pyproximal.L1#
- class pyproximal.L1(sigma=1.0, g=None)[source]#
L1 norm proximal operator.
Proximal operator of the \(\ell_1\) norm: \(\sigma\|\mathbf{x} - \mathbf{g}\|_1 = \sigma \sum |x_i - g_i|\).
- Parameters
- sigma
floatorlistornp.ndarrayorfunc, optional Multiplicative coefficient of L1 norm. This can be a constant number, a list of values (for multidimensional inputs, acting on the second dimension) or a function that is called passing a counter which keeps track of how many times the
proxmethod has been invoked before and returns a scalar (or a list of)sigmato be used.- g
np.ndarray, optional Vector to be subtracted
- sigma
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)Compute gradient
postcomposition(sigma)Postcomposition
precomposition(a, b)Precomposition
prox(*args, **kwargs)proxdual(**kwargs)
