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Shriram Sarvotham Dror Baron Richard Baraniuk

Measurements and Bits: Compressed Sensing meets Information Theory. Shriram Sarvotham Dror Baron Richard Baraniuk. ECE Department Rice University dsp.rice.edu/cs. CS encoding. Replace samples by more general encoder based on a few linear projections (inner products)

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Shriram Sarvotham Dror Baron Richard Baraniuk

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  1. Measurements and Bits: Compressed Sensing meets Information Theory Shriram Sarvotham Dror BaronRichard Baraniuk ECE DepartmentRice University dsp.rice.edu/cs

  2. CS encoding • Replace samples by more general encoderbased on a few linear projections (inner products) • Matrix vector multiplication sparsesignal measurements # non-zeros

  3. The CS revelation – • Of the infinitely many solutions seek the onewith smallest L1 norm

  4. The CS revelation – • Of the infinitely many solutions seek the onewith smallest L1 norm • If then perfect reconstructionw/ high probability [Candes et al.; Donoho] • Linear programming

  5. Compressible signals • Polynomial decay of signal components • Recovery algorithms • reconstruction performance: • also requires • polynomial complexity (BPDN) [Candes et al.] • Cannot reduce order of [Kashin,Gluskin] constant squared of best term approximation

  6. Fundamental goal: minimize • Compressed sensing aims to minimize resource consumption due to measurements • Donoho: “Why go to so much effort to acquire all the data when most of what we get will be thrown away?”

  7. Measurement reduction for sparse signals • Ideal CS reconstruction of -sparse signal • Of the infinitely many solutions seek sparsest one • If M · K then w/ high probability this can’t be done • If M ¸ K+1 then perfect reconstructionw/ high probability [Bresler et al.; Wakin et al.] • But not robust and combinatorial complexity number of nonzero entries

  8. Why is this a complicated problem?

  9. Rich design space • What performance metric to use? • Wainwright: determine support set of nonzero entries • this is distortion metric • but why let tiny nonzero entries spoil the fun? • metric? ? • What complexity class of reconstruction algorithms? • any algorithms? • polynomial complexity? • near-linear or better? • How to account for imprecisions? • noise in measurements? • compressible signal model?

  10. How many measurements do we need?

  11. Measurement noise • Measurement process is analog • Analog systems add noise, non-linearities, etc. • Assume Gaussian noise for ease of analysis

  12. Setup • Signal is iid • Additive white Gaussian noise • Noisy measurement process

  13. Measurement and reconstruction quality • Measurement signal to noise ratio • Reconstruct using decoder mapping • Reconstruction distortion metric • Goal: minimize CS measurement rate

  14. Measurement channel • Model process as measurement channel • Capacity of measurement channel • Measurements are bits!

  15. Main result • Theorem: For a sparse signal with rate-distortion function , lower bound on measurement rate subject to measurement quality and reconstruction distortion satisfies • Direct relationship to rate-distortion content

  16. Main result • Theorem: For a sparse signal with rate-distortion function , lower bound on measurement rate subject to measurement quality and reconstruction distortion satisfies • Proof sketch: • each measurement provides bits • information content of source bits • source-channel separation for continuous amplitude sources • minimal number of measurements • Obtain measurement rate via normalization by

  17. Example • Spike process - spikes of uniform amplitude • Rate-distortion function • Lower bound • Numbers: • signal of length 107 • 103 spikes • SNR=10 dB  • SNR=-20 dB 

  18. Upper bound (achievable) in progress…

  19. CS reconstruction meets channel coding

  20. Why is reconstruction expensive? Culprit: dense, unstructured sparsesignal measurements nonzeroentries

  21. Fast CS reconstruction • LDPC measurement matrix (sparse) • Only 0/1 in • Each row of contains randomly placed 1’s • Fast matrix multiplication  fast encoding and reconstruction sparsesignal measurements nonzeroentries

  22. Ongoing work: CS using BP [Sarvotham et al.] • Considering noisy CS signals • Application of Belief Propagation • BP over real number field • sparsity is modeled as prior in graph • Low complexity • Provable reconstruction with noisy measurements using • Success of LDPC+BP in channel coding carried over to CS!

  23. Summary • Determination of measurement rates in CS • measurements are bits: each measurement provides bits • lower bound on measurement rate • direct relationship to rate-distortion content • Compressed sensing meets information theory • Additional research directions • promising results with LDPC measurement matrices • upper bound (achievable) on number of measurements dsp.rice.edu/cs

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