13,086 research outputs found
A Nonconvex Projection Method for Robust PCA
Robust principal component analysis (RPCA) is a well-studied problem with the
goal of decomposing a matrix into the sum of low-rank and sparse components. In
this paper, we propose a nonconvex feasibility reformulation of RPCA problem
and apply an alternating projection method to solve it. To the best of our
knowledge, we are the first to propose a method that solves RPCA problem
without considering any objective function, convex relaxation, or surrogate
convex constraints. We demonstrate through extensive numerical experiments on a
variety of applications, including shadow removal, background estimation, face
detection, and galaxy evolution, that our approach matches and often
significantly outperforms current state-of-the-art in various ways.Comment: In the proceedings of Thirty-Third AAAI Conference on Artificial
Intelligence (AAAI-19
Non-convex Optimization for Machine Learning
A vast majority of machine learning algorithms train their models and perform
inference by solving optimization problems. In order to capture the learning
and prediction problems accurately, structural constraints such as sparsity or
low rank are frequently imposed or else the objective itself is designed to be
a non-convex function. This is especially true of algorithms that operate in
high-dimensional spaces or that train non-linear models such as tensor models
and deep networks.
The freedom to express the learning problem as a non-convex optimization
problem gives immense modeling power to the algorithm designer, but often such
problems are NP-hard to solve. A popular workaround to this has been to relax
non-convex problems to convex ones and use traditional methods to solve the
(convex) relaxed optimization problems. However this approach may be lossy and
nevertheless presents significant challenges for large scale optimization.
On the other hand, direct approaches to non-convex optimization have met with
resounding success in several domains and remain the methods of choice for the
practitioner, as they frequently outperform relaxation-based techniques -
popular heuristics include projected gradient descent and alternating
minimization. However, these are often poorly understood in terms of their
convergence and other properties.
This monograph presents a selection of recent advances that bridge a
long-standing gap in our understanding of these heuristics. The monograph will
lead the reader through several widely used non-convex optimization techniques,
as well as applications thereof. The goal of this monograph is to both,
introduce the rich literature in this area, as well as equip the reader with
the tools and techniques needed to analyze these simple procedures for
non-convex problems.Comment: The official publication is available from now publishers via
http://dx.doi.org/10.1561/220000005
Guaranteed Rank Minimization via Singular Value Projection
Minimizing the rank of a matrix subject to affine constraints is a
fundamental problem with many important applications in machine learning and
statistics. In this paper we propose a simple and fast algorithm SVP (Singular
Value Projection) for rank minimization with affine constraints (ARMP) and show
that SVP recovers the minimum rank solution for affine constraints that satisfy
the "restricted isometry property" and show robustness of our method to noise.
Our results improve upon a recent breakthrough by Recht, Fazel and Parillo
(RFP07) and Lee and Bresler (LB09) in three significant ways:
1) our method (SVP) is significantly simpler to analyze and easier to
implement,
2) we give recovery guarantees under strictly weaker isometry assumptions
3) we give geometric convergence guarantees for SVP even in presense of noise
and, as demonstrated empirically, SVP is significantly faster on real-world and
synthetic problems.
In addition, we address the practically important problem of low-rank matrix
completion (MCP), which can be seen as a special case of ARMP. We empirically
demonstrate that our algorithm recovers low-rank incoherent matrices from an
almost optimal number of uniformly sampled entries. We make partial progress
towards proving exact recovery and provide some intuition for the strong
performance of SVP applied to matrix completion by showing a more restricted
isometry property. Our algorithm outperforms existing methods, such as those of
\cite{RFP07,CR08,CT09,CCS08,KOM09,LB09}, for ARMP and the matrix-completion
problem by an order of magnitude and is also significantly more robust to
noise.Comment: An earlier version of this paper was submitted to NIPS-2009 on June
5, 200
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