Recovering low rank and sparse matrices from compressive measurements Aswin C Sankaranarayanan Rice University Richard G. Baraniuk Andrew E. Waters
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Recovering low rank and sparse matrices from compressive measurements
Aswin C SankaranarayananRice University
Richard G. Baraniuk Andrew E. Waters
Background subtraction in surveillance videos
static camera with foreground objects
rank 1 background
sparse foreground
More complex scenarios
Changing illumination + foreground motion
More complex scenarios
Changing illumination + foreground motion
Set of all images of a convex Lambertian scene under changing illumination is very close to a 9-dimensional subspace
[Basri and Jacobs, 2003]
More complex scenarios
Changing illumination + foreground motion
Video can be represented as a sum of a rank-9 matrix and a sparse matrix
Can we use such low rank+sparse model in a compressive recovery framework ?
Hyperspectral cube
450nm 550nm 720nm490nm 580nm
Rank approximately equal number of materials in the scene
Data courtesy Ayan Chakrabarti, http://vision.seas.harvard.edu/hyperspec/
Robust matrix completion
low rank matrix
low rank matrix with missing entries
Robust matrix completion
missing + corrupted
entries
low rank matrix
sparse corruptions
Problem formulation
• Noisy compressive measurements
L: r-rank matrix S: k-sparse matrix
• Measurement operator is different for different problems– Video CS: operates on each column of the matrix individually– Matrix completion: sampling operator– Hyperspectral
Problem formulation
• Noisy compressive measurements
L: r-rank matrix S: k-sparse matrix
Side note: Robust PCA “?”
• Recovery a low rank matrix L and a sparse matrix S, given M = L + S
Robust PCA [Candes et al, 2009]
Rank-sparsity incoherence [Chandrasekaran et al, 2011]
• We are interested in recovering a low rank matrix L and a sparse matrix S --- not from M --- but from compressive measurements of M
Connections to CS and Matrix Completion
• If we “remove” L from the optimization, then this reduces to traditional compressive recovery problem
• Similarly, if we “remove” S, then this reduces to the Affine rank minimization problem
Problem formulation
• Key questions– When can we recover L and S ?– Measurement bounds ?– Fast algorithms ?
SpaRCS
• SpaRCS: Sparse and low Rank recovery from CS – A greedy algorithm– It is an extension of CoSaMP [Tropp and Needell, 2009] and ADMiRA [Lee and Bresler, 2010]
SpaRCS
• SpaRCS: Sparse and low Rank recovery from CS– A greedy algorithm– It is an extension of CoSAMP [Tropp and Needell, 2009] and ADMiRA [Lee and Bresler, 2010]
SpaRCS
• SpaRCS: Sparse and low Rank recovery from CS – A greedy algorithm– It is an extension of CoSaMP [Tropp and Needell, 2009] and ADMiRA [Lee and Bresler, 2010]
• Claim– If satisfies both RIP and rank-RIP with small
constants,– and the low rank matrix is sufficiently dense, and sparse
matrix has random support (or bounded col/row degree)– then, SpaRCS converges exponentially to the right
answer
Phase transitions
• p = number of measurements• r = rank, K = sparsity• Matrix of size N x N; N = 512
r=5 r=10 r=15 r=20 r=25
Performance
Run time
CS IT: An alternating projection algorithm that uses soft thresholding at each step
CS APG: Variant of APG for RobustPCA problem.
Accuracy
Video CS
(a) Ground truth
(b) Estimated low rank matrix
(c) Estimated sparse component
Video: 128x128x201Compression 6.67xSNR = 31.1637 dB
Video CS
(a) Ground truth
(b) Compression 3x
Video 64x64x239Compression 3xSNR = 23.9 dB
Hyperspectral recovery results
128x128x128 HS cubeCompression 6.67xSNR = 31.1637 dB
Matrix completion
Run time
CVX: Interior point solver of convex formulation
OptSpace: Non-robust MC solver
Accuracy
Open questions
• Convergence results for the greedy algorithm
• Low rank component is sparse/compressible in a wavelet basis– Is it even possible ?
CS-LDS• [S, et al., SIAM J. IS*]
• Low rank model– Sparse rows (in a wavelet
transformation)
• Hyper-spectral data– 2300 Spectral bands– Spatial resolution 128 x 64– Rank 5
Ground Truth
2% 1%
25.2 dB 24.7 dB
Ground truth512x256x360 voxels
M/N = 10% M/N = 2% M/N = 1%
(rank = 20)
Open questions
• Convergence results for the greedy algorithm
• Low rank matrix is sparse/compressible in a wavelet basis– Is it even possible ?
• Streaming recovery etc…
dsp.rice.edu
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