes are labeled; the task the GCN paper solves (§6).
Sensitivity / false-positive rate	\(\Pr(+\mid S)\), the chance a test flags a true case / \(\Pr(+\mid\neg S)\), the chance it wrongly flags a healthy one (§5).
Sequential recommendation	Predict the next item from a user’s ordered history.
Serendipity	Relevant and surprising: hits a popularity baseline would not have shown, divided by \(K\).
Serving path / serving	Serving is answering a live request; the serving path is everything that runs between the request and the list shown (§1).
Session	The ordered items one person goes through in one sitting, a visit or an evening; short, and sometimes anonymous.
Session-based recommendation	Same, but from a short, often anonymous current session (no user id).
SGD	Stochastic Gradient Descent — gradient steps on random mini-batches.
SGL / SimGCL / XSimGCL / LightGCL	Self-supervised Graph Learning / Simple Graph Contrastive Learning / eXtremely Simple GCL / Light Graph Contrastive Learning — GCL methods differing in how the two views are built (edge-drop / embedding noise / cross-layer noise / SVD). Note the final letter: LightGCL is a training method, LightGCN the backbone.
Shape \(m\times n\)	\(m\) rows by \(n\) columns.
Shrink factor	\(1-2\eta\lambda\), what one L2 step multiplies a weight by (§5.1).
Shrinkage	Multiplying a similarity by \(n/(n+\kappa)\), so one backed by few co-ratings \(n\) is pulled toward \(0\) (§4).
Side features	Information about a user or film beyond clicks, such as genre or text (§4 box).
\(\sigma\) (nonlinearity)	An activation function: the rectified linear unit (ReLU), \(\mathrm{ReLU}(z)=\max(0,z)\), in GCN, and LeakyReLU (a ReLU that lets a small slope through for negative inputs; NGCF’s paper uses it) in NGCF; removed in LightGCN. In this chapter \(\sigma\) means the sigmoid only inside the BPR loss of §15: \(\sigma(z)=1/(1+e^{-z})\), the S-shaped function that squeezes any number into \(0\) to \(1\).
Sigmoid / logit	\(\sigma(z)=1/(1+e^{-z})\) squashes a raw score \(z\) (the logit) into \((0,1)\) (§2.2).
Sigmoid \(\sigma(z)\)	\(\frac{1}{1+e^{-z}}\): an S-shaped curve squashing any \(z\) into the range between \(0\) and \(1\); \(\sigma'=\sigma(1-\sigma)\le0.25\) (§3).
Significance (paired \(t\)-test)	Is a gain bigger than run-to-run seed noise? Report mean \(\pm\) standard deviation over seeds and a \(p\)-value.
Significance level \(\alpha\)	The pre-set bar (usually \(0.05\)); reject \(H_0\) if \(p<\alpha\).
SimCLR	A simple framework for contrastive learning of visual representations (Chen et al., 2020b, ICML); the source of the two-augmented-views recipe (§3.1).
Singular value \(\sigma\)	The strength of an SVD pattern, \(A\mathbf{v}_i=\sigma_i\mathbf{u}_i\); \(\sigma_i=\sqrt{\lambda_i(A^{\top}A)}\ge0\), biggest first.
Singular value decomposition (SVD)	A standard way to split a table of numbers into its main patterns (§13).
Singular value decomposition (SVD) \(A=U\Sigma V^{\top}\)	Factor any matrix into orthonormal patterns (\(U,V\)) scaled by singular values (\(\Sigma\)).
Singular value decomposition, SVD (\(M=U\Sigma V^\top\))	Factorizes a (rectangular) matrix into patterns \(U,V\) and strengths \(\Sigma\); keeping the top-\(q\) is an ideal low-pass (LightGCL).
Singular vectors	The columns of \(U\) (left, \(\mathbf{u}_i\): output directions) and of \(V\) (right, \(\mathbf{v}_i\): input directions).
Slate	A whole page/list of recommended items — the (combinatorial) action in slate RL.
SlateQ	Decomposes a slate’s value into per-item \(Q\)-values, making value-based slate RL tractable.
SLM	Small language model (\(\approx 0.5\)–9B) — cheap, high-throughput; the recsys feature-generation workhorse.
