The fastest ℓ_1,∞ prox in the west

by   Benjamín Béjar, et al.
University of Illinois at Urbana-Champaign

Proximal operators are of particular interest in optimization problems dealing with non-smooth objectives because in many practical cases they lead to optimization algorithms whose updates can be computed in closed form or very efficiently. A well-known example is the proximal operator of the vector ℓ_1 norm, which is given by the soft-thresholding operator. In this paper we study the proximal operator of the mixed ℓ_1,∞ matrix norm and show that it can be computed in closed form by applying the well-known soft-thresholding operator to each column of the matrix. However, unlike the vector ℓ_1 norm case where the threshold is constant, in the mixed ℓ_1,∞ norm case each column of the matrix might require a different threshold and all thresholds depend on the given matrix. We propose a general iterative algorithm for computing these thresholds, as well as two efficient implementations that further exploit easy to compute lower bounds for the mixed norm of the optimal solution. Experiments on large-scale synthetic and real data indicate that the proposed methods can be orders of magnitude faster than state-of-the-art methods.


page 1

page 2

page 3

page 4

page 5

page 6

page 7

page 8


Efficient Proximal Mapping of the 1-path-norm of Shallow Networks

We demonstrate two new important properties of the 1-path-norm of shallo...

Proximal and Federated Random Reshuffling

Random Reshuffling (RR), also known as Stochastic Gradient Descent (SGD)...

Sparse Inverse Covariance Estimation for Chordal Structures

In this paper, we consider the Graphical Lasso (GL), a popular optimizat...

l_2,p Matrix Norm and Its Application in Feature Selection

Recently, l_2,1 matrix norm has been widely applied to many areas such a...

Parseval Proximal Neural Networks

The aim of this paper is twofold. First, we show that a certain concaten...

From perspective maps to epigraphical projections

The projection onto the epigraph or a level set of a closed proper conve...

Local Kernels that Approximate Bayesian Regularization and Proximal Operators

In this work, we broadly connect kernel-based filtering (e.g. approaches...

Please sign up or login with your details

Forgot password? Click here to reset