Measuring Complexity of Learning Schemes Using Hessian-Schatten Total Variation

TL;DR

This paper introduces the Hessian-Schatten total variation (HTV), a new seminorm to measure the complexity of multivariate functions, with applications in assessing supervised learning schemes and analyzing function regularity.

Contribution

The paper defines HTV, proves its invariance properties, derives closed-form expressions for Sobolev and CPWL functions, and demonstrates its utility in evaluating learning model complexity.

Findings

HTV is invariant to rotations, scalings, and translations.

HTV coincides with the Hessian-Schatten seminorm for Sobolev functions.

HTV measures total slope change in CPWL functions.

Abstract

In this paper, we introduce the Hessian-Schatten total variation (HTV) -- a novel seminorm that quantifies the total "rugosity" of multivariate functions. Our motivation for defining HTV is to assess the complexity of supervised-learning schemes. We start by specifying the adequate matrix-valued Banach spaces that are equipped with suitable classes of mixed norms. We then show that the HTV is invariant to rotations, scalings, and translations. Additionally, its minimum value is achieved for linear mappings, which supports the common intuition that linear regression is the least complex learning model. Next, we present closed-form expressions of the HTV for two general classes of functions. The first one is the class of Sobolev functions with a certain degree of regularity, for which we show that the HTV coincides with the Hessian-Schatten seminorm that is sometimes used as a regularizer for image reconstruction. The second one is the class of continuous and piecewise-linear (CPWL) functions. In this case, we show that the HTV reflects the total change in slopes between linear regions that have a common facet. Hence, it can be viewed as a convex relaxation (l1-type) of the number of linear regions (l0-type) of CPWL mappings. Finally, we illustrate the use of our proposed seminorm.