Smooth / differentiable	Has a derivative everywhere — no kinks or jumps; looks straight up close.
Soft-thresholding	Move a weight a fixed amount toward zero and stop at \(0\) if it would cross: \(\lambda/2\) in §5.1’s one-weight solution.
Softmax	Multi-class sigmoid; scores → a probability distribution.
Softmax output	Multi-class output head: \(K\) scores → a probability distribution, paired with cross-entropy.
Sparse / one-hot feature	A category encoded as all-zeros with a single \(1\); CTR rows stack millions of these.
Sparse matrix	A table that is almost all zeros, stored as a list of its non-zero entries: \(10\) of the toy’s \(36\) (§15).
Sparse matrix-vector product	Multiplying a sparse matrix by a vector (or by each column of a matrix) while adding up only its stored non-zero entries (§15).
Sparsity	Most of \(R\) is unknown (MovieLens-20M: \({\sim}99.5\%\) empty); the core difficulty.
Spectrum	The list of a matrix’s singular values, biggest first (§6); a quickly falling spectrum means a few pieces capture most of the matrix.
SSL (self-supervised learning)	Training on an auxiliary task that needs no labels; supervision is manufactured from the data’s own structure.
Standard deviation \(\sigma\)	\(\sqrt{\operatorname{Var}[X]}\); the spread in the same units as \(X\) (§1).
Standard error (SE)	\(s_d/\sqrt n\) — how much an estimate wobbles across repeats; shrinks with more data.
State \(s\)	A summary of the current situation; what the agent conditions on.
Static vs contextual	word2vec gives one fixed vector per word; a Transformer gives a different vector per occurrence (context-dependent; §4.2).
Stochastic gradient descent (SGD)	Gradient descent on a noisy gradient estimated from a random minibatch, not the full dataset.
Structured output	Forcing an LLM to emit schema-valid JSON via constrained decoding — valid shape, not guaranteed-correct values.
Structured output / JSON mode	Forcing the LLM to emit schema-valid JSON, so the result is very likely parseable (§5 still validates).
STT / ASR / TTS / WER	Speech-to-text (= automatic speech recognition) / text-to-speech / word error rate — (substituted + deleted + inserted words) / words in the correct transcript. §7.
Student-\(t\) distribution	The spread of \(t\)-values pure luck produces — a bell with fatter tails than the Gaussian when there are few pairs, shaped by the df (§8.3).
Subgradient	A stand-in slope at a kink where the true derivative is undefined (e.g. any value in \([0,1]\) for ReLU at \(0\)).
Subword tokenization (BPE)	Split text into reusable sub-word pieces by greedily merging the most frequent adjacent pair; fixed vocabulary, and no <UNK> when the base vocabulary is all 256 bytes (byte-level BPE); WordPiece and Unigram are relatives; SentencePiece is a library that runs BPE or Unigram (§5.3).
Supervised / self-supervised learning (SSL)	Learning from examples that come with their correct answers / from a task the data sets for itself, with no human labels (§1, §11).
Surprise	\(-\log p\) of what happened (§2.3).
Surprise / self-information	\(-\log p(x)\); how surprising an outcome is (\(0\) if certain, \(\to\infty\) if rare).
SVD / singular value	Singular value decomposition: a table of numbers written as a sum of simple patterns, each with a strength, its singular value (Linear Algebra for Machine Learning §6; used in §5).
Symmetric	\(A=A^{\top}\); for a real symmetric matrix, the eigenvectors can be chosen mutually orthogonal (§5).
Symmetrically normalized adjacency \(\tilde{A}\)	\(\tilde A = D^{-1/2}AD^{-1/2}\): entry \((u,v)\) is \(A_{uv}/(\sqrt{\deg u}\sqrt{\deg v})\). One multiplication by it is one propagation layer (§7).

