# Infinite-Dimensional Compressed Sensing and Function Interpolation

Infinite-Dimensional Compressed Sensing and Function Interpolation We introduce and analyse a framework for function interpolation using compressed sensing. This framework—which is based on weighted $$\ell ^1$$ ℓ 1 minimization—does not require a priori bounds on the expansion tail in either its implementation or its theoretical guarantees and in the absence of noise leads to genuinely interpolatory approximations. We also establish a new recovery guarantee for compressed sensing with weighted $$\ell ^1$$ ℓ 1 minimization based on this framework. This guarantee conveys several benefits. First, unlike existing results, it is sharp (up to constants and log factors) for large classes of functions regardless of the choice of weights. Second, by examining the measurement condition in the recovery guarantee, we are able to suggest a good overall strategy for selecting the weights. In particular, when applied to the important case of multivariate approximation with orthogonal polynomials, this weighting strategy leads to provably optimal estimates on the number of measurements required, whenever the support set of the significant coefficients is a so-called lower set. Finally, this guarantee can also be used to theoretically confirm the benefits of alternative weighting strategies where the weights are chosen based on prior support information. This provides a theoretical basis for a number of recent numerical studies showing the effectiveness of such approaches. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Foundations of Computational Mathematics Springer Journals

# Infinite-Dimensional Compressed Sensing and Function Interpolation

, Volume 18 (3) – Apr 19, 2017
41 pages

/lp/springer_journal/infinite-dimensional-compressed-sensing-and-function-interpolation-Da90xQNQKp
Publisher
Springer Journals
Subject
Mathematics; Numerical Analysis; Economics, general; Applications of Mathematics; Linear and Multilinear Algebras, Matrix Theory; Math Applications in Computer Science; Computer Science, general
ISSN
1615-3375
eISSN
1615-3383
D.O.I.
10.1007/s10208-017-9350-3
Publisher site
See Article on Publisher Site

### Abstract

We introduce and analyse a framework for function interpolation using compressed sensing. This framework—which is based on weighted $$\ell ^1$$ ℓ 1 minimization—does not require a priori bounds on the expansion tail in either its implementation or its theoretical guarantees and in the absence of noise leads to genuinely interpolatory approximations. We also establish a new recovery guarantee for compressed sensing with weighted $$\ell ^1$$ ℓ 1 minimization based on this framework. This guarantee conveys several benefits. First, unlike existing results, it is sharp (up to constants and log factors) for large classes of functions regardless of the choice of weights. Second, by examining the measurement condition in the recovery guarantee, we are able to suggest a good overall strategy for selecting the weights. In particular, when applied to the important case of multivariate approximation with orthogonal polynomials, this weighting strategy leads to provably optimal estimates on the number of measurements required, whenever the support set of the significant coefficients is a so-called lower set. Finally, this guarantee can also be used to theoretically confirm the benefits of alternative weighting strategies where the weights are chosen based on prior support information. This provides a theoretical basis for a number of recent numerical studies showing the effectiveness of such approaches.

### Journal

Foundations of Computational MathematicsSpringer Journals

Published: Apr 19, 2017

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