3,406 research outputs found
A variational approach to stable principal component pursuit
We introduce a new convex formulation for stable principal component pursuit
(SPCP) to decompose noisy signals into low-rank and sparse representations. For
numerical solutions of our SPCP formulation, we first develop a convex
variational framework and then accelerate it with quasi-Newton methods. We
show, via synthetic and real data experiments, that our approach offers
advantages over the classical SPCP formulations in scalability and practical
parameter selection.Comment: 10 pages, 5 figure
Ensemble Joint Sparse Low Rank Matrix Decomposition for Thermography Diagnosis System
Composite is widely used in the aircraft industry and it is essential for manufacturers to monitor its health and quality. The most commonly found defects of composite are debonds and delamination. Different inner defects with complex irregular shape is difficult to be diagnosed by using conventional thermal imaging methods. In this paper, an ensemble joint sparse low rank matrix decomposition (EJSLRMD) algorithm is proposed by applying the optical pulse thermography (OPT) diagnosis system. The proposed algorithm jointly models the low rank and sparse pattern by using concatenated feature space. In particular, the weak defects information can be separated from strong noise and the resolution contrast of the defects has significantly been improved. Ensemble iterative sparse modelling are conducted to further enhance the weak information as well as reducing the computational cost. In order to show the robustness and efficacy of the model, experiments are conducted to detect the inner debond on multiple carbon fiber reinforced polymer (CFRP) composites. A comparative analysis is presented with general OPT algorithms. Not withstand above, the proposed model has been evaluated on synthetic data and compared with other low rank and sparse matrix decomposition algorithms
Non-Convex Rank Minimization via an Empirical Bayesian Approach
In many applications that require matrix solutions of minimal rank, the
underlying cost function is non-convex leading to an intractable, NP-hard
optimization problem. Consequently, the convex nuclear norm is frequently used
as a surrogate penalty term for matrix rank. The problem is that in many
practical scenarios there is no longer any guarantee that we can correctly
estimate generative low-rank matrices of interest, theoretical special cases
notwithstanding. Consequently, this paper proposes an alternative empirical
Bayesian procedure build upon a variational approximation that, unlike the
nuclear norm, retains the same globally minimizing point estimate as the rank
function under many useful constraints. However, locally minimizing solutions
are largely smoothed away via marginalization, allowing the algorithm to
succeed when standard convex relaxations completely fail. While the proposed
methodology is generally applicable to a wide range of low-rank applications,
we focus our attention on the robust principal component analysis problem
(RPCA), which involves estimating an unknown low-rank matrix with unknown
sparse corruptions. Theoretical and empirical evidence are presented to show
that our method is potentially superior to related MAP-based approaches, for
which the convex principle component pursuit (PCP) algorithm (Candes et al.,
2011) can be viewed as a special case.Comment: 10 pages, 6 figures, UAI 2012 pape
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
Sparse and stable Markowitz portfolios
We consider the problem of portfolio selection within the classical Markowitz
mean-variance framework, reformulated as a constrained least-squares regression
problem. We propose to add to the objective function a penalty proportional to
the sum of the absolute values of the portfolio weights. This penalty
regularizes (stabilizes) the optimization problem, encourages sparse portfolios
(i.e. portfolios with only few active positions), and allows to account for
transaction costs. Our approach recovers as special cases the
no-short-positions portfolios, but does allow for short positions in limited
number. We implement this methodology on two benchmark data sets constructed by
Fama and French. Using only a modest amount of training data, we construct
portfolios whose out-of-sample performance, as measured by Sharpe ratio, is
consistently and significantly better than that of the naive evenly-weighted
portfolio which constitutes, as shown in recent literature, a very tough
benchmark.Comment: Better emphasis of main result, new abstract, new examples and
figures. New appendix with full details of algorithm. 17 pages, 6 figure
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