pyproximal.Euclidean

class pyproximal.Euclidean(sigma: float = 1.0)

Euclidean norm proximal operator.

Proximal operator of the Euclidean norm: \(\sigma \|\mathbf{x}\|_2 = \sigma \sqrt{\sum x_i^2}\).

Parameters:

sigmaint, optional

Multiplicative coefficient of \(L_{2}\) norm

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)	Gradient of the Moreau envelope of the function.
postcomposition(sigma)	Postcomposition
precomposition(a, b)	Precomposition
prox(**kwargs)
proxdual(**kwargs)

Examples using pyproximal.Euclidean

Norms

Norms