Sparse Reconstruction viaBayesian Variable Selection andBayesian Model Averaging

Lee Potter, Phil Schniter, and Subhojit SomDepartment of Electrical & Computer EngineeringOhio State Universitywith support from AFOSR FA9550-06-1-0324

ATR Center Workshop, February 2009

2

Sparse Reconstruction

0.51 0.16 0.13 0.09

Variable Selection

[graphic adapted from R. Baraniuk]

Estimation

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Motivating Applications: Channel Estimation

(CW, from top left) Ohio Stadium at X-band; underwater acoustic communications; oximetrywith electronic spin resonance at L-band; ESR resonator; range-Doppler radar; roof-top rail-SAR; through-wall radar imaging.

[graphics: GD_ADS; M. Stojanovic, E. Ertin]

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The Variable Selection Problem

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Bayesian Variable Selection

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Typical Priors in Variable Selection

0

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Variable Selection: Posteriors

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Connection to AIC/BIC/RIC

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Bayesian Model Averaging

0.51 0.16 0.13 0.09

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Model Averaging: Implementation

0

11

Tipping’s Relevance Vector Machine (RVM)

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BMA versus RVM

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0 5 10 15 20 25 30 35 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

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Numerical Example: A compressible signal

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NMSE versus decay rate

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−26

−24

−22

−20

−18

−16

−14

−12

−10

−8

ρ

NM

SE

[dB

]

N = 512, M = 128, SNR = 15 dB, Dmax

= 5, Emax

= 20, T = 2000

FBMPmmse

(w/ EM update)

FBMPmap

(w/ EM update)

SparseBayesOMPStOMPGPSRBCSVB−BCS

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Sparsity of estimate versus decay rate

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

10

20

30

40

50

60

ρ

||xre

cove

ry|| 0

N = 512, M = 128, SNR = 15 dB, Dmax

= 5, Emax

= 20, T = 2000

FBMPmmse

(w/ EM update)

FBMPmap

(w/ EM update)

SparseBayesOMPStOMPGPSRBCS

16

Performance Guarantees: MAP variable selection

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Pair-wise Error Probability Analysis

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Conclusion