Super-Resolution Off the Grid

Super-resolution is the problem of recovering a superposition of point sources using bandlimited measurements, which may be corrupted with noise. This signal processing problem arises in numerous imaging problems, ranging from astronomy to biology to spectroscopy, where it is common to take (coarse) Fourier measurements of an object. Of particular interest is in obtaining estimation procedures which are robust to noise, with the following desirable statistical and computational properties: we seek to use coarse Fourier measurements (bounded by some cutoff frequency); we hope to take a (quantifiably) small number of measurements; we desire our algorithm to run quickly. Suppose we have k point sources in d dimensions, where the points are separated by at least Δ from each other (in Euclidean distance). This work provides an algorithm with the following favorable guarantees: • The algorithm uses Fourier measurements, whose frequencies are bounded by O(1/Δ) (up to log factors). Previous algorithms require a cutoff frequency which may be as large as Ω(√d/Δ). • The number of measurements taken by and the computational complexity of our algorithm are bounded by a polynomial in both the number of points k and the dimension d, with no dependence on the separation Δ. In contrast, previous algorithms depended inverse polynomially on the minimal separation and exponentially on the dimension for both of these quantities. Our estimation procedure itself is simple: we take random bandlimited measurements (as opposed to taking an exponential number of measurements on the hyper-grid). Furthermore, our analysis and algorithm are elementary (based on concentration bounds for sampling and the singular value decomposition).

[1]  K. Pearson.  Contributions to the Mathematical Theory of Evolution , 1894 .

[2]  Richard A. Harshman,et al.  Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis , 1970 .

[3]  D. Russell.  Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions , 1978 .

[4]  D. Donoho.  Superresolution via sparsity constraints , 1992 .

[5]  S. Leurgans,et al.  A Decomposition for Three-Way Arrays , 1993, SIAM J. Matrix Anal. Appl..

[6]  Sanjoy Dasgupta,et al.  Learning mixtures of Gaussians , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[7]  Sanjoy Dasgupta,et al.  A Two-Round Variant of EM for Gaussian Mixtures , 2000, UAI.

[8]  Sanjeev Arora,et al.  Learning mixtures of arbitrary gaussians , 2001, STOC '01.

[9]  Sanjoy Dasgupta,et al.  An elementary proof of a theorem of Johnson and Lindenstrauss , 2003, Random Struct. Algorithms.

[10]  Elchanan Mossel,et al.  Learning nonsingular phylogenies and hidden Markov models , 2005, STOC '05.

[11]  Vilmos Komornik,et al.  Fourier Series in Control Theory , 2005 .

[12]  Emmanuel J. Candès,et al.  Towards a Mathematical Theory of Super‐resolution , 2012, ArXiv.

[13]  Emmanuel J. Candès,et al.  Super-Resolution from Noisy Data , 2012, Journal of Fourier Analysis and Applications.

[14]  Anima Anandkumar,et al.  A Method of Moments for Mixture Models and Hidden Markov Models , 2012, COLT.

[15]  Debasis Kundu,et al.  Noise space decomposition method for two-dimensional sinusoidal model , 2013, Comput. Stat. Data Anal..

[16]  D. Potts,et al.  Parameter estimation for nonincreasing exponential sums by Prony-like methods , 2013 .

[17]  Badri Narayan Bhaskar,et al.  Compressed Sensing o the Grid , 2013 .

[18]  Parikshit Shah,et al.  Compressed Sensing Off the Grid , 2012, IEEE Transactions on Information Theory.

[19]  The Threshold for Super-resolution via Extremal Functions , 2014, ArXiv.

[20]  Yuxin Chen,et al.  Robust Spectral Compressed Sensing via Structured Matrix Completion , 2013, IEEE Transactions on Information Theory.

[21]  Wenjing Liao,et al.  MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-resolution , 2014, ArXiv.

[22]  Santosh S. Vempala,et al.  Max vs Min: Independent Component Analysis with nearly Linear Sample Complexity , 2014, ArXiv.

[23]  Anima Anandkumar,et al.  Tensor decompositions for learning latent variable models , 2012, J. Mach. Learn. Res..

[24]  Benjamin Recht,et al.  Superresolution without separation , 2015, 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[25]  Ankur Moitra,et al.  Super-resolution, Extremal Functions and the Condition Number of Vandermonde Matrices , 2014, STOC.

[26]  Santosh S. Vempala,et al.  Max vs Min: Tensor Decomposition and ICA with nearly Linear Sample Complexity , 2014, COLT.

[27]  Lena Jaeger,et al.  Fourier Series In Control Theory , 2016 .