T

Term	Meaning
t-SNE / variational inference	Drawing high-dimensional data as a 2-D map by making a KL divergence small / replacing a hard-to-compute posterior (§5) by the closest member of a simpler family, closest measured by KL (§7).
TALLRec	An LLM fine-tuned on recommendation instructions: a Tuning framework to Align LLMs with Recommendation (§2.1).
Tangent	The line that just touches a curve at a point; its slope is the derivative there (§1).
Tanh	Hyperbolic tangent, \(\tanh(z)=\frac{e^{z}-e^{-z}}{e^{z}+e^{-z}}=2\sigma(2z)-1\) — a sigmoid stretched to \((-1,1)\).
Taylor / linear approximation	\(f(x+\delta)\approx f(x)+f'(x)\delta\); a curve looks straight up close.
Temperature \(\tau\)	Divides the scores before a softmax: a small \(\tau\) gives a nearly one-hot output, a large \(\tau\) a nearly uniform one (§3).
Test set / fold	Data held back to measure a trained model / one of the \(k\) slices of cross-validation, each taking a turn as the test set (§8).
TF-IDF	A way to turn text into a feature vector (term frequency × inverse document frequency).
The rated-set leak	With explicit ratings, sending an update for item \(i\) reveals (to a server that knows whose it is) that the user rated \(i\), and the gradient is a scaled copy of the user’s vector; FedRec masks the rated set with decoy items (see §3).
Thompson sampling	(Thompson, 1933) Keep a posterior per arm; sample one value from each and pull the largest sample.
TIGER	The system that introduced Semantic-ID generative recommendation: Transformer Index for Generative Recommenders (§2.2).
Token	One element of an input sequence — a word, a sub-word piece (§5.3) or, in a recommender, one item of a history; the unit attention works on (§4.1).
Top-\(K\)	The \(K\) highest-scoring unseen items returned to the user.
TPR / FPR	True- / false-positive rate: the share of the relevant / irrelevant items scoring at or above a threshold (a cut-off score); lowering the threshold step by step traces the ROC curve.
Trace	Sum of a square matrix’s diagonal entries; equals the sum of its eigenvalues (shown here for \(2\times2\)).
Train/test split	How held-out test data is chosen: a random split, leave-one-out (LOO: each user’s chronologically last interaction), or a global temporal split (one cut-off date for everyone).
Train–serve skew	One case of the wider train/serve difference: the mismatch between the logged, sampled data a model is trained on and the whole live catalog it is served over (§8).
Transductive	A model that keeps one stored vector per node seen in training and cannot embed a node it never saw; a brand-new user or film has no row in \(E^{(0)}\) (§14). The GCN paper uses the word in a different sense: test nodes are visible during training.
Transductive / inductive	A model that can represent only the ids seen in training / one that can give a new id a vector from its content (§3).
Transformer	A stack of (self-attention + add&norm + feed-forward + add&norm) blocks; no recurrence, so training processes all positions in parallel (Vaswani 2017; §4.3).
Transition probability	\(P(\text{next}=i\mid\text{last}=l)\) — count of \(l\!\to\!i\) over count of \(l\)-as-from.
Transpose \(A^{\top}\)	The matrix mirrored across its diagonal (rows ↔︎ columns).
Triple	A training example \((u,i,j)\): a user, a film she liked, and a film she has no edge to (§13, §15).
Triple \((u,i^+,i^-)\)	A training example for pairwise ranking: user \(u\) prefers positive \(i^+\) over sampled negative \(i^-\).
Truncated SVD	Keep the top-\(q\) singular components = best low-rank approximation (least-squares / Frobenius sense — the Eckart–Young theorem).
Two-stage funnel	Retrieval → ranking: fast-and-forgiving then slow-and-sharp, so a rich model can run on only a few hundred candidates; the final cut applies the already-seen filter and any business rules, then shows the top \(K\).
Two-tower LLM-embedding retrieval	Embed users and items with an LLM-grade encoder offline, then retrieve by ANN over those vectors at serve time: enhancer-class cost, LLM semantics on the retrieval path (§4).
Two-tower model	Embeds user and item with two separate networks and scores by dot product; trained with sampled softmax (§10).
Type I / II error	False positive (reject true \(H_0\)) / false negative (miss a real effect).

