57,519 research outputs found
Schatten- Quasi-Norm Regularized Matrix Optimization via Iterative Reweighted Singular Value Minimization
In this paper we study general Schatten- quasi-norm (SPQN) regularized
matrix minimization problems. In particular, we first introduce a class of
first-order stationary points for them, and show that the first-order
stationary points introduced in [11] for an SPQN regularized
minimization problem are equivalent to those of an SPQN regularized
minimization reformulation. We also show that any local minimizer of the SPQN
regularized matrix minimization problems must be a first-order stationary
point. Moreover, we derive lower bounds for nonzero singular values of the
first-order stationary points and hence also of the local minimizers of the
SPQN regularized matrix minimization problems. The iterative reweighted
singular value minimization (IRSVM) methods are then proposed to solve these
problems, whose subproblems are shown to have a closed-form solution. In
contrast to the analogous methods for the SPQN regularized
minimization problems, the convergence analysis of these methods is
significantly more challenging. We develop a novel approach to establishing the
convergence of these methods, which makes use of the expression of a specific
solution of their subproblems and avoids the intricate issue of finding the
explicit expression for the Clarke subdifferential of the objective of their
subproblems. In particular, we show that any accumulation point of the sequence
generated by the IRSVM methods is a first-order stationary point of the
problems. Our computational results demonstrate that the IRSVM methods
generally outperform some recently developed state-of-the-art methods in terms
of solution quality and/or speed.Comment: This paper has been withdrawn by the author due to major revision and
correction
Risk hull method and regularization by projections of ill-posed inverse problems
We study a standard method of regularization by projections of the linear
inverse problem , where is a white Gaussian noise,
and is a known compact operator with singular values converging to zero
with polynomial decay. The unknown function is recovered by a projection
method using the singular value decomposition of . The bandwidth choice of
this projection regularization is governed by a data-driven procedure which is
based on the principle of risk hull minimization. We provide nonasymptotic
upper bounds for the mean square risk of this method and we show, in
particular, that in numerical simulations this approach may substantially
improve the classical method of unbiased risk estimation.Comment: Published at http://dx.doi.org/10.1214/009053606000000542 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Robust Principal Component Pursuit via Inexact Alternating Minimization on Matrix Manifolds
Robust principal component pursuit (RPCP) refers to a decomposition of a data matrix into a low-rank component and a sparse component. In this work, instead of invoking a convex-relaxation model based on the nuclear norm and the `1 -norm as is typically done in this context, RPCP is solved by considering a least-squares problem subject to rank and cardinality constraints. An inexact alternating minimization scheme, with guaranteed global convergence, is employed to solve the resulting constrained minimization problem. In particular, the low-rank matrix subproblem is resolved inexactly by a tailored Riemannian optimization technique, which favorably avoids singular value decompositions in full dimen- sion. For the overall method, a corresponding q-linear convergence theory is established. The numerical experiments show that the newly proposed method compares competitively with a popular convex-relaxation based approach.Peer Reviewe
Scalable Low-Rank Tensor Learning for Spatiotemporal Traffic Data Imputation
Missing value problem in spatiotemporal traffic data has long been a
challenging topic, in particular for large-scale and high-dimensional data with
complex missing mechanisms and diverse degrees of missingness. Recent studies
based on tensor nuclear norm have demonstrated the superiority of tensor
learning in imputation tasks by effectively characterizing the complex
correlations/dependencies in spatiotemporal data. However, despite the
promising results, these approaches do not scale well to large data tensors. In
this paper, we focus on addressing the missing data imputation problem for
large-scale spatiotemporal traffic data. To achieve both high accuracy and
efficiency, we develop a scalable tensor learning model -- Low-Tubal-Rank
Smoothing Tensor Completion (LSTC-Tubal) -- based on the existing framework of
Low-Rank Tensor Completion, which is well-suited for spatiotemporal traffic
data that is characterized by multidimensional structure of location
time of day day. In particular, the proposed LSTC-Tubal model involves
a scalable tensor nuclear norm minimization scheme by integrating linear
unitary transformation. Therefore, tensor nuclear norm minimization can be
solved by singular value thresholding on the transformed matrix of each day
while the day-to-day correlation can be effectively preserved by the unitary
transform matrix. We compare LSTC-Tubal with state-of-the-art baseline models,
and find that LSTC-Tubal can achieve competitive accuracy with a significantly
lower computational cost. In addition, the LSTC-Tubal will also benefit other
tasks in modeling large-scale spatiotemporal traffic data, such as
network-level traffic forecasting
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
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
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