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