Infinite-Dimensional Compressed Sensing and Function Interpolation
- 465 Downloads
We introduce and analyse a framework for function interpolation using compressed sensing. This framework—which is based on weighted \(\ell ^1\) 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 \(\ell ^1\) 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.
KeywordsHigh-dimensional approximation Interpolation Compressed sensing Structured sparsity Orthogonal polynomials
Mathematics Subject Classification41A05 41A10 41A63 65N12 65N15
The work was supported in part by the Natural Sciences and Engineering Research Council of Canada through Grant 611675 and an Alfred P. Sloan Research Fellowship. The author would particularly like to thank Abdellah Chkifa, Clayton Webster, Hoang Tran and Guannan Zhang for introducing him to the concept of lower sets. The results of Sect. 7.3 are due to their insight. He would also like to like to thank Rick Archibald, Nilima Nigam, Clarice Poon and Tao Zhou for useful discussions and comments.
- 1.B. Adcock. Infinite-dimensional \(\ell ^1\) minimization and function approximation from pointwise data. Constr. Approx. (to appear), 2016.Google Scholar
- 3.B. Adcock, R. Platte, and A. Shadrin. Optimal sampling rates for approximating analytic functions from pointwise samples. arXiv:1610.04769, 2016.Google Scholar
- 5.B. Bah and R. Ward. The sample complexity of weighted sparse approximation. arxiv:1507.0673, 2015.Google Scholar
- 6.J. Bigot, C. Boyer, and P. Weiss. An analysis of block sampling strategies in compressed sensing. IEEE Trans. Inform. Theory (to appear), 2016.Google Scholar
- 12.A. Chkifa, N. Dexter, H. Tran, and C. Webster. Polynomial approximation via compressed sensing of high-dimensional functions on lower sets. Technical Report ORNL/TM-2015/497, Oak Ridge National Laboratory (also available as arXiv:1602.05823), 2015.
- 13.I.-Y. Chun and B. Adcock. Compressed sensing and parallel acquisition. arXiv:1601.06214, 2016.Google Scholar
- 24.J. D. Jakeman, A. Narayan, and T. Zhou. A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions. arXiv:1602.06879, 2016.Google Scholar
- 26.L. Lorch. Alternative proof of a sharpened form of Bernstein’s inequality for legendre polynomials. Appl. Anal., 14:237–240, 1982/3.Google Scholar
- 28.G. Migliorati. Polynomial approximation by means of the random discrete \(L^2\) projection and application to inverse problems for PDEs with stochastic data. PhD thesis, Politecnico di Milano, 2013.Google Scholar
- 31.A. Narayan, J. D. Jakeman, and T. Zhou. A Christoffel function weighted least squares algorithm for collocation approximations. arXiv:1412.4305, 2014.Google Scholar
- 35.H. Rauhut and R. Ward. Sparse recovery for spherical harmonic expansions. In Proceedings of the 9th International Conference on Sampling Theory and Applications, 2011.Google Scholar
- 40.H. Tran, C. Webster, and G. Zhang. Analysis of quasi-optimal polynomial approximations for parameterized PDEs with deterministic and stochastic coefficients. ORNL/TM-2014/468, Oak Ridge National Laboratory (also available as arXiv:1508.01821), 2015.
- 41.E. van den Berg and M. P. Friedlander. SPGL1: A solver for large-scale sparse reconstruction. http://www.cs.ubc.ca/labs/scl/spgl1, June 2007.
- 42.E. van den Berg and M. P. Friedlander. Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput., 2(890–912), 31.Google Scholar
- 46.X. Yu and S. Baek. Sufficient conditions on stable recovery of sparse signals with partial support information. IEEE Signal Process. Letters, 20(5), 2013.Google Scholar