2,999 research outputs found
Compressed Factorization: Fast and Accurate Low-Rank Factorization of Compressively-Sensed Data
What learning algorithms can be run directly on compressively-sensed data? In
this work, we consider the question of accurately and efficiently computing
low-rank matrix or tensor factorizations given data compressed via random
projections. We examine the approach of first performing factorization in the
compressed domain, and then reconstructing the original high-dimensional
factors from the recovered (compressed) factors. In both the matrix and tensor
settings, we establish conditions under which this natural approach will
provably recover the original factors. While it is well-known that random
projections preserve a number of geometric properties of a dataset, our work
can be viewed as showing that they can also preserve certain solutions of
non-convex, NP-Hard problems like non-negative matrix factorization. We support
these theoretical results with experiments on synthetic data and demonstrate
the practical applicability of compressed factorization on real-world gene
expression and EEG time series datasets.Comment: Updates for ICML'19 camera-read
Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview
Substantial progress has been made recently on developing provably accurate
and efficient algorithms for low-rank matrix factorization via nonconvex
optimization. While conventional wisdom often takes a dim view of nonconvex
optimization algorithms due to their susceptibility to spurious local minima,
simple iterative methods such as gradient descent have been remarkably
successful in practice. The theoretical footings, however, had been largely
lacking until recently.
In this tutorial-style overview, we highlight the important role of
statistical models in enabling efficient nonconvex optimization with
performance guarantees. We review two contrasting approaches: (1) two-stage
algorithms, which consist of a tailored initialization step followed by
successive refinement; and (2) global landscape analysis and
initialization-free algorithms. Several canonical matrix factorization problems
are discussed, including but not limited to matrix sensing, phase retrieval,
matrix completion, blind deconvolution, robust principal component analysis,
phase synchronization, and joint alignment. Special care is taken to illustrate
the key technical insights underlying their analyses. This article serves as a
testament that the integrated consideration of optimization and statistics
leads to fruitful research findings.Comment: Invited overview articl
Matrix product operators and states: NP-hardness and undecidability
Tensor network states constitute an important variational set of quantum
states for numerical studies of strongly correlated systems in condensed-matter
physics, as well as in mathematical physics. This is specifically true for
finitely correlated states or matrix-product operators, designed to capture
mixed states of one-dimensional quantum systems. It is a well-known open
problem to find an efficient algorithm that decides whether a given
matrix-product operator actually represents a physical state that in particular
has no negative eigenvalues. We address and answer this question by showing
that the problem is provably undecidable in the thermodynamic limit and that
the bounded version of the problem is NP-hard in the system size. Furthermore,
we discuss numerous connections between tensor network methods and (seemingly)
different concepts treated before in the literature, such as hidden Markov
models and tensor trains.Comment: 7 pages, 2 figures; published version with improved presentatio
The Why and How of Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) has become a widely used tool for the
analysis of high-dimensional data as it automatically extracts sparse and
meaningful features from a set of nonnegative data vectors. We first illustrate
this property of NMF on three applications, in image processing, text mining
and hyperspectral imaging --this is the why. Then we address the problem of
solving NMF, which is NP-hard in general. We review some standard NMF
algorithms, and also present a recent subclass of NMF problems, referred to as
near-separable NMF, that can be solved efficiently (that is, in polynomial
time), even in the presence of noise --this is the how. Finally, we briefly
describe some problems in mathematics and computer science closely related to
NMF via the nonnegative rank.Comment: 25 pages, 5 figures. Some typos and errors corrected, Section 3.2
reorganize
Quartic First-Order Methods for Low Rank Minimization
We study a generalized nonconvex Burer-Monteiro formulation for low-rank
minimization problems. We use recent results on non-Euclidean first order
methods to provide efficient and scalable algorithms. Our approach uses
geometries induced by quartic kernels on matrix spaces; for unconstrained cases
we introduce a novel family of Gram kernels that considerably improves
numerical performances. Numerical experiments for Euclidean distance matrix
completion and symmetric nonnegative matrix factorization show that our
algorithms scale well and reach state of the art performance when compared to
specialized methods
