benchopt / benchmark_tv_2d

TV-regularized problem in 2D

2-dimensional Total Variation (TV) Benchmark

This benchmark is dedicated to solver of TV-2D regularised regression problem:

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

• $\boldsymbol{y} \in \mathbb{R}^{n \times m}$ is a vector of observations or targets.
• $A \in \mathbb{R}^{n \times n}$ is a design matrix or forward operator.
• $\lambda > 0$ is a regularization hyperparameter.
• $f(\boldsymbol{y}, A\boldsymbol{u}) = \sum\limits_{k=1}^{n} \sum\limits_{l=1}^{m} l(y_{k,l}, (A\boldsymbol{u})_{k,l})$ 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 $l(y, x) = h_{\delta} (y - x)$ defined by

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

• $D_1 \in \mathbb{R}^{(n-1) \times n}$ and $D_2 \in \mathbb{R}^{(m-1) \times m}$ are finite difference operators, such that the regularised TV-2D term $g(\boldsymbol{u}) = \lambda \| \boldsymbol{u} \|_{TV}$ expressed as follows.

In isotropic cases:

$$ g(\boldsymbol{u}) = \lambda \| \sqrt{ (D_1 \boldsymbol{u})^2 + (\boldsymbol{u} D_2^{\top})^2 } \|_{1} = \lambda \sum\limits_{k = 1}^{n-1} \sum\limits_{l = 1}^{m-1} \sqrt{\vert u_{k+1,l} - u_{k,l} \vert^2 + \vert u_{k,l+1} - u_{k,l} \vert^2} $$

In anisotropic cases:

$$ g(\boldsymbol{u}) = \lambda \| D_1 \boldsymbol{u} \|_{1} + \| \boldsymbol{u} D_2^{\top} \|_{1} = \lambda \sum\limits_{k = 1}^{n-1} \sum\limits_{l = 1}^{m-1} (\vert u_{k+1,l} - u_{k,l} \vert + \vert u_{k,l+1} - u_{k,l} \vert) $$

where n (or height) and m (or width) stand for the dimension of targeted vector.

Install

This benchmark can be run using the following commands:

$ pip install -U benchopt
$ git clone https://github.com/benchopt/benchmark_tv_2d
$ benchopt run benchmark_tv_2d

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_2d --config benchmark_tv_2d/example_config.yml

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