611 research outputs found
Exploiting the structure effectively and efficiently in low rank matrix recovery
Low rank model arises from a wide range of applications, including machine
learning, signal processing, computer algebra, computer vision, and imaging
science. Low rank matrix recovery is about reconstructing a low rank matrix
from incomplete measurements. In this survey we review recent developments on
low rank matrix recovery, focusing on three typical scenarios: matrix sensing,
matrix completion and phase retrieval. An overview of effective and efficient
approaches for the problem is given, including nuclear norm minimization,
projected gradient descent based on matrix factorization, and Riemannian
optimization based on the embedded manifold of low rank matrices. Numerical
recipes of different approaches are emphasized while accompanied by the
corresponding theoretical recovery guarantees
Finding Low-Rank Solutions via Non-Convex Matrix Factorization, Efficiently and Provably
A rank- matrix can be written as a product
, where and . One could exploit this observation in optimization: e.g., consider
the minimization of a convex function over rank- matrices, where the
set of rank- matrices is modeled via the factorization . Though
such parameterization reduces the number of variables, and is more
computationally efficient (of particular interest is the case ), it comes at a cost: becomes a non-convex function w.r.t.
and .
We study such parameterization for optimization of generic convex objectives
, and focus on first-order, gradient descent algorithmic solutions. We
propose the Bi-Factored Gradient Descent (BFGD) algorithm, an efficient
first-order method that operates on the factors. We show that when
is (restricted) smooth, BFGD has local sublinear convergence, and linear
convergence when is both (restricted) smooth and (restricted) strongly
convex. For several key applications, we provide simple and efficient
initialization schemes that provide approximate solutions good enough for the
above convergence results to hold.Comment: 45 page
Dropping Convexity for Faster Semi-definite Optimization
We study the minimization of a convex function over the set of
positive semi-definite matrices, but when the problem is recast as
, with and . We study the performance of gradient descent on ---which we refer to as
Factored Gradient Descent (FGD)---under standard assumptions on the original
function .
We provide a rule for selecting the step size and, with this choice, show
that the local convergence rate of FGD mirrors that of standard gradient
descent on the original : i.e., after steps, the error is for
smooth , and exponentially small in when is (restricted) strongly
convex. In addition, we provide a procedure to initialize FGD for (restricted)
strongly convex objectives and when one only has access to via a
first-order oracle; for several problem instances, such proper initialization
leads to global convergence guarantees.
FGD and similar procedures are widely used in practice for problems that can
be posed as matrix factorization. To the best of our knowledge, this is the
first paper to provide precise convergence rate guarantees for general convex
functions under standard convex assumptions.Comment: 40 page
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
Nonconvex Low-Rank Matrix Recovery with Arbitrary Outliers via Median-Truncated Gradient Descent
Recent work has demonstrated the effectiveness of gradient descent for
directly recovering the factors of low-rank matrices from random linear
measurements in a globally convergent manner when initialized properly.
However, the performance of existing algorithms is highly sensitive in the
presence of outliers that may take arbitrary values. In this paper, we propose
a truncated gradient descent algorithm to improve the robustness against
outliers, where the truncation is performed to rule out the contributions of
samples that deviate significantly from the {\em sample median} of measurement
residuals adaptively in each iteration. We demonstrate that, when initialized
in a basin of attraction close to the ground truth, the proposed algorithm
converges to the ground truth at a linear rate for the Gaussian measurement
model with a near-optimal number of measurements, even when a constant fraction
of the measurements are arbitrarily corrupted. In addition, we propose a new
truncated spectral method that ensures an initialization in the basin of
attraction at slightly higher requirements. We finally provide numerical
experiments to validate the superior performance of the proposed approach.Comment: 30 pages, 3 figure
Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent
We address the rectangular matrix completion problem by lifting the unknown
matrix to a positive semidefinite matrix in higher dimension, and optimizing a
nonconvex objective over the semidefinite factor using a simple gradient
descent scheme. With random
observations of a -incoherent matrix of rank and
condition number , where , the algorithm linearly
converges to the global optimum with high probability
Low-rank Solutions of Linear Matrix Equations via Procrustes Flow
In this paper we study the problem of recovering a low-rank matrix from
linear measurements. Our algorithm, which we call Procrustes Flow, starts from
an initial estimate obtained by a thresholding scheme followed by gradient
descent on a non-convex objective. We show that as long as the measurements
obey a standard restricted isometry property, our algorithm converges to the
unknown matrix at a geometric rate. In the case of Gaussian measurements, such
convergence occurs for a matrix of rank when the number of
measurements exceeds a constant times .Comment: Added new results for general rectangular matrice
How Much Restricted Isometry is Needed In Nonconvex Matrix Recovery?
