806 research outputs found
High-dimensional estimation with geometric constraints
Consider measuring an n-dimensional vector x through the inner product with
several measurement vectors, a_1, a_2, ..., a_m. It is common in both signal
processing and statistics to assume the linear response model y_i = +
e_i, where e_i is a noise term. However, in practice the precise relationship
between the signal x and the observations y_i may not follow the linear model,
and in some cases it may not even be known. To address this challenge, in this
paper we propose a general model where it is only assumed that each observation
y_i may depend on a_i only through . We do not assume that the
dependence is known. This is a form of the semiparametric single index model,
and it includes the linear model as well as many forms of the generalized
linear model as special cases. We further assume that the signal x has some
structure, and we formulate this as a general assumption that x belongs to some
known (but arbitrary) feasible set K. We carefully detail the benefit of using
the signal structure to improve estimation. The theory is based on the mean
width of K, a geometric parameter which can be used to understand its effective
dimension in estimation problems. We determine a simple, efficient two-step
procedure for estimating the signal based on this model -- a linear estimation
followed by metric projection onto K. We give general conditions under which
the estimator is minimax optimal up to a constant. This leads to the intriguing
conclusion that in the high noise regime, an unknown non-linearity in the
observations does not significantly reduce one's ability to determine the
signal, even when the non-linearity may be non-invertible. Our results may be
specialized to understand the effect of non-linearities in compressed sensing.Comment: This version incorporates minor revisions suggested by referee
Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions
We analyze a class of estimators based on convex relaxation for solving
high-dimensional matrix decomposition problems. The observations are noisy
realizations of a linear transformation of the sum of an
approximately) low rank matrix with a second matrix
endowed with a complementary form of low-dimensional structure;
this set-up includes many statistical models of interest, including factor
analysis, multi-task regression, and robust covariance estimation. We derive a
general theorem that bounds the Frobenius norm error for an estimate of the
pair obtained by solving a convex optimization
problem that combines the nuclear norm with a general decomposable regularizer.
Our results utilize a "spikiness" condition that is related to but milder than
singular vector incoherence. We specialize our general result to two cases that
have been studied in past work: low rank plus an entrywise sparse matrix, and
low rank plus a columnwise sparse matrix. For both models, our theory yields
non-asymptotic Frobenius error bounds for both deterministic and stochastic
noise matrices, and applies to matrices that can be exactly or
approximately low rank, and matrices that can be exactly or
approximately sparse. Moreover, for the case of stochastic noise matrices and
the identity observation operator, we establish matching lower bounds on the
minimax error. The sharpness of our predictions is confirmed by numerical
simulations.Comment: 41 pages, 2 figure
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
Rate Optimal Denoising of Simultaneously Sparse and Low Rank Matrices
We study minimax rates for denoising simultaneously sparse and low rank
matrices in high dimensions. We show that an iterative thresholding algorithm
achieves (near) optimal rates adaptively under mild conditions for a large
class of loss functions. Numerical experiments on synthetic datasets also
demonstrate the competitive performance of the proposed method
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