147,128 research outputs found
Matrix Completion With Noise
On the heels of compressed sensing, a remarkable new field has very recently
emerged. This field addresses a broad range of problems of significant
practical interest, namely, the recovery of a data matrix from what appears to
be incomplete, and perhaps even corrupted, information. In its simplest form,
the problem is to recover a matrix from a small sample of its entries, and
comes up in many areas of science and engineering including collaborative
filtering, machine learning, control, remote sensing, and computer vision to
name a few.
This paper surveys the novel literature on matrix completion, which shows
that under some suitable conditions, one can recover an unknown low-rank matrix
from a nearly minimal set of entries by solving a simple convex optimization
problem, namely, nuclear-norm minimization subject to data constraints.
Further, this paper introduces novel results showing that matrix completion is
provably accurate even when the few observed entries are corrupted with a small
amount of noise. A typical result is that one can recover an unknown n x n
matrix of low rank r from just about nr log^2 n noisy samples with an error
which is proportional to the noise level. We present numerical results which
complement our quantitative analysis and show that, in practice, nuclear norm
minimization accurately fills in the many missing entries of large low-rank
matrices from just a few noisy samples. Some analogies between matrix
completion and compressed sensing are discussed throughout.Comment: 11 pages, 4 figures, 1 tabl
Low Rank Matrix Completion with Exponential Family Noise
The matrix completion problem consists in reconstructing a matrix from a
sample of entries, possibly observed with noise. A popular class of estimator,
known as nuclear norm penalized estimators, are based on minimizing the sum of
a data fitting term and a nuclear norm penalization. Here, we investigate the
case where the noise distribution belongs to the exponential family and is
sub-exponential. Our framework alllows for a general sampling scheme. We first
consider an estimator defined as the minimizer of the sum of a log-likelihood
term and a nuclear norm penalization and prove an upper bound on the Frobenius
prediction risk. The rate obtained improves on previous works on matrix
completion for exponential family. When the sampling distribution is known, we
propose another estimator and prove an oracle inequality w.r.t. the
Kullback-Leibler prediction risk, which translates immediatly into an upper
bound on the Frobenius prediction risk. Finally, we show that all the rates
obtained are minimax optimal up to a logarithmic factor
Ad Hoc Microphone Array Calibration: Euclidean Distance Matrix Completion Algorithm and Theoretical Guarantees
This paper addresses the problem of ad hoc microphone array calibration where
only partial information about the distances between microphones is available.
We construct a matrix consisting of the pairwise distances and propose to
estimate the missing entries based on a novel Euclidean distance matrix
completion algorithm by alternative low-rank matrix completion and projection
onto the Euclidean distance space. This approach confines the recovered matrix
to the EDM cone at each iteration of the matrix completion algorithm. The
theoretical guarantees of the calibration performance are obtained considering
the random and locally structured missing entries as well as the measurement
noise on the known distances. This study elucidates the links between the
calibration error and the number of microphones along with the noise level and
the ratio of missing distances. Thorough experiments on real data recordings
and simulated setups are conducted to demonstrate these theoretical insights. A
significant improvement is achieved by the proposed Euclidean distance matrix
completion algorithm over the state-of-the-art techniques for ad hoc microphone
array calibration.Comment: In Press, available online, August 1, 2014.
http://www.sciencedirect.com/science/article/pii/S0165168414003508, Signal
Processing, 201
Restricted strong convexity and weighted matrix completion: Optimal bounds with noise
We consider the matrix completion problem under a form of row/column weighted
entrywise sampling, including the case of uniform entrywise sampling as a
special case. We analyze the associated random observation operator, and prove
that with high probability, it satisfies a form of restricted strong convexity
with respect to weighted Frobenius norm. Using this property, we obtain as
corollaries a number of error bounds on matrix completion in the weighted
Frobenius norm under noisy sampling and for both exact and near low-rank
matrices. Our results are based on measures of the "spikiness" and
"low-rankness" of matrices that are less restrictive than the incoherence
conditions imposed in previous work. Our technique involves an -estimator
that includes controls on both the rank and spikiness of the solution, and we
establish non-asymptotic error bounds in weighted Frobenius norm for recovering
matrices lying with -"balls" of bounded spikiness. Using
information-theoretic methods, we show that no algorithm can achieve better
estimates (up to a logarithmic factor) over these same sets, showing that our
conditions on matrices and associated rates are essentially optimal
Matrix Completion with Noise via Leveraged Sampling
Many matrix completion methods assume that the data follows the uniform
distribution. To address the limitation of this assumption, Chen et al.
\cite{Chen20152999} propose to recover the matrix where the data follows the
specific biased distribution. Unfortunately, in most real-world applications,
the recovery of a data matrix appears to be incomplete, and perhaps even
corrupted information. This paper considers the recovery of a low-rank matrix,
where some observed entries are sampled in a \emph{biased distribution}
suitably dependent on \emph{leverage scores} of a matrix, and some observed
entries are uniformly corrupted. Our theoretical findings show that we can
provably recover an unknown matrix of rank from just about
entries even when the few observed entries are corrupted with a
small amount of noisy information. Empirical studies verify our theoretical
results
Exponential Family Matrix Completion under Structural Constraints
We consider the matrix completion problem of recovering a structured matrix
from noisy and partial measurements. Recent works have proposed tractable
estimators with strong statistical guarantees for the case where the underlying
matrix is low--rank, and the measurements consist of a subset, either of the
exact individual entries, or of the entries perturbed by additive Gaussian
noise, which is thus implicitly suited for thin--tailed continuous data.
Arguably, common applications of matrix completion require estimators for (a)
heterogeneous data--types, such as skewed--continuous, count, binary, etc., (b)
for heterogeneous noise models (beyond Gaussian), which capture varied
uncertainty in the measurements, and (c) heterogeneous structural constraints
beyond low--rank, such as block--sparsity, or a superposition structure of
low--rank plus elementwise sparseness, among others. In this paper, we provide
a vastly unified framework for generalized matrix completion by considering a
matrix completion setting wherein the matrix entries are sampled from any
member of the rich family of exponential family distributions; and impose
general structural constraints on the underlying matrix, as captured by a
general regularizer . We propose a simple convex regularized
--estimator for the generalized framework, and provide a unified and novel
statistical analysis for this general class of estimators. We finally
corroborate our theoretical results on simulated datasets.Comment: 20 pages, 9 figure
Poisson Matrix Completion
We extend the theory of matrix completion to the case where we make Poisson
observations for a subset of entries of a low-rank matrix. We consider the
(now) usual matrix recovery formulation through maximum likelihood with proper
constraints on the matrix , and establish theoretical upper and lower bounds
on the recovery error. Our bounds are nearly optimal up to a factor on the
order of . These bounds are obtained by adapting
the arguments used for one-bit matrix completion \cite{davenport20121}
(although these two problems are different in nature) and the adaptation
requires new techniques exploiting properties of the Poisson likelihood
function and tackling the difficulties posed by the locally sub-Gaussian
characteristic of the Poisson distribution. Our results highlight a few
important distinctions of Poisson matrix completion compared to the prior work
in matrix completion including having to impose a minimum signal-to-noise
requirement on each observed entry. We also develop an efficient iterative
algorithm and demonstrate its good performance in recovering solar flare
images.Comment: Submitted to IEEE for publicatio
Constructing confidence sets for the matrix completion problem
In the present note we consider the problem of constructing honest and
adaptive confidence sets for the matrix completion problem. For the Bernoulli
model with known variance of the noise we provide a realizable method for
constructing confidence sets that adapt to the unknown rank of the true matrix
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