The Gelfand widths of ℓp-balls for 0<p≤1

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets K ⊂ H of a Hilbert space H by the metric entropy of the set K where the covering numbers N(K, ") of K by "-balls of H satisfy the Lorentz condition ∫ ∞ 0 ( log2N(K, ") )r/s d" <∞ for some fixed 0 < r, s ≤ ∞ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 <...

Aicke Hinrichs, Simon Foucart, Alain Pajor, Holger Rauhut, Tino Ullrich win the 2010 Best Paper Award

The Award Committee – Steffen Dereich, TU Berlin, Germany, and Frances Kuo, University of New South Wales, Australia — determined that the following two papers exhibited exceptional merit and therefore awarded the prize to: AickeHinrichs, for paper ‘‘Optimal importance sampling for the approximation of integrals’’, which appeared in April, 2010, vol. 26, pp. 125–134. Simon Foucart, Alain Pajor,...

Performance evaluation of typical approximation algorithms for nonconvex ℓp-minimization in diffuse optical tomography.

The sparse estimation methods that utilize the ℓp-norm, with p being between 0 and 1, have shown better utility in providing optimal solutions to the inverse problem in diffuse optical tomography. These ℓp-norm-based regularizations make the optimization function nonconvex, and algorithms that implement ℓp-norm minimization utilize approximations to the original ℓp-norm function. In this work, ...

Stability of low-rank matrix recovery and its connections to Banach space geometry

Abstract. There are well-known relationships between compressed sensing and the geometry of the finite-dimensional lp spaces. A result of Kashin and Temlyakov [20] can be described as a characterization of the stability of the recovery of sparse vectors via l1minimization in terms of the Gelfand widths of certain identity mappings between finitedimensional l1 and l2 spaces, whereas a more recen...

1 6 Fe b 20 08 Widths of l p balls