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Recovering low rank and sparse matrices from compressive measurements

Recovering low rank and sparse matrices from compressive measurements. Aswin C Sankaranarayanan Rice University. Richard G. Baraniuk. Andrew E. Waters. Background subtraction in surveillance videos. s tatic camera with foreground objects. r ank 1 background. s parse foreground.

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Recovering low rank and sparse matrices from compressive measurements

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  1. Recovering low rank and sparse matrices from compressive measurements Aswin C Sankaranarayanan Rice University Richard G. Baraniuk Andrew E. Waters

  2. Background subtraction in surveillance videos static camera with foreground objects rank 1 background sparseforeground

  3. More complex scenarios Changingillumination+ foreground motion

  4. More complex scenarios Changingillumination+ 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]

  5. More complex scenarios Changingillumination+ 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 compressiverecovery framework ?

  6. Hyperspectral cube 450nm 490nm 550nm 580nm 720nm Rank approximately equal number of materials in the scene Data courtesy AyanChakrabarti, http://vision.seas.harvard.edu/hyperspec/

  7. Robust matrix completion low rank matrix with missing entries low rank matrix

  8. Robust matrix completion missing + corruptedentries low rank matrix sparse corruptions

  9. Problem formulation • Noisy compressive measurements L: r-rank matrix S: k-sparsematrix • Measurement operator is different for different problems • Video CS: operates on each column of the matrix individually • Matrix completion: sampling operator • Hyperspectral

  10. Problem formulation • Noisy compressive measurements L: r-rank matrix S: k-sparsematrix

  11. 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

  12. 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

  13. Problem formulation • Key questions • When can we recover L and S ? • Measurement bounds ? • Fast algorithms ?

  14. 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]

  15. 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]

  16. 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, SpaRCSconverges exponentially to the right answer

  17. Phase transitions • p = number of measurements • r = rank, K = sparsity • Matrix of size N x N; N = 512 r=25 r=10 r=15 r=20 r=5

  18. Accuracy Performance Run time CS IT: An alternating projection algorithm that uses soft thresholding at each step CS APG: Variant of APG for RobustPCA problem.

  19. Video CS (a) Ground truth (b) Estimated low rank matrix (c) Estimated sparse component Video: 128x128x201 Compression 6.67x SNR = 31.1637 dB

  20. Video CS (a) Ground truth (b) Compression 3x Video 64x64x239 Compression 3x SNR = 23.9 dB

  21. Hyperspectral recovery results 128x128x128 HS cube Compression 6.67x SNR = 31.1637 dB

  22. Accuracy Matrix completion Run time CVX: Interior point solver of convex formulation OptSpace: Non-robust MC solver

  23. Open questions • Convergenceresults for the greedy algorithm • Low rank component is sparse/compressible in a wavelet basis • Is it even possible ?

  24. 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 22.8 dB Ground Truth 200x 25.2 dB 24.7 dB 2% 1%

  25. M/N = 2% M/N = 10% M/N = 1% (rank = 20) Ground truth 512x256x360 voxels

  26. Open questions • Convergenceresults 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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