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<h1>Available benchmarks</h1>
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<section id="available-benchmarks">
<span id="id1"></span><h1>Available benchmarks<a class="headerlink" href="#available-benchmarks" title="Link to this heading">#</a></h1>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Some benchmarks are briefly described in the list below. For a complete
list of benchmarks, see GitHub repositories of the form <a class="reference external" href="https://github.com/orgs/benchopt/repositories?q=benchmark_&type=all&language=&sort=stargazers/">benchopt/benchmark_*</a>.</p>
</div>
<p><strong>Notation:</strong> In what follows, <span class="math notranslate nohighlight">\(n\)</span> (or <code class="docutils literal notranslate"><span class="pre">n_samples</span></code>) stands for the number of samples and <span class="math notranslate nohighlight">\(p`\)</span> (or <code class="docutils literal notranslate"><span class="pre">n_features</span></code>) stands for the number of features.</p>
<div class="math notranslate nohighlight">
\[y \in \mathbb{R}^n, X = [x_1^\top, \dots, x_n^\top]^\top \in \mathbb{R}^{n \times p}\]</div>
<ul class="simple">
<li><p><a class="reference external" href="https://github.com/benchopt/benchmark_ols">Ordinary Least Squares (OLS)</a>: <a class="reference external" href="https://github.com/benchopt/benchmark_ols/actions"><img alt="Build Status OLS" src="https://github.com/benchopt/benchmark_ols/actions/workflows/main.yml/badge.svg" /></a></p></li>
</ul>
<div class="math notranslate nohighlight">
\[\min_w \frac{1}{2} \|y - Xw\|^2_2\]</div>
<ul class="simple">
<li><p><a class="reference external" href="https://github.com/benchopt/benchmark_nnls">Non-Negative Least Squares (NNLS)</a>: <a class="reference external" href="https://github.com/benchopt/benchmark_nnls/actions"><img alt="Build Status NNLS" src="https://github.com/benchopt/benchmark_nnls/actions/workflows/main.yml/badge.svg" /></a></p></li>
</ul>
<div class="math notranslate nohighlight">
\[\min_{w \geq 0} \frac{1}{2} \|y - Xw\|^2_2\]</div>
<ul class="simple">
<li><p><a class="reference external" href="https://github.com/benchopt/benchmark_lasso">LASSO: L1-regularized least squares</a>: <a class="reference external" href="https://github.com/benchopt/benchmark_lasso/actions"><img alt="Build Status Lasso" src="https://github.com/benchopt/benchmark_lasso/actions/workflows/main.yml/badge.svg" /></a></p></li>
</ul>
<div class="math notranslate nohighlight">
\[\min_w \frac{1}{2} \|y - Xw\|^2_2 + \lambda \|w\|_1\]</div>
<ul class="simple">
<li><p><a class="reference external" href="https://github.com/benchopt/benchmark_logreg_l2">L2-regularized logistic regression</a>: <a class="reference external" href="https://github.com/benchopt/benchmark_logreg_l2/actions"><img alt="Build Status LogRegL2" src="https://github.com/benchopt/benchmark_logreg_l2/actions/workflows/main.yml/badge.svg" /></a></p></li>
</ul>
<div class="math notranslate nohighlight">
\[\min_w \sum_{i=1}^{n} \log(1 + \exp(-y_i x_i^\top w)) + \frac{\lambda}{2} \|w\|_2^2\]</div>
<ul class="simple">
<li><p><a class="reference external" href="https://github.com/benchopt/benchmark_logreg_l1">L1-regularized logistic regression</a>: <a class="reference external" href="https://github.com/benchopt/benchmark_logreg_l1/actions"><img alt="Build Status LogRegL1" src="https://github.com/benchopt/benchmark_logreg_l1/actions/workflows/main.yml/badge.svg" /></a></p></li>
</ul>
<div class="math notranslate nohighlight">
\[\min_w \sum_{i=1}^{n} \log(1 + \exp(-y_i x_i^\top w)) + \lambda \|w\|_1\]</div>
<ul class="simple">
