Optimal $$C^\infty $$-approximation of functions with exponentially or sub-exponentially integrable derivative

Exponentially Vanishing Sub-optimal Local Minima in Multilayer Neural Networks

Background: Statistical mechanics results (Dauphin et al. (2014); Choromanska et al. (2015)) suggest that local minima with high error are exponentially rare in high dimensions. However, to prove low error guarantees for Multilayer Neural Networks (MNNs), previous works so far required either a heavily modified MNN model or training method, strong assumptions on the labels (e.g., “near” linear ...

Tight frames of exponentially decaying Wannier functions

Let L be a Schrödinger operator ( i ∂ ∂x −A(x))2+V (x) with periodic magnetic and electric potentials A,V , a Maxwell operator ∇× 1 ε(x)∇× in a periodic medium, or an arbitrary self-adjoint elliptic linear partial differential operator in R with coefficients periodic with respect to a lattice Γ. Let also S be a finite part of its spectrum separated by gaps from the rest of the spectrum. We cons...

Exponentially Convex Functions on Hypercomplex Systems

Regularity of Refinable Functions with Exponentially Decaying Masks

The smoothness property of refinable functions is an important issue in all multiresolution analysis and has a strong impact on applications of wavelets to image processing, geometric and numerical solutions of elliptic partial differential equations. The purpose of this paper is to characterize the smoothness properties of refinable functions with exponentially decaying masks and an isotropic ...

Sub-exponentially many 3-colorings of triangle-free planar graphs

Thomassen conjectured that every triangle-free planar graph on n vertices has exponentially many 3-colorings, and proved that it has at least 2 1/12/20000 distinct 3-colorings. We show that it has at least 2 √ n/362 distinct 3-colorings. April 2010, revised 8 July 2010. An extended abstract of this paper appeared in [1]. Partially supported by an NSF Graduate Research Fellowship. Partially supp...