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Distilled Sensing: Selective Sampling for Sparse Signal Recovery 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 by

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

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Introduction 3/14 Sparse signal recovery problem 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={X 1, …, X n }

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Introduction 4/14 In this paper: 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 Existing method: False-discovery rate (FDR) analysis [Benjamini and Hochberg, 1995] Cannot recover weak signals 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.

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

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Non-Adaptive Sensing 6/14 Considering sparse signals have signal components, each of amplitude, for some, and r > 0. 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→∞. Requiring that the nonzero components obey Recovery procedure

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Distilled Sensing 7/14 Allowing multiple observations, indexed by j with the restriction, limiting the sensing energy. 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

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

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Distilled Sensing 9/14 Source: Truth First observation Second observation Distillation

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

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

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

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

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Conclusion 14/14 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

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