# pyproximal.ETP#

class pyproximal.ETP(sigma, gamma=1.0)[source]#

Exponential-type penalty (ETP).

The exponential-type penalty is defined as

$\mathrm{ETP}_{\sigma,\gamma}(\mathbf{x}) = \sum_i \frac{\sigma}{1-e^{-\gamma}}(1-e^{-\gamma|x_i|})$

for $${\sigma>0}$$, and $${\gamma>0}$$.

Parameters
sigmafloat

Regularization parameter.

gammafloat, optional

Regularization parameter. Default is 1.0.

Notes

As $${\gamma\rightarrow 0}$$ the exponential-type penalty approaches the $$\ell_1$$-penalty and when $${\gamma\rightarrow\infty}$$ tends to the $$\ell_0$$-penalty .

As for the proximal operator, consider the one-dimensional case

$\prox_{\tau \mathrm{ETP}(\cdot)}(x) = \argmin_{z} \mathrm{ETP}(z) + \frac{1}{2\tau}(x - z)^2$

and assume that $$x\geq 0$$. The minima can be obtained when $$z=0$$ or at a stationary point, where the latter must satisfy

$x = z + \frac{\gamma \sigma \tau}{1-e^{-\gamma}} e^{-\gamma z} .$

The solution to the above equation can be expressed using the Lambert W function.

1

Gao, C. et al. “A Feasible Nonconvex Relaxation Approach to Feature Selection”, In the Proceedings of the Conference on Artificial Intelligence (AAAI), 2011.

Methods

 __init__(sigma[, gamma]) affine_addition(v) Affine addition chain(g) Chain elementwise(x) grad(x) Compute gradient postcomposition(sigma) Postcomposition precomposition(a, b) Precomposition prox(**kwargs) proxdual(**kwargs)

## Examples using pyproximal.ETP# Concave penalties

Concave penalties