Universal discretization and sparse sampling recovery

Abstract

Recently, it was discovered that for a given function class F the error of best linear recovery in the square norm can be bounded above by the Kolmogorov width of F in the uniform norm. That analysis is based on deep results in discretization of the square norm of functions from finite dimensional subspaces. In this paper we show how very recent results on universal discretization of the square norm of functions from a collection of finite dimensional subspaces lead to an inequality between optimal sparse recovery in the square norm and best sparse approximations in the uniform norm with respect to appropriate dictionaries.

Random points are good for universal discretization

“…We now discuss application of Remark 4.1 to optimal sampling recovery of periodic functions. This discussion complements the one from [6], Section 5. Let s = (s 1 , .…”

“…We proved in [6] the conditional Theorem 4.3. Recall that given a finite dimensional subspace X of continuous functions on Ω, and a vector ξ = (ξ 1 , • • • , ξ m ) ∈ Ω m , the classical least squares recovery operator (algorithm) (see, for instance, [3]) is defined as…”

“…Namely, it gives the corresponding Lebesgue-type inequality in the norm L 2 (Ω m , µ m ). We now discuss some results from [6], which provide good error of sparse sampling recovery in the norm L 2 (Ω, µ).…”

“…In [5] we only discuss the problem of existence of good points for discretization. In the very recent paper [6] we showed that points of good universal discretization can be used for sparse recovery of multivariate smooth functions. This motivated us to address the problem of finding points for good universal discretization.…”

“…In the very recent paper [6] we used the universal discretization results for the L 2 norm for analyzing a theoretical sparse recovery algorithm, based on a special dictionary D N . We showed that in the sense of accuracy it is a very good algorithm -we proved the corresponding Lebesgue-type inequalities for it.…”