94 research outputs found
1-Bit Matrix Completion under Exact Low-Rank Constraint
We consider the problem of noisy 1-bit matrix completion under an exact rank
constraint on the true underlying matrix . Instead of observing a subset
of the noisy continuous-valued entries of a matrix , we observe a subset
of noisy 1-bit (or binary) measurements generated according to a probabilistic
model. We consider constrained maximum likelihood estimation of , under a
constraint on the entry-wise infinity-norm of and an exact rank
constraint. This is in contrast to previous work which has used convex
relaxations for the rank. We provide an upper bound on the matrix estimation
error under this model. Compared to the existing results, our bound has faster
convergence rate with matrix dimensions when the fraction of revealed 1-bit
observations is fixed, independent of the matrix dimensions. We also propose an
iterative algorithm for solving our nonconvex optimization with a certificate
of global optimality of the limiting point. This algorithm is based on low rank
factorization of . We validate the method on synthetic and real data with
improved performance over existing methods.Comment: 6 pages, 3 figures, to appear in CISS 201
A Bayesian Approach for Noisy Matrix Completion: Optimal Rate under General Sampling Distribution
Bayesian methods for low-rank matrix completion with noise have been shown to
be very efficient computationally. While the behaviour of penalized
minimization methods is well understood both from the theoretical and
computational points of view in this problem, the theoretical optimality of
Bayesian estimators have not been explored yet. In this paper, we propose a
Bayesian estimator for matrix completion under general sampling distribution.
We also provide an oracle inequality for this estimator. This inequality proves
that, whatever the rank of the matrix to be estimated, our estimator reaches
the minimax-optimal rate of convergence (up to a logarithmic factor). We end
the paper with a short simulation study
Randomized Low-Memory Singular Value Projection
Affine rank minimization algorithms typically rely on calculating the
gradient of a data error followed by a singular value decomposition at every
iteration. Because these two steps are expensive, heuristic approximations are
often used to reduce computational burden. To this end, we propose a recovery
scheme that merges the two steps with randomized approximations, and as a
result, operates on space proportional to the degrees of freedom in the
problem. We theoretically establish the estimation guarantees of the algorithm
as a function of approximation tolerance. While the theoretical approximation
requirements are overly pessimistic, we demonstrate that in practice the
algorithm performs well on the quantum tomography recovery problem.Comment: 13 pages. This version has a revised theorem and new numerical
experiment
A variational approach to stable principal component pursuit
We introduce a new convex formulation for stable principal component pursuit
(SPCP) to decompose noisy signals into low-rank and sparse representations. For
numerical solutions of our SPCP formulation, we first develop a convex
variational framework and then accelerate it with quasi-Newton methods. We
show, via synthetic and real data experiments, that our approach offers
advantages over the classical SPCP formulations in scalability and practical
parameter selection.Comment: 10 pages, 5 figure
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