  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.
