defined over Sobolev spaces is connected to the convexity of the potential w. In the scalar case, that is for functions u with domain or range in R, the functional I is weakly W 1,p lower semi-continuous (weakly * W ) if and only if w is convex, provided it is continuous and satisfies some growth conditions. The notion which replaces convexity in the vector case is quasi-convexity (introduced b...

Variational autoencoders (VAE) often use Gaussian or category distribution to model the inference process. This puts a limit on variational learning because this simplified assumption does not match the true posterior distribution, which is usually much more sophisticated. To break this limitation and apply arbitrary parametric distribution during inference, this paper derives a semi-continuous...

We discuss the dependence of set-valued dynamical systems on parameters. Under mild assumptions which are often satisfied for random dynamical systems with bounded noise and control systems, we establish the fact that topological bifurcations of minimal invariant sets are discontinuous with respect to the Hausdorff metric, taking the form of lower semi-continuous explosions and instantaneous ap...

Linear discriminant or Karhunen-Lo eve transforms are established techniques for mapping features into a lower dimensional subspace. This paper introduces a uniform statistical framework, where the computation of the optimal feature reduction is formalized as a Maximum-Likelihood estimation problem. The experimental evaluation of this suggested extension of linear selection methods shows a slig...

The theory of compensated compactness of Murat and Tartar links the algebraic condition of rank-r convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank-r convex forms arises. In the present paper, we def...

The descriptional complexity of semi-conditional grammars is studied. It is proved that every recursively enumerable language is generated by a semi-conditional grammar of degree (2, 1) with no more than seven conditional productions and eight nonterminals.