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Hybrid Dense /Sparse Matrices in Compressed Sensing Reconstruction

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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

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

sparsesignal

measurements

# non-zeros

Caveats

- Input x strictly sparse w/ real values
- Noiseless measurements
- noise can be addressed (later)
- Assumptions relevant to content distribution (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

- 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

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

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

Challenge

- With measurements parts of signal still not reconstructed
- How do we recover the rest of the signal?

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 (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

- 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]

- 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

- 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

- Multiple iterations alias noise into next iteration!
- Use one iteration
- Requires small threshold (large SNR)
- Contribution 3:Provable reconstruction
- deterministic & random variants

Summary

- Hybrid Dense/Sparse Matrix
- Two-part reconstruction
- Simple (cute?) algorithm
- Fast
- Applicable to content distribution
- Expandable to measurement noise

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