2,863 research outputs found
Estimation of high-dimensional low-rank matrices
Suppose that we observe entries or, more generally, linear combinations of
entries of an unknown -matrix corrupted by noise. We are
particularly interested in the high-dimensional setting where the number
of unknown entries can be much larger than the sample size . Motivated by
several applications, we consider estimation of matrix under the assumption
that it has small rank. This can be viewed as dimension reduction or sparsity
assumption. In order to shrink toward a low-rank representation, we investigate
penalized least squares estimators with a Schatten- quasi-norm penalty term,
. We study these estimators under two possible assumptions---a modified
version of the restricted isometry condition and a uniform bound on the ratio
"empirical norm induced by the sampling operator/Frobenius norm." The main
results are stated as nonasymptotic upper bounds on the prediction risk and on
the Schatten- risk of the estimators, where . The rates that we
obtain for the prediction risk are of the form (for ), up to
logarithmic factors, where is the rank of . The particular examples of
multi-task learning and matrix completion are worked out in detail. The proofs
are based on tools from the theory of empirical processes. As a by-product, we
derive bounds for the th entropy numbers of the quasi-convex Schatten class
embeddings , , which are of independent
interest.Comment: Published in at http://dx.doi.org/10.1214/10-AOS860 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
CVXR: An R Package for Disciplined Convex Optimization
CVXR is an R package that provides an object-oriented modeling language for
convex optimization, similar to CVX, CVXPY, YALMIP, and Convex.jl. It allows
the user to formulate convex optimization problems in a natural mathematical
syntax rather than the restrictive form required by most solvers. The user
specifies an objective and set of constraints by combining constants,
variables, and parameters using a library of functions with known mathematical
properties. CVXR then applies signed disciplined convex programming (DCP) to
verify the problem's convexity. Once verified, the problem is converted into
standard conic form using graph implementations and passed to a cone solver
such as ECOS or SCS. We demonstrate CVXR's modeling framework with several
applications.Comment: 34 pages, 9 figure
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