hybrid dense/sparse matrices in compressed sensing reconstruction ilya poltorak dror baron deanna...
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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• Replace samples by more general encoder
based on a few linear projections (inner products)
measurements sparsesignal
# non-zeros
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?
measurementssparsesignal
nonzeroentries
Culprit: dense, unstructured
Sparse Measurement Matrices (dense later!)
measurementssparsesignal
nonzeroentries
• LDPC measurement matrix (sparse)• Only {-1,0,+1} in • Each row of contains L randomly placed nonzeros • Fast matrix-vector multiplication
fast encoding & decoding
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
?
?
?
?
?
?
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
?
?
?
?
?
?
• What does zero measurement imply?• Hint: x strictly sparse
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
?
0
0
?
?
?
• Graph reduction!
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
?
0
0
?
?
?
• What do matching measurements imply?• Hint: non-zeros in x are real numbers
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
0
0
0
0
1
?
• What is the last entry of x?
Noiseless Algorithm [Luby & Mitzenmacher 2005]
[Sarvotham, Baron, & Baraniuk 2006][Zhang & Pfister 2008]
Phase1: zero measurements
Phase2: matching measurements
Phase3: singleton measurements
Initialize
Done? Arrange outputyesno
typically iterate 2-3 times
Numbers (4 seconds)
• N=40,000• 5% non-zeros• M=0.22N• L=20 ones per row
Only 2-3 iterations
Iteration, Phase
Zeros Non-zeros Total
1,1 30615 0 30615
1,2 35224 977 36201
1,3 35224 1500 36724
2,1 36800 1500 38300
2,2 37180 1833 39013
2,3 37180 2063 39243
3,1 37268 2063 39331
3,2 37289 2074 39363
3,3 37289 2083 39372
4,1 37289 2083 39372
4,2 37291 2084 39375
4,3 37291 2084 39375
iteration #1
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
sudo decoder
Residual via matrix inversion
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
residual columns
sparse solver
residual via dense solver
Runtimes (K=0.05N, M=0.22N)
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
THE END