Unidimensional Total variation (TV) Benchmark

This benchmark is dedicated to solver of TV-1D regularised regression problem:

$$\mathit{u}\in \underset{\mathit{u}\in {\mathbb{R}}^p}{\mathrm{argmin}}f(\mathit{y},A\mathit{u})+g(D\mathit{u})$$

• $\mathit{y}\in {\mathbb{R}}^n$ is a vector of observations or targets.
• $A\in {\mathbb{R}}^{n\times p}$ is a design matrix or forward operator.
• $\lambda >0$ is a regularization hyperparameter.
• $f(\mathit{y},A\mathit{u})=\sum _{k}l(y_k,(A\mathit{u}{)}_k)$ is a loss function, where $l$ can be quadratic loss as $l(y,x)=\frac{1}{2}|y-x{|}_2^2$, or Huber loss as $l(y,x)=h_{\delta }(y-x)$ defined by

$$h_{\delta }(t)=\{\begin{array}{ll}\frac{1}{2}{t}^2& \mathrm{if}|t|\le \delta \\ \delta |t|-\frac{1}{2}{\delta }^2& \mathrm{otherwise}\end{array}$$

• $D\in {\mathbb{R}}^{(p-1)\times p}$ is a finite difference operator, such that the regularised TV-1D term $g(D\mathit{u})=\lambda \parallel \mathit{u}{\parallel }_{TV}$ expressed as follows.

$$g(D\mathit{u})=\lambda \parallel D\mathit{u}{\parallel }_1=\lambda \sum _{k=1}^{p-1}|u_{k+1}-u_k|$$

where n (or n_samples) stands for the number of samples, p (or n_features) stands for the number of features.

Install

This benchmark can be run using the following commands:

$ pip install -U benchopt
$ git clone https://github.com/benchopt/benchmark_tv_1d
$ benchopt run benchmark_tv_1d

Apart from the problem, options can be passed to benchopt run, to restrict the benchmarks to some solvers or datasets, e.g.:

$ benchopt run benchmark_tv_1d --config benchmark_tv_1d/example_config.yml

Use benchopt run -h for more details about these options, or visit https://benchopt.github.io/api.html.