13 research outputs found
Partial Sum Minimization of Singular Values in Robust PCA: Algorithm and Applications
Robust Principal Component Analysis (RPCA) via rank minimization is a
powerful tool for recovering underlying low-rank structure of clean data
corrupted with sparse noise/outliers. In many low-level vision problems, not
only it is known that the underlying structure of clean data is low-rank, but
the exact rank of clean data is also known. Yet, when applying conventional
rank minimization for those problems, the objective function is formulated in a
way that does not fully utilize a priori target rank information about the
problems. This observation motivates us to investigate whether there is a
better alternative solution when using rank minimization. In this paper,
instead of minimizing the nuclear norm, we propose to minimize the partial sum
of singular values, which implicitly encourages the target rank constraint. Our
experimental analyses show that, when the number of samples is deficient, our
approach leads to a higher success rate than conventional rank minimization,
while the solutions obtained by the two approaches are almost identical when
the number of samples is more than sufficient. We apply our approach to various
low-level vision problems, e.g. high dynamic range imaging, motion edge
detection, photometric stereo, image alignment and recovery, and show that our
results outperform those obtained by the conventional nuclear norm rank
minimization method.Comment: Accepted in Transactions on Pattern Analysis and Machine Intelligence
(TPAMI). To appea
Low-Rank Matrix Approximations with Flip-Flop Spectrum-Revealing QR Factorization
We present Flip-Flop Spectrum-Revealing QR (Flip-Flop SRQR) factorization, a
significantly faster and more reliable variant of the QLP factorization of
Stewart, for low-rank matrix approximations. Flip-Flop SRQR uses SRQR
factorization to initialize a partial column pivoted QR factorization and then
compute a partial LQ factorization. As observed by Stewart in his original QLP
work, Flip-Flop SRQR tracks the exact singular values with "considerable
fidelity". We develop singular value lower bounds and residual error upper
bounds for Flip-Flop SRQR factorization. In situations where singular values of
the input matrix decay relatively quickly, the low-rank approximation computed
by SRQR is guaranteed to be as accurate as truncated SVD. We also perform a
complexity analysis to show that for the same accuracy, Flip-Flop SRQR is
faster than randomized subspace iteration for approximating the SVD, the
standard method used in Matlab tensor toolbox. We also compare Flip-Flop SRQR
with alternatives on two applications, tensor approximation and nuclear norm
minimization, to demonstrate its efficiency and effectiveness
Fast randomized Singular Value Thresholding for Nuclear Norm Minimization
Rank minimization problem can be boiled down to either Nuclear Norm Minimization (NNM) or Weighted NNM (WNNM) problem. The problems related to NNM (or WNNM) can be solved iteratively by applying a closed-form proximal operator, called Singular Value Thresholding (SVT) (or Weighted SVT), but they suffer from high computational cost to compute a Singular Value Decomposition (SVD) at each iteration. In this paper, we propose an accurate and fast approximation method for SVT, called fast randomized SVT (FRSVT), where we avoid direct computation of SVD. The key idea is to extract an approximate basis for the range of a matrix from its compressed matrix. Given the basis, we compute the partial singular values of the original matrix from a small factored matrix. While the basis approximation is the bottleneck, our method is already severalfold faster than thin SVD. By adopting a range propagation technique, we can further avoid one of the bottleneck at each iteration. Our theoretical analysis provides a stepping stone between the approximation bound of SVD and its effect to NNM via SVT. Along with the analysis, our empirical results on both quantitative and qualitative studies show our approximation rarely harms the convergence behavior of the host algorithms. We apply it and validate the efficiency of our method on various vision problems, e.g. subspace clustering, weather artifact removal, simultaneous multi-image alignment and rectification.1