List of Symbols
A quick reference for the symbols that recur across the book. Each is defined in full the first time it appears; this list is a reminder, grouped by where it is introduced. Where the same object is written two ways across the book, the change is flagged.
Linear algebra (Part I).
| Symbol | Meaning |
|---|---|
| \(x,\ \alpha\) | a scalar (italic lower-case) |
| \(\mathbf{x},\ \mathbf{w}\) | a vector (bold lower-case; a Greek list such as all of a model’s weights is bold, \(\boldsymbol\theta\), where a chapter writes it out, though some formulas print the plain \(\theta\) for it, and the text says when \(\theta\) is the whole list) |
| \(M,\ A\) | a matrix (upper-case) |
| \(\mathbf{u}\cdot\mathbf{v}\) | dot product (the universal “score two things” operation) |
| \(\lVert\mathbf{x}\rVert\) | Euclidean (\(L_2\)) norm / length of a vector |
| \(\cos(\mathbf{u},\mathbf{v})\) | cosine similarity, \(\dfrac{\mathbf{u}\cdot\mathbf{v}}{\lVert\mathbf{u}\rVert\,\lVert\mathbf{v}\rVert}\) |
| \(M^{\top}\) | transpose; \(I\) — identity matrix |
| \(\lambda,\ \mathbf{v}\) | eigenvalue and its eigenvector (\(M\mathbf{v}=\lambda\mathbf{v}\)) |
Calculus and optimization (Part I).
| Symbol | Meaning |
|---|---|
| \(f'(x),\ \frac{df}{dx}\) | derivative (slope / rate of change) |
| \(\partial,\ \nabla f\) | partial derivative; gradient (the vector of partials) |
| \(H\) | Hessian (matrix of second derivatives) |
| \(\eta\) | learning rate (gradient-descent step size) |
| \(\mathcal{L}\) | the loss (objective being minimized) |
| \(z,\ \mathbf{z}\) | pre-activation: \(z=\mathbf{w}\cdot\mathbf{x}+b\) for one neuron, \(\mathbf{z}=W\mathbf{h}+\mathbf{b}\) for a layer (before the activation) |
| \(a,\ \mathbf{h}\) | post-activation: \(a=\phi(z)\) for one neuron; \(\mathbf{h}\) for a whole layer’s output |
| \(\mathbf{b}\) | a layer’s bias vector, added after the weight multiply |
| \(\hat{\mathbf{y}}\) | the network’s output / prediction — the last layer, \(\mathbf{h}^{(L)}=\hat{\mathbf{y}}\) |
Probability (Part I).
| Symbol | Meaning |
|---|---|
| \(\Pr(\cdot),\ \mathbb{E}[\cdot]\) | probability; expectation |
| \(X,\ x\) | a random variable (capital) and one value it can take (lower-case) |
| \(\mu,\ \sigma^2,\ \sigma\) | mean, variance (\(\operatorname{Var}[X]\)), standard deviation — this \(\sigma\) is not the sigmoid \(\sigma(z)\) below; context tells them apart |
| \(\theta\) | the parameters of a distribution or a model |
| \(\propto\) | “is proportional to”: equal up to a constant multiplier |
| \(x\sim p,\ \mathbb{E}_{x\sim p}[\cdot]\) | \(x\) drawn from \(p\); the average over such draws |
| \(\neg B\) | “not \(B\)”, the complement of an event |
| \(x\in S\) | “\(x\) is a member of the set \(S\)”: \(x\in\{0,1\}\) reads “\(x\) is \(0\) or \(1\)” |
| \(H(p),\ H(p,q)\) | entropy; cross-entropy — this \(H\) is not the Hessian above |
| \(D_{\mathrm{KL}}(p\Vert q),\ I(X;Y)\) | Kullback–Leibler (KL) divergence; mutual information |
| \(H_0,\ \alpha\) | the null hypothesis; the significance level of a test (not the Beta’s \(\alpha\); \(\alpha_{\text{post}},\beta_{\text{post}}\) are a posterior Beta’s) |
| \(t^{*}\), df, SE, CI | a critical value; degrees of freedom; standard error (SE); confidence interval (CI) |
| \(\sigma(z)\) | the sigmoid (logistic) function, \(1/(1+e^{-z})\) |
| \(\tau\) | temperature (softmax / contrastive sharpness) |
| \(\mathcal{N},\ \text{Beta}(\alpha,\beta)\) | Gaussian; Beta distribution |
Recommender systems (Parts III onward).
| Symbol | Meaning |
|---|---|
| \(u,\ i,\ j\) | a user; an item; a second (often negative) item |
| \(r_{ui},\ \hat r_{ui}\) | observed rating; model-predicted score for \((u,i)\) |
| \(R\) | the user–item interaction / rating matrix |
| \(\mathbf{p}_u,\ \mathbf{q}_i\) | matrix-factorization user/item latent vectors (Part III), \(\hat r_{ui}=\mathbf{p}_u\cdot\mathbf{q}_i\) |
| \(\mathcal{R}\) | the set of relevant items (evaluation) |
| @\(K\) | “evaluated at the top \(K\) positions” of the ranked list |
Sequence-based recommenders (Part IV).
