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Recovering low rank and sparse matrices from compressive measurements

This study focuses on recovering low-rank and sparse matrices from compressive measurements in various scenarios, such as changing illumination and foreground motion. It explores the application of a compressive recovery framework to recover video data represented as a sum of a rank-9 matrix and a sparse matrix. The research includes robust matrix completion and robust PCA, with a focus on noisy compressive measurements and different measurement operators for various problems. The SpaRCS algorithm is introduced for sparse and low-rank recovery from compressive sensing, with connections to the Compressed Sensing (CS) and Matrix Completion fields. The study also discusses phase transitions, accuracy performance metrics, and open questions for further exploration, such as convergence results for the greedy algorithm and the compressibility of low-rank components in wavelet bases.

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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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