Table of Notation
The table below collects the most frequently used notation in this book for quick reference. Some symbols are reused across chapters for closely related purposes (e.g. \({\bf W}\) denotes a graph adjacency matrix in the manifold learning and spectral clustering sections, but an NMF feature matrix in Chapter 4 and a neural network weight matrix in the autoencoder section); in these cases, the “Context” column indicates where each usage applies. A more detailed discussion of notation and conventions is given in Chapter 2.
| Symbol | Meaning | Context |
|---|---|---|
| \(N\) | Number of samples/observations | General |
| \(d\) | Dimension of the original data | General |
| \(t\) | Reduced or intrinsic dimension | Manifold learning |
| \(k\) | Number of clusters | Clustering |
| \(\vec{x},\vec{y},\vec{z}\) | Vectors (lowercase, arrow) | General |
| \(\vec{0},\, \vec{1}\) | Vector of all zeros / all ones | General |
| \(\vec{x}^T\) | Transpose of \(\vec{x}\) (a row vector) | General |
| \(\vec{x}^T\vec{y}\) | Inner (dot) product | General |
| \(\vec{x}\vec{y}^T\) | Outer product (a matrix) | General |
| \(\|\vec{x}\|\) | Euclidean norm of \(\vec{x}\) | General |
| \({\bf A},{\bf X},{\bf \Sigma}\) | Matrices (bold) | General |
| \({\bf A}_{ij}\) | Entry in row \(i\), column \(j\) | General |
| \({\bf A}^T\) | Transpose of \({\bf A}\) | General |
| \({\bf I}\) | Identity matrix | General |
| \(Tr({\bf B})\) | Trace of \({\bf B}\) | General |
| \(\det({\bf B})\) | Determinant, also written \(|{\bf B}|\) | General |
| \({\bf H}\) | Centering matrix \({\bf I}-\frac{1}{N}\vec{1}\vec{1}^T\) | Ch. 2, 4 |
| \(\nabla f\) | Gradient of \(f\) | Ch. 2 |
| \(\mathcal{H}f\) | Hessian matrix of \(f\) | Ch. 2 |
| \(\hat{\mu},\, \hat{\bf \Sigma}\) | Sample mean / sample covariance | Ch. 2, 4 |
| \(\lambda_1,\dots,\lambda_d\) | Eigenvalues, decreasing order | Ch. 4 |
| \(\vec{w}_1,\dots,\vec{w}_d\) | Eigenvectors, e.g. PCA loadings | Ch. 4 |
| \(k(\vec{x},\vec{y})\) | Kernel function | Ch. 5-6 |
| \({\bf K}\) | Kernel matrix, \({\bf K}_{ij}=k(\vec{x}_i,\vec{x}_j)\) | Ch. 5-6 |
| \(\varphi\) | Feature map for a kernel | Ch. 5 |
| \(\mathcal{H}\) | Feature space for a kernel | Ch. 5 |
| \(C_1,\dots,C_k\) | A partition of the data into clusters | Ch. 6 |
| \(\vec{\mu}_\ell,\, \vec{c}_\ell\) | Center of cluster \(\ell\) | Ch. 6 |
| \(\Psi\) | A manifold map, \(\vec{x}_i = \Psi(\vec{z}_i)\) | Ch. 5 |
| \(\mathcal{M}\) | A manifold | Ch. 5 |
| \(\phi\) | A chart on a manifold | Ch. 5 |
| \({\bf W}\) | Graph adjacency/weight matrix; also NMF and neural net weights | Ch. 5-6 |
| \({\bf D}\) | Diagonal degree matrix of a graph | Ch. 5-6 |
| \({\bf L}\) | Graph Laplacian, \({\bf L}={\bf D}-{\bf W}\) | Ch. 5-6 |
| \(\mathbb{1}(\cdot)\) | Indicator function (1 if true, else 0) | General |
| \(\mathcal{N}(\vec{\mu},{\bf \Sigma})\) | Multivariate normal, mean \(\vec{\mu}\), cov. \({\bf \Sigma}\) | General |
| \(Z_i\) | Latent cluster label for sample \(i\) | Ch. 6 |