Distilled Sensing: Selective Sampling for Sparse Signal Recovery

1. Distilled Sensing: Selective Sampling for Sparse Signal Recovery by Jarvis Haupt, Rui Castro, and Robert Nowak (International Conference on Artificial Intelligence and Statistics, 2009) Presented by Lihan He ECE, Duke University April 16, 2009

2. Outline • Introduction • Sparse recovery by non-adaptive sensing • Distilled sensing • Main theorems • Experimental results • Conclusion 2/14

3. n is very large • Signal • Most of the components are zero – sparse signal • Objective: Identifying the locations of the non-zero components based on the data X={X1, …, Xn} Introduction Sparse signal recovery problem 3/14

4. Introduction Existing method: • False-discovery rate (FDR) analysis [Benjamini and Hochberg, 1995] • Cannot recover weak signals In this paper: Given a collection of noisy observations of the components of a sparse vector, it is far easier to identify a large set of null components (where the signal is absent) than it is to identify a small set of signal components. • A sequential adaptive sensing procedure • Taking multiple “looks” • At each look, finding out the locations where the signal is probably not present, and focusing sensing resources into the remaining locations for the next look • Can recover very weak signals than non-adaptive methods 4/14

5. Define is the estimated S from a given signal recovery procedure Non-Adaptive Sensing FDP and NDP False discovery proportion number of false discovered components total number of discovered components Non-discovery proportion NDP = number of undiscovered components total number of signal components 5/14

6. Considering sparse signals have signal components, each of amplitude , for some , and r > 0. Requiring that the nonzero components obey Non-Adaptive Sensing Recovery procedure Consider a coordinate-wise thresholding procedure to estimate the locations of the signal components. Previous research [Abramovich et al., 2006] shows that if r>β, with a threshold τ that may depend on r, β and n, the procedure drives both the FDP and NDP to zero simultaneously with probability tending to one as n→∞. Conversely, if r<β, no such coordinate-wise thresholding procedure can drive the FDP and NDP to zero simultaneously with probability tending to one as n→∞. 6/14

7. with the restriction , limiting the sensing energy. Distilled Sensing Allowing multiple observations, indexed by j • Adaptive, sequential designs of • depends explicitly on the past • Iteratively allocate more sensing resources to locations that are most promising while ignoring locations that are unlikely to contain signal • components 7/14

8. Distilled Sensing assuming signal is positive Each distillation step retains almost all of the locations corresponding to signal components, but only about half of the locations corresponding to null components. 8/14

9. Distilled Sensing Truth First observation Distillation Second observation 9/14 Source: http://homepages.cae.wisc.edu/~jhaupt/ds.html

10. Main Theorems Define energy allocation strategy Requiring that the nonzero components obey 10/14

11. Given k above, only requiring that the nonzero components obey Main Theorems 11/14

12. Experiments Astronomical survey 256 x 256 pixels; 533 pixels have nonzero amplitude of 2.98 (β=0.43, r=0.4) Noisy observation Truth Recovered by non-adaptive sensing Recovered by distilled sensing (Δ=0.9, k=5) 12/14

13. Experiments 10 independent trials Solid: non-adaptive sensing Dashed: distilled sensing 13/14

14. Conclusion • Proposed a sequential adaptive sensing procedure – distilled sensing • Recover sparse signal in noise • Iteratively focus sensing resources towards the signal subspace containing nonzero components • Can recover dramatically weaker sparse signals compared with traditional non-adaptive sensing procedure 14/14