| Symbol | Meaning |
|---|---|
| \((i_1,\dots,i_{t-1})\) | a user’s ordered history / session prefix; the task is to predict the next item \(i_t\) |
| \(P(\text{next}=i\mid\text{last}=l)\) | a first-order Markov transition probability (estimated by counting) |
| \(\mathbf{h}_t\) | the hidden state of a recurrent neural network (RNN) — the running summary of the session up to step \(t\) |
| \(\alpha_{ij}\) | a self-attention weight: how much position \(i\) attends to position \(j\) (softmax of scaled dot products); the sequence chapter writes it \(\alpha_{tj}\), position \(t\) attending to \(j\) |
Graphs and graph collaborative filtering (Part V).
| Symbol | Meaning |
|---|---|
| \(A,\ \tilde A\) | adjacency matrix; its symmetric-normalized form \(D^{-1/2}AD^{-1/2}\) |
| \(\deg(u)\) | degree (number of neighbors) of a node |
| \(\mathbf{e}_u,\ \mathbf{e}_i\) | the learned identifier (ID) embedding of a user/item (graph era) — the same kind of object as \(\mathbf{p}_u,\mathbf{q}_i\) above, now refined by graph propagation rather than plain factorization |
| \(\mathbf{e}_u^{(k)}\) | the embedding after \(k\) propagation layers; \(\alpha_k\) — layer-combination weight |
Contrastive learning and large language models (Parts V–VI).
| Symbol | Meaning |
|---|---|
| \(\mathbf{z},\ \mathbf{z}^{+},\ \mathbf{z}_k\) | an anchor; its positive (matching view); a negative — embeddings, so bold like every other vector |
| \(\operatorname{sim}(\cdot,\cdot)\) | a similarity (usually cosine) |
| \(\mathbf{s}_i\) | a semantic embedding (e.g. a frozen text-encoder applied to a profile that a large language model wrote), aligned with the collaborative \(\mathbf{e}_i\) |
Two letters that carry a different job in different parts. Both are flagged here rather than forced into one symbol, because in each case the two uses are genuinely different objects and each chapter says so where it uses them.
- \(d\) and \(\Delta\). In Loss Functions & Regularizers the pairwise score gap is \(d\) (as in \(-\ln\sigma(d)\)). Everywhere else \(d\) is the embedding dimension (it already is in the linear-algebra, Transformer and matrix-factorization chapters), and Training and Serving a Recommender writes the score gap \(\Delta\). Read \(d\) as “gap” only inside the losses chapter.
- \(K\) and \(L\). \(L\) counts the layers of a neural network (so \(\mathbf{h}^{(L)}=\hat{\mathbf{y}}\)); the neural graph collaborative filtering (NGCF) sketch in From Graphs to LightGCN also writes \(L\) for the number of graph layers (the same \(K\) hops as in the other graph chapters), and in The Spectral / Graph-Filter View \(L=I-\tilde A\) is the graph Laplacian, a matrix. \(K\) counts propagation steps on a graph, and is chosen because it equals the number of hops a signal travels — \(K\) layers means reach \(K\) hops. Note also \(@K\) above, which is the cut-off of a ranked list and unrelated to both.
Other letters that are reused, each named where it changes meaning.
- \(H\): entropy (Probability), the Hessian (Calculus), the number of attention heads (Representation Learning & the Transformer) and the graph filter (The Spectral / Graph-Filter View).
- \(\mathcal R\): the set of relevant items in Evaluation Metrics, and the regularizer \(\mathcal R(\theta)\) in Loss Functions & Regularizers.
- \(\sigma\) is the sigmoid, a standard deviation, a singular value (Linear Algebra, the spectral chapter), and, as \(\sigma_\downarrow,\sigma_\uparrow\), a projection in Self-Supervised & Contrastive Learning; the chapters on large language models write that projection \(g\).
- \(\gamma\): the discount factor (Reinforcement Learning), a focal exponent (Loss Functions & Regularizers), the federated server’s step, an angle. \(\epsilon\): an exploration rate (Bandits), a perturbation radius (SimGCL, in Self-Supervised & Contrastive Learning) and a privacy budget (Federated).
- \(z\): the pre-activation of a neuron or layer (Neural Networks & Back-propagation) and, in bold, the anchor embedding of contrastive learning (Self-Supervised & Contrastive Learning).
- \(I\): the identity matrix (Linear Algebra, and in the graph Laplacian \(L=I-\tilde A\)) and, in \(I(X;Y)\), mutual information.
- The model’s score for \((u,i)\) is \(\hat r_{ui}\) in the losses, matrix-factorization and serving chapters; the graph chapters write the same score as \(\hat y_{ui}\), the sequence chapters write \(\hat s\), and the click-prediction chapter writes \(\hat p\) (a probability).
- \(L\) also counts the propagation layers in Training and Serving (there \(K\) is the list length) and the cross layers in Click-Through Rate Prediction; the graph Laplacian is written \(L=I-\tilde A\) in plain type in the spectral chapter.
The cast. Ann, Bob and Cara are the three users of most chapters (the calculus and probability primers use none of them, and a few chapters use only one or two); Dave appears only in Appendix C. A few chapters add a one-off user for one worked example (Dan, Dana, Eve, Zed); each is a different person in each chapter and is introduced where used.
One notation change to keep in mind. Through the matrix-factorization chapter the user/item vectors are written \(\mathbf{p}_u,\mathbf{q}_i\); from From Graphs to LightGCN onward the learned user/item vector is written \(\mathbf{e}_u,\mathbf{e}_i\) (the ID embedding). They are the same kind of object — a vector whose location carries meaning — differing only in how it is learned.