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Hybrid Dense /Sparse Matrices in Compressed Sensing Reconstruction. Ilya Poltorak Dror Baron Deanna Needell. The work has been supported by the Israel Science Foundation and National Science Foundation. CS Measurement.

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Hybrid Dense /Sparse Matrices in Compressed Sensing Reconstruction

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Hybrid dense sparse matrices in compressed sensing reconstruction

Hybrid Dense/Sparse Matrices

in Compressed Sensing Reconstruction

Ilya Poltorak

Dror Baron

Deanna Needell

The work has been supported by the Israel Science Foundation and National Science Foundation.


Cs measurement

CS Measurement

  • Replace samples by more general encoderbased on a few linear projections (inner products)

sparsesignal

measurements

# non-zeros


Caveats

Caveats

  • Input x strictly sparse w/ real values

  • Noiseless measurements

    • noise can be addressed (later)

  • Assumptions relevant to content distribution (later)


Why is decoding expensive

Why is Decoding Expensive?

Culprit: dense, unstructured

sparsesignal

measurements

nonzeroentries


Sparse measurement matrices dense later

Sparse Measurement Matrices (dense later!)

  • LDPC measurement matrix (sparse)

  • Only {-1,0,+1} in 

  • Each row of  contains L randomly placed nonzeros

  • Fast matrix-vector multiplication

    • fast encoding & decoding

sparsesignal

measurements

nonzeroentries


Example

Example

0

1

1

4

?

?

?

?

?

?

0 1 1 0 0 0

0 0 0 1 1 0

1 1 0 0 1 0

0 0 0 0 1 1


Example1

Example

  • What does zero measurement imply?

  • Hint: x strictly sparse

0

1

1

4

?

?

?

?

?

?

0 1 1 0 0 0

0 0 0 1 1 0

1 1 0 0 1 0

0 0 0 0 1 1


Example2

Example

  • Graph reduction!

0

1

1

4

?

0

0

?

?

?

0 1 1 0 0 0

0 0 0 1 1 0

1 1 0 0 1 0

0 0 0 0 1 1


Example3

Example

  • What do matching measurements imply?

  • Hint: non-zeros in x are real numbers

0

1

1

4

?

0

0

?

?

?

0 1 1 0 0 0

0 0 0 1 1 0

1 1 0 0 1 0

0 0 0 0 1 1


Example4

Example

  • What is the last entry of x?

0

1

1

4

0

0

0

0

1

?

0 1 1 0 0 0

0 0 0 1 1 0

1 1 0 0 1 0

00 00 1 1


Noiseless algorithm luby mitzenmacher 2005 sarvotham baron baraniuk 2006 zhang pfister 2008

Noiseless Algorithm[Luby & Mitzenmacher 2005] [Sarvotham, Baron, & Baraniuk 2006][Zhang & Pfister 2008]

Phase1:zero measurements

Initialize

Phase2: matchingmeasurements

typically iterate 2-3 times

Phase3: singleton measurements

Arrange output

Done?

yes

no


Numbers 4 seconds

Numbers (4 seconds)

  • N=40,000

  • 5% non-zeros

  • M=0.22N

  • L=20 ones per row

  • Only 2-3 iterations

iteration #1


Challenge

Challenge

  • Withmeasurements parts of signal still not reconstructed

  • How do we recover the rest of the signal?


Solution hybrid dense sparse matrix

Solution: Hybrid Dense/Sparse Matrix

  • Withmeasurements parts of signal still not reconstructed

  • Add extra dense measurements

  • Residual of signal w/ residual dense columns

residual columns


Sudocodes with two part decoding sarvotham baron baraniuk 2006

Sudocodes with Two-Part Decoding[Sarvotham, Baron, & Baraniuk 2006]

  • Sudocodes (related to sudoku)

  • Graph reduction solves most of CS problem

  • Residual solved via matrix inversion

Residual via matrix inversion

sudo decoder

residual columns


Contribution 1 two part reconstruction

Contribution 1: Two-Part Reconstruction

  • Many CS algorithms for sparse matrices

    [Gilbert et al., Berinde & Indyk, Sarvotham et al.]

  • Many CS algorithms for dense matrices

    [Cormode & Muthukrishnan, Candes et al., Donoho et al., Gilbert et al., Milenkovic et al., Berinde & Indyk, Zhang & Pfister, Hale et al.,…]

  • Solve each part with appropriate algorithm

sparse solver

residual via dense solver

residual columns


Runtimes k 0 05n m 0 22n

Runtimes (K=0.05N, M=0.22N)


Theoretical results sarvotham baron baraniuk 2006

Theoretical Results [Sarvotham, Baron, & Baraniuk 2006]

  • Fast encoder and decoder

    • sub-linear decoding complexity

    • caveat: constructing data structure

  • Distributed content distribution

    • sparsified data

    • measurements stored on different servers

    • any M measurements suffice

  • Strictly sparse signals, noiseless measurements


Contribution 2 noisy measurements

Contribution 2: Noisy Measurements

  • Results can be extended to noisy measurements

  • Part 1 (zero measurements): measurement |ym|<

  • Part 2 (matching): |yi-yj|<

  • Part 3 (singleton): unchanged


Problems with noisy measurements

Problems with Noisy Measurements

  • Multiple iterations alias noise into next iteration!

  • Use one iteration

  • Requires small threshold  (large SNR)

  • Contribution 3:Provable reconstruction

    • deterministic & random variants


Summary

Summary

  • Hybrid Dense/Sparse Matrix

    • Two-part reconstruction

  • Simple (cute?) algorithm

  • Fast

  • Applicable to content distribution

  • Expandable to measurement noise


The end

THE END


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