322,925 research outputs found
Completing Low-Rank Matrices with Corrupted Samples from Few Coefficients in General Basis
Subspace recovery from corrupted and missing data is crucial for various
applications in signal processing and information theory. To complete missing
values and detect column corruptions, existing robust Matrix Completion (MC)
methods mostly concentrate on recovering a low-rank matrix from few corrupted
coefficients w.r.t. standard basis, which, however, does not apply to more
general basis, e.g., Fourier basis. In this paper, we prove that the range
space of an matrix with rank can be exactly recovered from few
coefficients w.r.t. general basis, though and the number of corrupted
samples are both as high as . Our model covers
previous ones as special cases, and robust MC can recover the intrinsic matrix
with a higher rank. Moreover, we suggest a universal choice of the
regularization parameter, which is . By our
filtering algorithm, which has theoretical guarantees, we can
further reduce the computational cost of our model. As an application, we also
find that the solutions to extended robust Low-Rank Representation and to our
extended robust MC are mutually expressible, so both our theory and algorithm
can be applied to the subspace clustering problem with missing values under
certain conditions. Experiments verify our theories.Comment: To appear in IEEE Transactions on Information Theor
Stokes Phenomena and Non-perturbative Completion in the Multi-cut Two-matrix Models
The Stokes multipliers in the matrix models are invariants in the
string-theory moduli space and related to the D-instanton chemical potentials.
They not only represent non-perturbative information but also play an important
role in connecting various perturbative string theories in the moduli space.
They are a key concept to the non-perturbative completion of string theory and
also expected to imply some remnant of strong coupling dynamics in M theory. In
this paper, we investigate the non-perturbative completion problem consisting
of two constraints on the Stokes multipliers. As the first constraint, Stokes
phenomena which realize the multi-cut geometry are studied in the Z_k symmetric
critical points of the multi-cut two-matrix models. Sequence of solutions to
the constraints are obtained in general k-cut critical points. A discrete set
of solutions and a continuum set of solutions are explicitly shown, and they
can be classified by several constrained configurations of the Young diagram.
As the second constraint, we discuss non-perturbative stability of backgrounds
in terms of the Riemann-Hilbert problem. In particular, our procedure in the
2-cut (1,2) case (pure-supergravity case) completely fixes the D-instanton
chemical potentials and results in the Hastings-McLeod solution to the
Painlev\'e II equation. It is also stressed that the Riemann-Hilbert approach
realizes an off-shell background independent formulation of non-critical string
theory.Comment: 71 pages, v3: organization of Sec. 3, Sec. 4, App. C and App. D
improved, final version to be published in Nucl. Phys.
Recovering Missing Data via Matrix Completion in Electricity Distribution Systems
The performance of matrix completion based recovery of missing data in electricity distribution systems is analyzed. Under the assumption that the state variables follow a multivariate Gaussian distribution the matrix completion recovery is compared to estimation and information theoretic limits. The assumption about the distribution of the state variables is validated by the data shared by Electricity North West Limited. That being the case, the achievable distortion using minimum mean square error (MMSE) estimation is assessed for both random sampling and optimal linear encoding acquisition schemes. Within this setting, the impact of imperfect second order source statistics is numerically evaluated. The fundamental limit of the recovery process is characterized using Rate-Distortion theory to obtain the optimal performance theoretically attainable. Interestingly, numerical results show that matrix completion based recovery outperforms MMSE estimator when the number of available observations is low and access to perfect source statistics is not availabl
Improving compressed sensing with the diamond norm
In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a
minimal number of linear measurements. Within the paradigm of compressed
sensing, this is made computationally efficient by minimizing the nuclear norm
as a convex surrogate for rank.
In this work, we identify an improved regularizer based on the so-called
diamond norm, a concept imported from quantum information theory. We show that
-for a class of matrices saturating a certain norm inequality- the descent cone
of the diamond norm is contained in that of the nuclear norm. This suggests
superior reconstruction properties for these matrices. We explicitly
characterize this set of matrices. Moreover, we demonstrate numerically that
the diamond norm indeed outperforms the nuclear norm in a number of relevant
applications: These include signal analysis tasks such as blind matrix
deconvolution or the retrieval of certain unitary basis changes, as well as the
quantum information problem of process tomography with random measurements.
The diamond norm is defined for matrices that can be interpreted as order-4
tensors and it turns out that the above condition depends crucially on that
tensorial structure. In this sense, this work touches on an aspect of the
notoriously difficult tensor completion problem.Comment: 25 pages + Appendix, 7 Figures, published versio
Online Matrix Completion and Online Robust PCA
This work studies two interrelated problems - online robust PCA (RPCA) and
online low-rank matrix completion (MC). In recent work by Cand\`{e}s et al.,
RPCA has been defined as a problem of separating a low-rank matrix (true data),
and a sparse
matrix (outliers), from their
sum, . Our work uses this definition of RPCA. An important application
where both these problems occur is in video analytics in trying to separate
sparse foregrounds (e.g., moving objects) and slowly changing backgrounds.
While there has been a large amount of recent work on both developing and
analyzing batch RPCA and batch MC algorithms, the online problem is largely
open. In this work, we develop a practical modification of our recently
proposed algorithm to solve both the online RPCA and online MC problems. The
main contribution of this work is that we obtain correctness results for the
proposed algorithms under mild assumptions. The assumptions that we need are:
(a) a good estimate of the initial subspace is available (easy to obtain using
a short sequence of background-only frames in video surveillance); (b) the
's obey a `slow subspace change' assumption; (c) the basis vectors for
the subspace from which is generated are dense (non-sparse); (d) the
support of changes by at least a certain amount at least every so often;
and (e) algorithm parameters are appropriately setComment: Presented at ISIT (IEEE Intnl. Symp. on Information Theory), 2015.
Submitted to IEEE Transactions on Information Theory. This version: changes
are in blue; the main changes are just to explain the model assumptions
better (added based on ISIT reviewers' comments
Analysis of a Collaborative Filter Based on Popularity Amongst Neighbors
In this paper, we analyze a collaborative filter that answers the simple
question: What is popular amongst your friends? While this basic principle
seems to be prevalent in many practical implementations, there does not appear
to be much theoretical analysis of its performance. In this paper, we partly
fill this gap. While recent works on this topic, such as the low-rank matrix
completion literature, consider the probability of error in recovering the
entire rating matrix, we consider probability of an error in an individual
recommendation (bit error rate (BER)). For a mathematical model introduced in
[1],[2], we identify three regimes of operation for our algorithm (named
Popularity Amongst Friends (PAF)) in the limit as the matrix size grows to
infinity. In a regime characterized by large number of samples and small
degrees of freedom (defined precisely for the model in the paper), the
asymptotic BER is zero; in a regime characterized by large number of samples
and large degrees of freedom, the asymptotic BER is bounded away from 0 and 1/2
(and is identified exactly except for a special case); and in a regime
characterized by a small number of samples, the algorithm fails. We also
present numerical results for the MovieLens and Netflix datasets. We discuss
the empirical performance in light of our theoretical results and compare with
an approach based on low-rank matrix completion.Comment: 47 pages. Submitted to IEEE Transactions on Information Theory
(revised in July 2011). A shorter version would be presented at ISIT 201
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