Precise Undersampling Theorems

• Published in: Proceedings of the IEEE Vol. 98; no. 6; pp. 913 - 924
• Main Authors: Donoho, David L., Tanner, Jared
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2010; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Bandlimited measurements; Combinatorial analysis; Compressed; Compressed sensing; ell_{1} -minimization; Geometry; Heart; Image reconstruction; Image sampling; Length measurement; Magnetic resonance imaging; Mathematical analysis; Particle measurements; Phase transformations; Phase transitions; Psychology; random measurements; random polytopes; Sampling; Sampling methods; Sparsity; Studies; superresolution; Theorems; undersampling; universality of matrix ensembles
• ISSN: 0018-9219; 1558-2256
• DOI: 10.1109/JPROC.2010.2045630

Undersampling theorems state that we may gather far fewer samples than the usual sampling theorem while exactly reconstructing the object of interest-provided the object in question obeys a sparsity condition, the samples measure appropriate linear combinations of signal values, and we reconstruct with a particular nonlinear procedure. While there are many ways to crudely demonstrate such undersampling phenomena, we know of only one mathematically rigorous approach which precisely quantifies the true sparsity-undersampling tradeoff curve of standard algorithms and standard compressed sensing matrices. That approach, based on combinatorial geometry, predicts the exact location in sparsity-undersampling domain where standard algorithms exhibit phase transitions in performance. We review the phase transition approach here and describe the broad range of cases where it applies. We also mention exceptions and state challenge problems for future research. Sample result: one can efficiently reconstruct a k-sparse signal of length N from n measurements, provided n ¿ 2k · log(N/n), for (k,n,N) large.k ¿ N. AMS 2000 subject classifications . Primary: 41A46, 52A22, 52B05, 62E20, 68P30, 94A20; Secondary: 15A52, 60F10, 68P25, 90C25, 94B20.