When the linear measurements of an instance of low-rank matrix recovery
satisfy a restricted isometry property (RIP)---i.e. they are approximately
norm-preserving---the problem is known to contain no spurious local minima, so
exact recovery is guaranteed. In this paper, we show that moderate RIP is not
enough to eliminate spurious local minima, so existing results can only hold
for near-perfect RIP. In fact, counterexamples are ubiquitous: we prove that
every x is the spurious local minimum of a rank-1 instance of matrix recovery
that satisfies RIP. One specific counterexample has RIP constant ,
but causes randomly initialized stochastic gradient descent (SGD) to fail 12%
of the time. SGD is frequently able to avoid and escape spurious local minima,
but this empirical result shows that it can occasionally be defeated by their
existence. Hence, while exact recovery guarantees will likely require a proof
of no spurious local minima, arguments based solely on norm preservation will
only be applicable to a narrow set of nearly-isotropic instances.Comment: 32nd Conference on Neural Information Processing Systems (NIPS 2018
Coordinate Descent Algorithms for Phase Retrieval
Phase retrieval aims at recovering a complex-valued signal from
magnitude-only measurements, which attracts much attention since it has
numerous applications in many disciplines. However, phase recovery involves
solving a system of quadratic equations, indicating that it is a challenging
nonconvex optimization problem. To tackle phase retrieval in an effective and
efficient manner, we apply coordinate descent (CD) such that a single unknown
is solved at each iteration while all other variables are kept fixed. As a
result, only minimization of a univariate quartic polynomial is needed which is
easily achieved by finding the closed-form roots of a cubic equation. Three
computationally simple algorithms referred to as cyclic, randomized and greedy
CDs, based on different updating rules, are devised. It is proved that the
three CDs globally converge to a stationary point of the nonconvex problem, and
specifically, the randomized CD locally converges to the global minimum and
attains exact recovery at a geometric rate with high probability if the sample
size is large enough. The cyclic and randomized CDs are also modified via
minimization of the -regularized quartic polynomial for phase retrieval
of sparse signals. Furthermore, a novel application of the three CDs, namely,
blind equalization in digital communications, is proposed. It is demonstrated
that the CD methodology is superior to the state-of-the-art techniques in terms
of computational efficiency and/or recovery performance
Low-Rank Positive Semidefinite Matrix Recovery from Corrupted Rank-One Measurements
We study the problem of estimating a low-rank positive semidefinite (PSD)
matrix from a set of rank-one measurements using sensing vectors composed of
i.i.d. standard Gaussian entries, which are possibly corrupted by arbitrary
outliers. This problem arises from applications such as phase retrieval,
covariance sketching, quantum space tomography, and power spectrum estimation.
We first propose a convex optimization algorithm that seeks the PSD matrix with
the minimum -norm of the observation residual. The advantage of our
algorithm is that it is free of parameters, therefore eliminating the need for
tuning parameters and allowing easy implementations. We establish that with
high probability, a low-rank PSD matrix can be exactly recovered as soon as the
number of measurements is large enough, even when a fraction of the
measurements are corrupted by outliers with arbitrary magnitudes. Moreover, the
recovery is also stable against bounded noise. With the additional information
of an upper bound of the rank of the PSD matrix, we propose another non-convex
algorithm based on subgradient descent that demonstrates excellent empirical
performance in terms of computational efficiency and accuracy.Comment: 12 pages, 7 figure
- …