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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.

We introduce a variational framework to learn the activation functions of deep neural networks. Our aim is to increase the capacity of the network while controlling an upper-bound of the actual Lipschitz constant of the input-output relation. To that end, we first establish a global bound for the Lipschitz constant of neural networks. Based on the obtained bound, we then formulate a variational problem for learning activation functions. Our variational problem is infinite-dimensional and is not computationally tractable. However, we prove that there always exists a solution that has continuous and piecewise-linear (linear-spline) activations. This reduces the original problem to a finite-dimensional minimization where an l1 penalty on the parameters of the activations favors the learning of sparse nonlinearities. We numerically compare our scheme with standard ReLU network and its variations, PReLU and LeakyReLU and we empirically demonstrate the practical aspects of our framework.