Machine Learning & Signals Learning
Part IV Appendix
A Notation
Numbers and indexing
| \(a\) | Scalar | |
| \(\ba \) | Vector | |
| \(a_i\) | Element \(i\) of a vector \(a\), indexing starting at 1 | |
| \(\mathbf {A}\) | Matrix | |
| \(a_{ij}\) | Element \(i,j\) of a matrix \(\mathbf {A}\), indexing starting at 1 | |
| \(\real \) | Real numbers domain | |
| \(\real ^D\) | \(D\)-dimensional vector | |
| \(\real ^{D_1\times D_2}\) | matrix of a dimension \(D_1\times D_2\) | |
| \(\bI \) | Identity matrix | |
| \(\bOne \) | Vector/matrix of ones | |
| \(\bZero \) | Vector/matrix of zeros | |
| \(\indFunc \) | Indicator function (Sec. B.2) |
Datasets
| \(L\) | Model complexity | |
| \(N\) | Number of features | |
| \(M\) | Number of entries in the dataset | |
| \(K\) | Number of classes | |
| \(\Delta ^{K-1}\) | Probability simplex: \(\{\bp \in \real ^K:\, p_i\ge 0,\; \sum _i p_i = 1\}\) | |
| \(\bw \) or \(w_i\) | Model parameters (vector form) | |
| \(f(\cdot ;\bw )\) | Model | |
| \(h(\bx )\) or \(h(x)\) | True unknown function | |
| \(x_{ij}\) | Single data value | |
| \(\bx _i\) | Single data vector (sample \(i\)); \(\bx _i^T\) is the \(i\)-th row of \(\bX \) | |
| \(\btx _j\) | \(j\)-th column (feature) of \(\bX \) | |
| \(\bX \) | Data matrix | |
| \(\by \) | Target vector for the data in \(\bX \) | |
| \(\hat {\by }\) | Prediction vector of \(\by \) | |
| \(y_i\) | Target value | |
| \(\hat {y}_i\) | Predicted target value | |
| \(\loss (\by ,\hat {\by })\) or \(\mathcal {L}(y_i,\hat {y}_i)\) | Loss function | |
| \(\lambda \) | Regularization parameter | |
| \(\ba ^{[k]}\) | Activation of layer \(k\) | |
| \(\bz ^{[k]}\) | Output of layer \(k\) | |
| \(g_k(\cdot )\) | Activation function of layer \(k\) | |
| \(\bth \) or \(\theta _i\) | Model parameters (general form) | |
| \(\balpha \) | Kernel/dual coefficients vector | |
| \(\be \) | Error/residual vector | |
| \(\bepsilon \) or \(\epsilon _i\) | Noise vector/term | |
| \(\bn \) | Noise vector (signal processing) | |
| \(\bh \) | Impulse response / filter coefficients | |
| \(\bP \) | Projection matrix | |
| \(\bK \) | Kernel matrix | |
| \(\bR \) | Autocorrelation matrix | |
| \(\phi (\cdot )\) | Feature mapping / basis function | |
| \(\alpha \) | Learning rate (gradient descent step size) |
Statistics
| \(x\) | Sample set | |
| \(\bar x\) | Sample mean | |
| \(s_x^2\) | Sample variance (biased or unbiased) | |
| \(s_x\) | Sample std (biased or unbiased) | |
| \(s_{xy}\) | Sample covariance (biased or unbiased) | |
| \(r_{xy}\) | Sample correlation coefficient | |
| \(\mu \) | Population mean | |
| \(\sigma ^2\) | Population variance | |
| \(\sigma \) | Population standard deviation | |
| \(\E [\cdot ]\) | Expectation operator | |
| \(\Var [\cdot ]\) | Variance operator | |
| \(\Cov [\cdot ]\) | Covariance operator |
Signals
| \(\omega \) | Angular frequency (discrete) | |
| \(\theta \) | Phase angle | |
| \(A\) | Amplitude | |
| \(F\) | Frequency [Hz] | |
| \(F_s\) | Sampling frequency | |
| \(T\) | Period [sec] |