pyproximal.L1¶

class pyproximal.L1(sigma: float | ndarray[tuple[Any, ...], dtype[_ScalarT]] | Callable[[int], float | ndarray[tuple[Any, ...], dtype[_ScalarT]]] = 1.0, g: ndarray[tuple[Any, ...], dtype[_ScalarT]] | None = None)[source]¶

L1 norm proximal operator.

Proximal operator of the \(\ell_1\) norm:

\(\sigma\|\mathbf{x} - \mathbf{g}\|_1 = \sigma \sum_i |x_i - g_i|\) for 1-dimensional arrays;
\(\sum_j \sigma_j \|\mathbf{X}_j - \mathbf{g}\|_1 = \sum_j \sigma_j \sum_i |x_{i,j} - g_i|\) for 2-dimensional arrays.

Parameters:

sigmafloat or numpy.ndarray or func, optional: Multiplicative coefficient of L1 norm. This can be a constant number, a list of values (for 2-dimensional inputs, acting on the second dimension) or a function that is called passing a counter which keeps track of how many times the prox method has been invoked before and returns a scalar (or a list of) sigma to be used.
gnumpy.ndarray, optional: Vector to be subtracted

Notes

The \(\ell_1\) proximal operator is defined as [1]:

\[\begin{split}\prox_{\tau \sigma \|\cdot\|_1}(\mathbf{x}) = \operatorname{soft}(\mathbf{x}, \tau \sigma) = \begin{cases} x_i + \tau \sigma, & x_i - g_i < -\tau \sigma \\ g_i, & -\tau\sigma \leq x_i - g_i \leq \tau\sigma \\ x_i - \tau\sigma, & x_i - g_i > \tau\sigma\\ \end{cases}\end{split}\]

where \(\operatorname{soft}\) is the so-called called soft thresholding.

Moreover, as the conjugate of the \(\ell_1\) norm is the orthogonal projection of its dual norm (i.e., \(\ell_\inf\) norm) onto a unit ball, its dual operator (when \(\mathbf{g}=\mathbf{0}\)) is defined as:

\[\begin{split}\prox^*_{\tau \sigma \|\cdot\|_1}(\mathbf{x}) = P_{\|\cdot\|_{\infty} <=\sigma}(\mathbf{x}) = \begin{cases} -\sigma, & x_i < -\sigma \\ x_i,& -\sigma \leq x_i \leq \sigma \\ \sigma, & x_i > \sigma\\ \end{cases}\end{split}\]

[1]

Chambolle, and A., Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging”, Journal of Mathematical Imaging and Vision, 40, 8pp. 120–145. 2011.

Methods

`__init__`([sigma, g])
`affine_addition`(v)	Affine addition
`chain`(g)	Chain
`grad`(x)	Gradient of the Moreau envelope of the function.
`postcomposition`(sigma)	Postcomposition
`precomposition`(a, b)	Precomposition
`prox`(args, *kwargs)
`proxdual`(**kwargs)