23,326 research outputs found
Phase Retrieval via Randomized Kaczmarz: Theoretical Guarantees
We consider the problem of phase retrieval, i.e. that of solving systems of
quadratic equations. A simple variant of the randomized Kaczmarz method was
recently proposed for phase retrieval, and it was shown numerically to have a
computational edge over state-of-the-art Wirtinger flow methods. In this paper,
we provide the first theoretical guarantee for the convergence of the
randomized Kaczmarz method for phase retrieval. We show that it is sufficient
to have as many Gaussian measurements as the dimension, up to a constant
factor. Along the way, we introduce a sufficient condition on measurement sets
for which the randomized Kaczmarz method is guaranteed to work. We show that
Gaussian sampling vectors satisfy this property with high probability; this is
proved using a chaining argument coupled with bounds on VC dimension and metric
entropy.Comment: Revised after comments from referee
Polynomial Time and Sample Complexity for Non-Gaussian Component Analysis: Spectral Methods
The problem of Non-Gaussian Component Analysis (NGCA) is about finding a
maximal low-dimensional subspace in so that data points
projected onto follow a non-gaussian distribution. Although this is an
appropriate model for some real world data analysis problems, there has been
little progress on this problem over the last decade.
In this paper, we attempt to address this state of affairs in two ways.
First, we give a new characterization of standard gaussian distributions in
high-dimensions, which lead to effective tests for non-gaussianness. Second, we
propose a simple algorithm, \emph{Reweighted PCA}, as a method for solving the
NGCA problem. We prove that for a general unknown non-gaussian distribution,
this algorithm recovers at least one direction in , with sample and time
complexity depending polynomially on the dimension of the ambient space. We
conjecture that the algorithm actually recovers the entire
Online Stochastic Gradient Descent with Arbitrary Initialization Solves Non-smooth, Non-convex Phase Retrieval
In recent literature, a general two step procedure has been formulated for
solving the problem of phase retrieval. First, a spectral technique is used to
obtain a constant-error initial estimate, following which, the estimate is
refined to arbitrary precision by first-order optimization of a non-convex loss
function. Numerical experiments, however, seem to suggest that simply running
the iterative schemes from a random initialization may also lead to
convergence, albeit at the cost of slightly higher sample complexity. In this
paper, we prove that, in fact, constant step size online stochastic gradient
descent (SGD) converges from arbitrary initializations for the non-smooth,
non-convex amplitude squared loss objective. In this setting, online SGD is
also equivalent to the randomized Kaczmarz algorithm from numerical analysis.
Our analysis can easily be generalized to other single index models. It also
makes use of new ideas from stochastic process theory, including the notion of
a summary state space, which we believe will be of use for the broader field of
non-convex optimization
An asymptotic preserving scheme for kinetic models with singular limit
We propose a new class of asymptotic preserving schemes to solve kinetic
equations with mono-kinetic singular limit. The main idea to deal with the
singularity is to transform the equations by appropriate scalings in velocity.
In particular, we study two biologically related kinetic systems. We derive the
scaling factors and prove that the rescaled solution does not have a singular
limit, under appropriate spatial non-oscillatory assumptions, which can be
verified numerically by a newly developed asymptotic preserving scheme. We set
up a few numerical experiments to demonstrate the accuracy, stability,
efficiency and asymptotic preserving property of the schemes.Comment: 24 pages, 6 figure
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