198 research outputs found
Exploring Algorithmic Limits of Matrix Rank Minimization under Affine Constraints
Many applications require recovering a matrix of minimal rank within an
affine constraint set, with matrix completion a notable special case. Because
the problem is NP-hard in general, it is common to replace the matrix rank with
the nuclear norm, which acts as a convenient convex surrogate. While elegant
theoretical conditions elucidate when this replacement is likely to be
successful, they are highly restrictive and convex algorithms fail when the
ambient rank is too high or when the constraint set is poorly structured.
Non-convex alternatives fare somewhat better when carefully tuned; however,
convergence to locally optimal solutions remains a continuing source of
failure. Against this backdrop we derive a deceptively simple and
parameter-free probabilistic PCA-like algorithm that is capable, over a wide
battery of empirical tests, of successful recovery even at the theoretical
limit where the number of measurements equal the degrees of freedom in the
unknown low-rank matrix. Somewhat surprisingly, this is possible even when the
affine constraint set is highly ill-conditioned. While proving general recovery
guarantees remains evasive for non-convex algorithms, Bayesian-inspired or
otherwise, we nonetheless show conditions whereby the underlying cost function
has a unique stationary point located at the global optimum; no existing cost
function we are aware of satisfies this same property. We conclude with a
simple computer vision application involving image rectification and a standard
collaborative filtering benchmark
A Simplified Approach to Recovery Conditions for Low Rank Matrices
Recovering sparse vectors and low-rank matrices from noisy linear
measurements has been the focus of much recent research. Various reconstruction
algorithms have been studied, including and nuclear norm minimization
as well as minimization with . These algorithms are known to
succeed if certain conditions on the measurement map are satisfied. Proofs of
robust recovery for matrices have so far been much more involved than in the
vector case.
In this paper, we show how several robust classes of recovery conditions can
be extended from vectors to matrices in a simple and transparent way, leading
to the best known restricted isometry and nullspace conditions for matrix
recovery. Our results rely on the ability to "vectorize" matrices through the
use of a key singular value inequality.Comment: 6 pages, This is a modified version of a paper submitted to ISIT
2011; Proc. Intl. Symp. Info. Theory (ISIT), Aug 201
Guaranteed clustering and biclustering via semidefinite programming
Identifying clusters of similar objects in data plays a significant role in a
wide range of applications. As a model problem for clustering, we consider the
densest k-disjoint-clique problem, whose goal is to identify the collection of
k disjoint cliques of a given weighted complete graph maximizing the sum of the
densities of the complete subgraphs induced by these cliques. In this paper, we
establish conditions ensuring exact recovery of the densest k cliques of a
given graph from the optimal solution of a particular semidefinite program. In
particular, the semidefinite relaxation is exact for input graphs corresponding
to data consisting of k large, distinct clusters and a smaller number of
outliers. This approach also yields a semidefinite relaxation for the
biclustering problem with similar recovery guarantees. Given a set of objects
and a set of features exhibited by these objects, biclustering seeks to
simultaneously group the objects and features according to their expression
levels. This problem may be posed as partitioning the nodes of a weighted
bipartite complete graph such that the sum of the densities of the resulting
bipartite complete subgraphs is maximized. As in our analysis of the densest
k-disjoint-clique problem, we show that the correct partition of the objects
and features can be recovered from the optimal solution of a semidefinite
program in the case that the given data consists of several disjoint sets of
objects exhibiting similar features. Empirical evidence from numerical
experiments supporting these theoretical guarantees is also provided
Subspace System Identification via Weighted Nuclear Norm Optimization
We present a subspace system identification method based on weighted nuclear
norm approximation. The weight matrices used in the nuclear norm minimization
are the same weights as used in standard subspace identification methods. We
show that the inclusion of the weights improves the performance in terms of fit
on validation data. As a second benefit, the weights reduce the size of the
optimization problems that need to be solved. Experimental results from
randomly generated examples as well as from the Daisy benchmark collection are
reported. The key to an efficient implementation is the use of the alternating
direction method of multipliers to solve the optimization problem.Comment: Submitted to IEEE Conference on Decision and Contro
A Non-Convex Relaxation for Fixed-Rank Approximation
This paper considers the problem of finding a low rank matrix from
observations of linear combinations of its elements. It is well known that if
the problem fulfills a restricted isometry property (RIP), convex relaxations
using the nuclear norm typically work well and come with theoretical
performance guarantees. On the other hand these formulations suffer from a
shrinking bias that can severely degrade the solution in the presence of noise.