U

Term	Meaning
UCB	Pull \(\arg\max_i\,[\hat\mu_i + \sqrt{2\ln t / n_i}\,]\), with \(t\) the pulls so far and \(n_i\) the pulls of arm \(i\): highest upper bound, optimism under uncertainty.
Uniform / popularity-weighted / in-batch / hard negatives	Four sampling distributions for \(i^-\): equal-probability; \(\propto\text{pop}^{0.75}\); reuse the batch’s other positives; pick high-scoring (confusing) items.
Uniformity	Embeddings spread evenly over the sphere. It is the term that scores collapse as worst (§3.3); that it also helps the long tail is the papers’ claim, not tested here.
Unit circle / semi-axes	The unit circle is all vectors of length \(1\); a matrix sends it to an ellipse, whose longest and shortest radii (the semi-axes) have lengths \(\sigma_1,\sigma_2\) (§6).
Unit vector	A vector of length \(1\); “normalizing” = dividing by your own length.
Unit vector / unit hypersphere	A vector of length 1; the set of all such vectors, a circle in 2-D (§3.3).
Universal approximation	A wide enough net can approximate any continuous function on a bounded region.
User profile	The average of the feature vectors of the items a user liked; content-based filtering scores items against it (§2).
User simulator	A model of user behavior used to train RL without experimenting on real users.
User–item matrix \(R\)	Rows = users, cols = items, entries = ratings/clicks (mostly unknown).

V

Term	Meaning
Validation loss	The loss on held-out examples the network never trains on — a stand-in for new data (§6).
Validation set	A slice of the data held out from training and used only to measure progress (§5.2).
Value / \(Q\)-value	Expected return from a state (\(V\)) or a state–action pair (\(Q\)).
Vanishing gradient	A chain-rule product of many small factors shrinks toward zero, so early layers stop learning (§3, §8; see also the §5 box on gradient pathologies).
Variance \(\sigma^2\)	Expected squared distance from the mean; spread.
Variational autoencoder (VAE) / ELBO	A network that learns to generate new data / its training objective, the evidence lower bound, which can never exceed the log of the evidence (§5); it contains a KL term that keeps each example’s code near a prior (§7).
Vector	An ordered list of numbers; equivalently, an arrow from the origin to a point.
Vector-wise vs bit-wise	A cross with one weight per pair of whole field vectors (xDeepFM’s CIN) versus one weight per coordinate (DCN, an MLP).
View	The embedding a node gets after an augmented input is encoded; contrastive learning needs two per node (§3.1).

W

Term	Meaning
\(W\) (weight matrix)	Learnable feature transform in GCN/NGCF; removed in LightGCN.
Walk	A path of \(K\) steps along edges that may turn back (Ann, Titanic, Ann); layer \(K\) mixes the starts of nodes a \(K\)-step walk reaches (§5).
Weight	One adjustable multiplier in a layer’s weighted sum; the main kind of parameter in a neural network (§3).
Weight / bias	Learned multiplier per input / learned constant offset.
Weight decay	The L2 penalty \(\lambda\lVert W\rVert^2\) on the weights; its gradient \(2\lambda W\) shrinks every weight a little at each step (§7).
Weighted / unweighted	Whether edges carry a number (rating/count) or are just 0/1.
Weighted sum	Scale several vectors by numbers and add the results, e.g. \(1\cdot[2,1]+2\cdot[0,3]=[2,7]\).
Weights	A model’s learned parameters, collected in one list \(\theta\) (§6), the numbers a regularizer keeps small.
Wide & Deep	Joint linear-with-hand-crosses over the raw features (memorize) \(+\) MLP-over-embeddings (generalize) model.
Wilcoxon signed-rank	Non-parametric paired test (ranks, not values); use when differences aren’t normal and there are at least \(6\) pairs (below that it cannot reach \(p<0.05\); §8.6).
word2vec	Predict-based word embeddings trained on word–context prediction: skip-gram predicts the context from a word, CBOW the word from its context (Mikolov 2013a, 2013b; §2.2).
WRMF / ALS	Weighted Regularized MF / Alternating Least Squares (implicit-feedback, squared error).

X

Term	Meaning
Xavier / Glorot init	The same idea for symmetric activations (tanh, sigmoid): variance \(2/(\text{fan\_in}+\text{fan\_out})\), with fan_out = neurons in the layer (§6).
xDeepFM / AutoInt / DIN	Vector-wise crosses (CIN) / attention-learned interactions / per-candidate behavior attention.

Z

Term	Meaning
Zero vector \(\mathbf{0}\)	The vector whose components are all \(0\); it has no direction, so it cannot be normalized and has no cosine.
Zero-shot / few-shot	Zero-shot: a frozen model is simply asked, with no examples and no extra training. Few-shot: still frozen, but a handful of solved examples are pasted into the prompt (§2.1).
Zero-shot NER	Extracting arbitrary entity types with no task-specific training (GLiNER, NuNER-Zero).