Unidimensional Total variation (TV) Benchmark

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

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

• $\boldsymbol{y} \in \mathbb{R}^{n}$ 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.
• $f(\boldsymbol{y}, A\boldsymbol{u}) = \sum\limits_{k} l(y_{k}, (A\boldsymbol{u})_{k})$ is a loss function, where $l$ can be quadratic loss as $l(y, x) = \frac{1}{2} \vert y - x \vert_2^2$, or Huber loss as $l(y, x) = h_{\delta} (y - x)$ defined by

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

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

$$g(D\boldsymbol{u}) = \lambda \| D \boldsymbol{u} \|_{1} = \lambda \sum\limits_{k = 1}^{p-1} \vert u_{k+1} - u_{k} \vert $$

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.