<li><p><a class="reference external" href="https://github.com/benchopt/benchmark_huber_l2">L2-regularized Huber regression</a>: <a class="reference external" href="https://github.com/benchopt/benchmark_huber_l2/actions"><img alt="Build Status HuberL2" src="https://github.com/benchopt/benchmark_huber_l2/actions/workflows/main.yml/badge.svg" /></a></p></li>
</ul>
<div class="math notranslate nohighlight">
\[\min_{w, \sigma} {\sum_{i=1}^n \left(\sigma + H_{\epsilon}\left(\frac{X_{i}w - y_{i}}{\sigma}\right)\sigma\right) + \lambda {\|w\|_2}^2}\]</div>
<p>where</p>
<div class="math notranslate nohighlight">
\[\begin{split}H_{\epsilon}(z) = \begin{cases}
z^2, & \text {if } |z| < \epsilon, \\
2\epsilon|z| - \epsilon^2, & \text{otherwise}
\end{cases}\end{split}\]</div>
<ul class="simple">
<li><p><a class="reference external" href="https://github.com/benchopt/benchmark_quantile_regression">L1-regularized quantile regression</a>: <a class="reference external" href="https://github.com/benchopt/benchmark_quantile_regression/actions"><img alt="Build Status QuantileRegL1" src="https://github.com/benchopt/benchmark_quantile_regression/actions/workflows/main.yml/badge.svg" /></a></p></li>
</ul>
<div class="math notranslate nohighlight">
\[\min_{w} \frac{1}{n} \sum_{i=1}^{n} PB_q(y_i - X_i w) + \lambda ||w||_1.\]</div>
<p>where <span class="math notranslate nohighlight">\(PB_q\)</span> is the pinball loss:</p>
<div class="math notranslate nohighlight">
\[\begin{split}PB_q(t) = q \max(t, 0) + (1 - q) \max(-t, 0) =
\begin{cases}
q t, & t > 0, \\
0, & t = 0, \\
(q - 1) t, & t < 0
\end{cases}\end{split}\]</div>
<ul class="simple">
<li><p><a class="reference external" href="https://github.com/benchopt/benchmark_linear_ica">Linear ICA</a>: <a class="reference external" href="https://github.com/benchopt/benchmark_linear_ica/actions"><img alt="Build Status LinearICA" src="https://github.com/benchopt/benchmark_linear_ica/actions/workflows/main.yml/badge.svg" /></a></p></li>
</ul>
<p>Given some data <span class="math notranslate nohighlight">\(X \in \mathbb{R}^{d \times n}\)</span> assumed to be linearly
related to unknown independent sources <span class="math notranslate nohighlight">\(S \in \mathbb{R}^{d \times n}\)</span> with</p>
<div class="math notranslate nohighlight">
\[X = A S\]</div>
<p>where <span class="math notranslate nohighlight">\(A \in \mathbb{R}^{d \times d}\)</span> is also unknown, the objective of
linear ICA is to recover <span class="math notranslate nohighlight">\(A\)</span> up to permutation and scaling of its columns.
The objective in this benchmark is related to some estimation on <span class="math notranslate nohighlight">\(A\)</span>
quantified with the so-called AMARI distance.</p>
<ul class="simple">
<li><p><a class="reference external" href="https://github.com/benchopt/benchmark_jointdiag">Approximate Joint Diagonalization (AJD)</a>: <a class="reference external" href="https://github.com/benchopt/benchmark_jointdiag/actions"><img alt="Build Status JointDiag" src="https://github.com/benchopt/benchmark_jointdiag/actions/workflows/main.yml/badge.svg" /></a></p></li>
</ul>
<p>Given n square symmetric positive matrices <span class="math notranslate nohighlight">\(C^i\)</span>, it consists of solving
the following problem:</p>
<div class="math notranslate nohighlight">
\[\min_B \frac{1}{2n} \sum_{i=1}^n \log |\textrm{diag} (B C^i B^{\top}) | - \log | B C^i B^{\top} |\]</div>
<p>where <span class="math notranslate nohighlight">\(|\cdot|\)</span> stands for the matrix determinant and <span class="math notranslate nohighlight">\(\textrm{diag}\)</span> stands
for the operator that keeps only the diagonal elements of a matrix. Optionally, the
matrix <span class="math notranslate nohighlight">\(B\)</span> can be enforced to be orthogonal.</p>
<p>See <a class="reference external" href="https://github.com/benchopt/">benchmark_* repositories on GitHub</a> for more.</p>
</section>
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