pyproximal.Euclidean#
- class pyproximal.Euclidean(sigma=1.0)[source]#
Euclidean norm proximal operator.
Proximal operator of the Euclidean norm: \(\sigma \|\mathbf{x}\|_2 = \sigma \sqrt{\sum x_i^2}\).
- Parameters
- sigma
int, optional Multiplicative coefficient of \(L_{2}\) norm
- sigma
Notes
The Euclidean proximal operator is defined as:
\[\prox_{\tau \sigma \|\cdot\|_2}(\mathbf{x}) = \left(1 - \frac{\tau \sigma }{\max\{\|\mathbf{x}\|_2, \tau \sigma \}}\right) \mathbf{x}\]This operator is sometimes called block soft thresholding.
Moreover, as the conjugate of the Euclidean norm is the orthogonal projection of its dual norm (i.e., Euclidean norm) onto a unit ball, its dual operator is defined as:
\[\prox^*_{\tau \sigma \|\cdot\|_2}(\mathbf{x}) = \frac{\sigma \mathbf{x}}{\max\{\|\mathbf{x}\|_2, \sigma\}}\]Methods
__init__([sigma])affine_addition(v)Affine addition
chain(g)Chain
grad(x)Compute gradient
postcomposition(sigma)Postcomposition
precomposition(a, b)Precomposition
prox(**kwargs)proxdual(**kwargs)
