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
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
490nm
550nm
580nm
720nm
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 with
missing entries
low rank
matrix
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
r=5
r=10
r=15
• p = number of measurements
• r = rank, K = sparsity
• Matrix of size N x N; N = 512
r=20
r=25
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.
Video CS
(a) Ground truth
(b) Estimated low rank matrix
(c) Estimated sparse component
Video: 128x128x201
Compression 6.67x
SNR = 31.1637 dB
Video CS
(a) Ground truth
(b) Compression 3x
Video 64x64x239
Compression 3x
SNR = 23.9 dB
Hyperspectral recovery results
128x128x128 HS cube
Compression 6.67x
SNR = 31.1637 dB
Accuracy
Matrix
completion
Run time
CVX: Interior point solver of
convex formulation
OptSpace: Non-robust MC solver
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
2%
Ground
Truth
1%
M/N = 10%
M/N = 2%
(rank = 20)
M/N = 1%
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