Best Convex Lower Approximations of the l 0 Pseudonorm on Unit Balls

TL;DR

This paper develops a framework for finding the best convex lower bounds of the l0 pseudonorm on various unit balls, revealing that the l1-norm is the tightest minorant across all lp-norms.

Contribution

It introduces conjugacies for analyzing 0-homogeneous functions and characterizes the optimal convex lower approximations of the l0 pseudonorm on different unit balls.

Findings

The tightest convex lower approximation of l0 is derived for any unit ball.

The l1-norm is the tightest norm that minorizes l0 on lp-norm unit balls.

Explicit expressions for convex approximations of l0 are provided.

Abstract

Whereas the norm of a vector measures amplitude (and is a 1-homogeneous function), sparsity is measured by the 0-homogeneous l0 pseudonorm, which counts the number of nonzero components. We propose a family of conjugacies suitable for the analysis of 0-homogeneous functions. These conjugacies are derived from couplings between vectors, given by their scalar product divided by a 1-homogeneous normalizing factor. With this, we characterize the best convex lower approximation of a 0-homogeneous function on the unit ''ball'' of a normalization function (i.e. a norm without the requirement of subadditivity). We do the same with the best convex and 1-homogeneous lower approximation. In particular, we provide expressions for the tightest convex lower approximation of the l0 pseudonorm on any unit ball, and we show that the tightest norm which minorizes the l0 pseudonorm on the unit ball of any lp-norm is the l1-norm. We also provide the tightest convex lower convex approximation of the l0 pseudonorm on the unit ball of any norm.