1,711 research outputs found
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
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
Large Deviations for Hamiltonian Systems on Intermediate Time Scales
We consider a two-dimensional Hamiltonian system perturbed by a small
diffusion term, whose coefficient is state-dependent and non-degenerate. As a
result, the process consists of the fast motion along the level curves and slow
motion across them. On finite time intervals, the large deviation principle
applies, while on time scales that are inversely proportional to the size of
the perturbation, the averaging principle holds, i.e., the projection of the
process onto the Reeb graph converges to a Markov process. In our paper, we
consider the intermediate time scales and prove the large deviation principle,
with the action functional determined in terms of the averaged process on the
graph
Research on the Formulation and Implementation of Development Plans for Primary and Secondary Schools
The formulation and implementation of development plans for primary and secondary schools is an important task to ensure that schools can adapt to the constantly changing educational environment, provide high-quality education, and meet the needs of students. With the continuous promotion of education reform and the increasing demand for school development, the formulation and implementation of school development plans have become particularly crucial. This study explores the process of formulating and implementing development plans for primary and secondary schools. In addition, it will also explore the future trends and challenges faced by primary and secondary school planning, such as the future direction of educational policies and planning, the role of technology and innovation in planning, the challenges of social change and diversification, and sustainable development and environmental factors. The research aims to provide useful references and suggestions for planners and decision-makers in primary and secondary schools, in order to promote sustainable development of schools and improve educational quality
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