Super-resolution, Extremal Functions and the Condition Number of Vandermonde Matrices
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Moitra_Super-resolution.pdf
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Author(s)
Moitra, Ankur
Date Issued
June 2015
Journal
Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15
Publisher
ACM Symposium on Theory of Computing. Proceedings
Citation
Moitra, Ankur. “Super-Resolution, Extremal Functions and the Condition Number of Vandermonde Matrices.” ACM Press, 2015. 821–830.
Version
Author's final manuscript
Abstract
Super-resolution is a fundamental task in imaging, where the goal is to extract fine-grained structure from coarse-grained measurements. Here we are interested in a popular mathematical abstraction of this problem that has been widely studied in the statistics, signal processing and machine learning communities. We exactly resolve the threshold at which noisy super-resolution is possible. In particular, we establish a sharp phase transition for the relationship between the cutoff frequency (m) and the
separation (∆). If m > 1/∆ + 1, our estimator converges to the true values at an inverse polynomial rate in terms of the magnitude of the noise. And when m < (1 − epsilon )/∆ no estimator can distinguish between a particular pair of ∆-separated signals even if the magnitude of the noise is exponentially small. Our results involve making novel connections between extremal functions and the spectral properties of Vandermonde matrices. We establish a sharp phase transition for their condition number which in
turn allows us to give the first noise tolerance bounds for the matrix pencil method. Moreover we show that our methods can be interpreted as giving preconditioners for Vandermonde matrices, and we use this observation to design faster algorithms for super-resolution. We believe that these ideas may have other applications in designing faster algorithms for other basic tasks in signal processing.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1145/2746539.2746561
