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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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slide1

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
  • With measurements 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
  • With measurements 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

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
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