1,856 research outputs found
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
The affine rank minimization problem consists of finding a matrix of minimum
rank that satisfies a given system of linear equality constraints. Such
problems have appeared in the literature of a diverse set of fields including
system identification and control, Euclidean embedding, and collaborative
filtering. Although specific instances can often be solved with specialized
algorithms, the general affine rank minimization problem is NP-hard. In this
paper, we show that if a certain restricted isometry property holds for the
linear transformation defining the constraints, the minimum rank solution can
be recovered by solving a convex optimization problem, namely the minimization
of the nuclear norm over the given affine space. We present several random
ensembles of equations where the restricted isometry property holds with
overwhelming probability. The techniques used in our analysis have strong
parallels in the compressed sensing framework. We discuss how affine rank
minimization generalizes this pre-existing concept and outline a dictionary
relating concepts from cardinality minimization to those of rank minimization
Random matrix theory and symmetric spaces
In this review we discuss the relationship between random matrix theories and
symmetric spaces. We show that the integration manifolds of random matrix
theories, the eigenvalue distribution, and the Dyson and boundary indices
characterizing the ensembles are in strict correspondence with symmetric spaces
and the intrinsic characteristics of their restricted root lattices. Several
important results can be obtained from this identification. In particular the
Cartan classification of triplets of symmetric spaces with positive, zero and
negative curvature gives rise to a new classification of random matrix
ensembles. The review is organized into two main parts. In Part I the theory of
symmetric spaces is reviewed with particular emphasis on the ideas relevant for
appreciating the correspondence with random matrix theories. In Part II we
discuss various applications of symmetric spaces to random matrix theories and
in particular the new classification of disordered systems derived from the
classification of symmetric spaces. We also review how the mapping from
integrable Calogero--Sutherland models to symmetric spaces can be used in the
theory of random matrices, with particular consequences for quantum transport
problems. We conclude indicating some interesting new directions of research
based on these identifications.Comment: 161 pages, LaTeX, no figures. Revised version with major additions in
the second part of the review. Version accepted for publication on Physics
Report
Recovery of Low-Rank Plus Compressed Sparse Matrices with Application to Unveiling Traffic Anomalies
Given the superposition of a low-rank matrix plus the product of a known fat
compression matrix times a sparse matrix, the goal of this paper is to
establish deterministic conditions under which exact recovery of the low-rank
and sparse components becomes possible. This fundamental identifiability issue
arises with traffic anomaly detection in backbone networks, and subsumes
compressed sensing as well as the timely low-rank plus sparse matrix recovery
tasks encountered in matrix decomposition problems. Leveraging the ability of
- and nuclear norms to recover sparse and low-rank matrices, a convex
program is formulated to estimate the unknowns. Analysis and simulations
confirm that the said convex program can recover the unknowns for sufficiently
low-rank and sparse enough components, along with a compression matrix
possessing an isometry property when restricted to operate on sparse vectors.
When the low-rank, sparse, and compression matrices are drawn from certain
random ensembles, it is established that exact recovery is possible with high
probability. First-order algorithms are developed to solve the nonsmooth convex
optimization problem with provable iteration complexity guarantees. Insightful
tests with synthetic and real network data corroborate the effectiveness of the
novel approach in unveiling traffic anomalies across flows and time, and its
ability to outperform existing alternatives.Comment: 38 pages, submitted to the IEEE Transactions on Information Theor
Multilinear Subspace Clustering
In this paper we present a new model and an algorithm for unsupervised
clustering of 2-D data such as images. We assume that the data comes from a
union of multilinear subspaces (UOMS) model, which is a specific structured
case of the much studied union of subspaces (UOS) model. For segmentation under
this model, we develop Multilinear Subspace Clustering (MSC) algorithm and
evaluate its performance on the YaleB and Olivietti image data sets. We show
that MSC is highly competitive with existing algorithms employing the UOS model
in terms of clustering performance while enjoying improvement in computational
complexity
Stable Principal Component Pursuit
In this paper, we study the problem of recovering a low-rank matrix (the
principal components) from a high-dimensional data matrix despite both small
entry-wise noise and gross sparse errors. Recently, it has been shown that a
convex program, named Principal Component Pursuit (PCP), can recover the
low-rank matrix when the data matrix is corrupted by gross sparse errors. We
further prove that the solution to a related convex program (a relaxed PCP)
gives an estimate of the low-rank matrix that is simultaneously stable to small
entrywise noise and robust to gross sparse errors. More precisely, our result
shows that the proposed convex program recovers the low-rank matrix even though
a positive fraction of its entries are arbitrarily corrupted, with an error
bound proportional to the noise level. We present simulation results to support
our result and demonstrate that the new convex program accurately recovers the
principal components (the low-rank matrix) under quite broad conditions. To our
knowledge, this is the first result that shows the classical Principal
Component Analysis (PCA), optimal for small i.i.d. noise, can be made robust to
gross sparse errors; or the first that shows the newly proposed PCP can be made
stable to small entry-wise perturbations.Comment: 5-page paper submitted to ISIT 201
- …