This paper investigates how hidden layers of deep rectifier networks are capable of transforming two or more pattern sets to be linearly separable while preserving the distances with a guaranteed degree, and proves the universal classification power of such distance preserving rectifier networks. Through the nearly isometric nonlinear transformation in the hidden layers, the margin of the linea...

In this paper, we give a complete description of the structure of zero product and orthogonality preserving linear maps between W*-algebras. In particular, two W*-algebras are *-isomorphic if and only if there is a bijective linear map between them preserving their zero product or orthogonality structure in two directions. It is also the case when they have equivalent linear and left (right) id...

Many dispersive processes have moments of displacements with large-t behavior 〈|x|p〉 ∼ tp . The study of γp as a function of p provides a more complete characterization of the process than does the single number γ2. Also at long times, the core of the concentration relaxes to a self-similar profile, while the large-x tails, consisting of particles which have experienced exceptional displacement...