Harnessing Structures in Big Data via Guaranteed Low-Rank Matrix Estimation
Low-rank modeling plays a pivotal role in signal processing and machine
learning, with applications ranging from collaborative filtering, video
surveillance, medical imaging, to dimensionality reduction and adaptive
filtering. Many modern high-dimensional data and interactions thereof can be
modeled as lying approximately in a low-dimensional subspace or manifold,
possibly with additional structures, and its proper exploitations lead to
significant reduction of costs in sensing, computation and storage. In recent
years, there is a plethora of progress in understanding how to exploit low-rank
structures using computationally efficient procedures in a provable manner,
including both convex and nonconvex approaches. On one side, convex relaxations
such as nuclear norm minimization often lead to statistically optimal
procedures for estimating low-rank matrices, where first-order methods are
developed to address the computational challenges; on the other side, there is
emerging evidence that properly designed nonconvex procedures, such as
projected gradient descent, often provide globally optimal solutions with a
much lower computational cost in many problems. This survey article will
provide a unified overview of these recent advances on low-rank matrix
estimation from incomplete measurements. Attention is paid to rigorous
characterization of the performance of these algorithms, and to problems where
the low-rank matrix have additional structural properties that require new
algorithmic designs and theoretical analysis.Comment: To appear in IEEE Signal Processing Magazin
Optimal Solutions for Sparse Principal Component Analysis
Given a sample covariance matrix, we examine the problem of maximizing the
variance explained by a linear combination of the input variables while
constraining the number of nonzero coefficients in this combination. This is
known as sparse principal component analysis and has a wide array of
applications in machine learning and engineering. We formulate a new
semidefinite relaxation to this problem and derive a greedy algorithm that
computes a full set of good solutions for all target numbers of non zero
coefficients, with total complexity O(n^3), where n is the number of variables.
We then use the same relaxation to derive sufficient conditions for global
optimality of a solution, which can be tested in O(n^3) per pattern. We discuss
applications in subset selection and sparse recovery and show on artificial
examples and biological data that our algorithm does provide globally optimal
solutions in many cases.Comment: Revised journal version. More efficient optimality conditions and new
examples in subset selection and sparse recovery. Original version is in ICML
proceeding
Sparse PCA: Convex Relaxations, Algorithms and Applications
Given a sample covariance matrix, we examine the problem of maximizing the
variance explained by a linear combination of the input variables while
constraining the number of nonzero coefficients in this combination. This is
known as sparse principal component analysis and has a wide array of
applications in machine learning and engineering. Unfortunately, this problem
is also combinatorially hard and we discuss convex relaxation techniques that
efficiently produce good approximate solutions. We then describe several
algorithms solving these relaxations as well as greedy algorithms that
iteratively improve the solution quality. Finally, we illustrate sparse PCA in
several applications, ranging from senate voting and finance to news data.Comment: To appear in "Handbook on Semidefinite, Cone and Polynomial
Optimization", M. Anjos and J.B. Lasserre, editors. This revision includes
ROC curves for greedy algorithm
A polynomial-time algorithm for the ground state of 1D gapped local Hamiltonians
Computing ground states of local Hamiltonians is a fundamental problem in
condensed matter physics. We give the first randomized polynomial-time
algorithm for finding ground states of gapped one-dimensional Hamiltonians: it
outputs an (inverse-polynomial) approximation, expressed as a matrix product
state (MPS) of polynomial bond dimension. The algorithm combines many
ingredients, including recently discovered structural features of gapped 1D
systems, convex programming, insights from classical algorithms for 1D
satisfiability, and new techniques for manipulating and bounding the complexity
of MPS. Our result provides one of the first major classes of Hamiltonians for
which computing ground states is provably tractable despite the exponential
nature of the objects involved.Comment: 17 page
Introduction to Nonnegative Matrix Factorization
In this paper, we introduce and provide a short overview of nonnegative
matrix factorization (NMF). Several aspects of NMF are discussed, namely, the
application in hyperspectral imaging, geometry and uniqueness of NMF solutions,
complexity, algorithms, and its link with extended formulations of polyhedra.
In order to put NMF into perspective, the more general problem class of
constrained low-rank matrix approximation problems is first briefly introduced.Comment: 18 pages, 4 figure
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