# Available benchmarks#

Note

Some benchmarks are briefly described in the list below. For a complete list of benchmarks, see GitHub repositories of the form benchopt/benchmark_*.

Notation: In what follows, $$n$$ (or n_samples) stands for the number of samples and $$p$$ (or n_features) stands for the number of features.

$y \in \mathbb{R}^n, X = [x_1^\top, \dots, x_n^\top]^\top \in \mathbb{R}^{n \times p}$
$\min_w \frac{1}{2} \|y - Xw\|^2_2$
$\min_{w \geq 0} \frac{1}{2} \|y - Xw\|^2_2$
$\min_w \frac{1}{2} \|y - Xw\|^2_2 + \lambda \|w\|_1$
$\min_w \sum_{i=1}^{n} \log(1 + \exp(-y_i x_i^\top w)) + \frac{\lambda}{2} \|w\|_2^2$
$\min_w \sum_{i=1}^{n} \log(1 + \exp(-y_i x_i^\top w)) + \lambda \|w\|_1$
$\min_{w, \sigma} {\sum_{i=1}^n \left(\sigma + H_{\epsilon}\left(\frac{X_{i}w - y_{i}}{\sigma}\right)\sigma\right) + \lambda {\|w\|_2}^2}$

where

$\begin{split}H_{\epsilon}(z) = \begin{cases} z^2, & \text {if } |z| < \epsilon, \\ 2\epsilon|z| - \epsilon^2, & \text{otherwise} \end{cases}\end{split}$
$\min_{w} \frac{1}{n} \sum_{i=1}^{n} PB_q(y_i - X_i w) + \lambda ||w||_1.$

where $$PB_q$$ is the pinball loss:

$\begin{split}PB_q(t) = q \max(t, 0) + (1 - q) \max(-t, 0) = \begin{cases} q t, & t > 0, \\ 0, & t = 0, \\ (q - 1) t, & t < 0 \end{cases}\end{split}$

Given some data $$X \in \mathbb{R}^{d \times n}$$ assumed to be linearly related to unknown independent sources $$S \in \mathbb{R}^{d \times n}$$ with

$X = A S$

where $$A \in \mathbb{R}^{d \times d}$$ is also unknown, the objective of linear ICA is to recover $$A$$ up to permutation and scaling of its columns. The objective in this benchmark is related to some estimation on $$A$$ quantified with the so-called AMARI distance.

Given n square symmetric positive matrices $$C^i$$, it consists of solving the following problem:

$\min_B \frac{1}{2n} \sum_{i=1}^n \log |\textrm{diag} (B C^i B^{\top}) | - \log | B C^i B^{\top} |$

where $$|\cdot|$$ stands for the matrix determinant and $$\textrm{diag}$$ stands for the operator that keeps only the diagonal elements of a matrix. Optionally, the matrix $$B$$ can be enforced to be orthogonal.

See benchmark_* repositories on GitHub for more.