2,075 research outputs found
A Singular Value Thresholding Algorithm for Matrix Completion
This paper introduces a novel algorithm to approximate the matrix with minimum
nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood
as the convex relaxation of a rank minimization problem and arises in many important
applications as in the task of recovering a large matrix from a small subset of its entries (the famous
Netflix problem). Off-the-shelf algorithms such as interior point methods are not directly amenable
to large problems of this kind with over a million unknown entries. This paper develops a simple
first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in
which the optimal solution has low rank. The algorithm is iterative, produces a sequence of matrices
{X^k,Y^k}, and at each step mainly performs a soft-thresholding operation on the singular values
of the matrix Y^k. There are two remarkable features making this attractive for low-rank matrix
completion problems. The first is that the soft-thresholding operation is applied to a sparse matrix;
the second is that the rank of the iterates {X^k} is empirically nondecreasing. Both these facts allow
the algorithm to make use of very minimal storage space and keep the computational cost of each
iteration low. On the theoretical side, we provide a convergence analysis showing that the sequence
of iterates converges. On the practical side, we provide numerical examples in which 1,000 × 1,000
matrices are recovered in less than a minute on a modest desktop computer. We also demonstrate
that our approach is amenable to very large scale problems by recovering matrices of rank about
10 with nearly a billion unknowns from just about 0.4% of their sampled entries. Our methods are
connected with the recent literature on linearized Bregman iterations for ℓ_1 minimization, and we
develop a framework in which one can understand these algorithms in terms of well-known Lagrange
multiplier algorithms
OptShrink: An algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage
The truncated singular value decomposition (SVD) of the measurement matrix is
the optimal solution to the_representation_ problem of how to best approximate
a noisy measurement matrix using a low-rank matrix. Here, we consider the
(unobservable)_denoising_ problem of how to best approximate a low-rank signal
matrix buried in noise by optimal (re)weighting of the singular vectors of the
measurement matrix. We exploit recent results from random matrix theory to
exactly characterize the large matrix limit of the optimal weighting
coefficients and show that they can be computed directly from data for a large
class of noise models that includes the i.i.d. Gaussian noise case.
Our analysis brings into sharp focus the shrinkage-and-thresholding form of
the optimal weights, the non-convex nature of the associated shrinkage function
(on the singular values) and explains why matrix regularization via singular
value thresholding with convex penalty functions (such as the nuclear norm)
will always be suboptimal. We validate our theoretical predictions with
numerical simulations, develop an implementable algorithm (OptShrink) that
realizes the predicted performance gains and show how our methods can be used
to improve estimation in the setting where the measured matrix has missing
entries.Comment: Published version. The algorithm can be downloaded from
http://www.eecs.umich.edu/~rajnrao/optshrin
Finding a low-rank basis in a matrix subspace
For a given matrix subspace, how can we find a basis that consists of
low-rank matrices? This is a generalization of the sparse vector problem. It
turns out that when the subspace is spanned by rank-1 matrices, the matrices
can be obtained by the tensor CP decomposition. For the higher rank case, the
situation is not as straightforward. In this work we present an algorithm based
on a greedy process applicable to higher rank problems. Our algorithm first
estimates the minimum rank by applying soft singular value thresholding to a
nuclear norm relaxation, and then computes a matrix with that rank using the
method of alternating projections. We provide local convergence results, and
compare our algorithm with several alternative approaches. Applications include
data compression beyond the classical truncated SVD, computing accurate
eigenvectors of a near-multiple eigenvalue, image separation and graph
Laplacian eigenproblems
Weighted Schatten -Norm Minimization for Image Denoising and Background Subtraction
Low rank matrix approximation (LRMA), which aims to recover the underlying
low rank matrix from its degraded observation, has a wide range of applications
in computer vision. The latest LRMA methods resort to using the nuclear norm
minimization (NNM) as a convex relaxation of the nonconvex rank minimization.
However, NNM tends to over-shrink the rank components and treats the different
rank components equally, limiting its flexibility in practical applications. We
propose a more flexible model, namely the Weighted Schatten -Norm
Minimization (WSNM), to generalize the NNM to the Schatten -norm
minimization with weights assigned to different singular values. The proposed
WSNM not only gives better approximation to the original low-rank assumption,
but also considers the importance of different rank components. We analyze the
solution of WSNM and prove that, under certain weights permutation, WSNM can be
equivalently transformed into independent non-convex -norm subproblems,
whose global optimum can be efficiently solved by generalized iterated
shrinkage algorithm. We apply WSNM to typical low-level vision problems, e.g.,
image denoising and background subtraction. Extensive experimental results
show, both qualitatively and quantitatively, that the proposed WSNM can more
effectively remove noise, and model complex and dynamic scenes compared with
state-of-the-art methods.Comment: 13 pages, 11 figure
PEAR: PEriodic And fixed Rank separation for fast fMRI
In functional MRI (fMRI), faster acquisition via undersampling of data can
improve the spatial-temporal resolution trade-off and increase statistical
robustness through increased degrees-of-freedom. High quality reconstruction of
fMRI data from undersampled measurements requires proper modeling of the data.
We present an fMRI reconstruction approach based on modeling the fMRI signal as
a sum of periodic and fixed rank components, for improved reconstruction from
undersampled measurements. We decompose the fMRI signal into a component which
a has fixed rank and a component consisting of a sum of periodic signals which
is sparse in the temporal Fourier domain. Data reconstruction is performed by
solving a constrained problem that enforces a fixed, moderate rank on one of
the components, and a limited number of temporal frequencies on the other. Our
approach is coined PEAR - PEriodic And fixed Rank separation for fast fMRI.
Experimental results include purely synthetic simulation, a simulation with
real timecourses and retrospective undersampling of a real fMRI dataset.
Evaluation was performed both quantitatively and visually versus ground truth,
comparing PEAR to two additional recent methods for fMRI reconstruction from
undersampled measurements. Results demonstrate PEAR's improvement in estimating
the timecourses and activation maps versus the methods compared against at
acceleration ratios of R=8,16 (for simulated data) and R=6.66,10 (for real
data). PEAR results in reconstruction with higher fidelity than when using a
fixed-rank based model or a conventional Low-rank+Sparse algorithm. We have
shown that splitting the functional information between the components leads to
better modeling of fMRI, over state-of-the-art methods
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