In this theoretical paper we study an alternative non-convex relaxation that
in contrast to the nuclear norm does not penalize the leading singular values
and thereby avoids this bias. We show that despite its non-convexity the
proposed formulation will in many cases have a single local minimizer if a RIP
holds. Our numerical tests show that our approach typically converges to a
better solution than nuclear norm based alternatives even in cases when the RIP
does not hold
Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting
In this paper we establish links between, and new results for, three problems
that are not usually considered together. The first is a matrix decomposition
problem that arises in areas such as statistical modeling and signal
processing: given a matrix formed as the sum of an unknown diagonal matrix
and an unknown low rank positive semidefinite matrix, decompose into these
constituents. The second problem we consider is to determine the facial
structure of the set of correlation matrices, a convex set also known as the
elliptope. This convex body, and particularly its facial structure, plays a
role in applications from combinatorial optimization to mathematical finance.
The third problem is a basic geometric question: given points
(where ) determine whether there is a centered
ellipsoid passing \emph{exactly} through all of the points.
We show that in a precise sense these three problems are equivalent.
Furthermore we establish a simple sufficient condition on a subspace that
ensures any positive semidefinite matrix with column space can be
recovered from for any diagonal matrix using a convex
optimization-based heuristic known as minimum trace factor analysis. This
result leads to a new understanding of the structure of rank-deficient
correlation matrices and a simple condition on a set of points that ensures
there is a centered ellipsoid passing through them.Comment: 20 page
Computational Complexity versus Statistical Performance on Sparse Recovery Problems
We show that several classical quantities controlling compressed sensing
performance directly match classical parameters controlling algorithmic
complexity. We first describe linearly convergent restart schemes on
first-order methods solving a broad range of compressed sensing problems, where
sharpness at the optimum controls convergence speed. We show that for sparse
recovery problems, this sharpness can be written as a condition number, given
by the ratio between true signal sparsity and the largest signal size that can
be recovered by the observation matrix. In a similar vein, Renegar's condition
number is a data-driven complexity measure for convex programs, generalizing
classical condition numbers for linear systems. We show that for a broad class
of compressed sensing problems, the worst case value of this algorithmic
complexity measure taken over all signals matches the restricted singular value
of the observation matrix which controls robust recovery performance. Overall,
this means in both cases that, in compressed sensing problems, a single
parameter directly controls both computational complexity and recovery
performance. Numerical experiments illustrate these points using several
classical algorithms.Comment: Final version, to appear in information and Inferenc
The Convex Geometry of Linear Inverse Problems
In applications throughout science and engineering one is often faced with
the challenge of solving an ill-posed inverse problem, where the number of
available measurements is smaller than the dimension of the model to be
estimated. However in many practical situations of interest, models are
constrained structurally so that they only have a few degrees of freedom
relative to their ambient dimension. This paper provides a general framework to
convert notions of simplicity into convex penalty functions, resulting in
convex optimization solutions to linear, underdetermined inverse problems. The
class of simple models considered are those formed as the sum of a few atoms
from some (possibly infinite) elementary atomic set; examples include
well-studied cases such as sparse vectors and low-rank matrices, as well as
several others including sums of a few permutations matrices, low-rank tensors,
orthogonal matrices, and atomic measures. The convex programming formulation is
based on minimizing the norm induced by the convex hull of the atomic set; this
norm is referred to as the atomic norm. The facial structure of the atomic norm
ball carries a number of favorable properties that are useful for recovering
simple models, and an analysis of the underlying convex geometry provides sharp
estimates of the number of generic measurements required for exact and robust
recovery of models from partial information. These estimates are based on
computing the Gaussian widths of tangent cones to the atomic norm ball. When
the atomic set has algebraic structure the resulting optimization problems can
be solved or approximated via semidefinite programming. The quality of these
approximations affects the number of measurements required for recovery. Thus
this work extends the catalog of simple models that can be recovered from
limited linear information via tractable convex programming
Robust Subspace System Identification via Weighted Nuclear Norm Optimization
Subspace identification is a classical and very well studied problem in
system identification. The problem was recently posed as a convex optimization
problem via the nuclear norm relaxation. Inspired by robust PCA, we extend this
framework to handle outliers. The proposed framework takes the form of a convex
optimization problem with an objective that trades off fit, rank and sparsity.
As in robust PCA, it can be problematic to find a suitable regularization
parameter. We show how the space in which a suitable parameter should be sought
can be limited to a bounded open set of the two dimensional parameter space. In
practice, this is very useful since it restricts the parameter space that is
needed to be surveyed.Comment: Submitted to the IFAC World Congress 201
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