3,071 research outputs found
Revisiting the Nystrom Method for Improved Large-Scale Machine Learning
We reconsider randomized algorithms for the low-rank approximation of
symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel
matrices that arise in data analysis and machine learning applications. Our
main results consist of an empirical evaluation of the performance quality and
running time of sampling and projection methods on a diverse suite of SPSD
matrices. Our results highlight complementary aspects of sampling versus
projection methods; they characterize the effects of common data preprocessing
steps on the performance of these algorithms; and they point to important
differences between uniform sampling and nonuniform sampling methods based on
leverage scores. In addition, our empirical results illustrate that existing
theory is so weak that it does not provide even a qualitative guide to
practice. Thus, we complement our empirical results with a suite of worst-case
theoretical bounds for both random sampling and random projection methods.
These bounds are qualitatively superior to existing bounds---e.g. improved
additive-error bounds for spectral and Frobenius norm error and relative-error
bounds for trace norm error---and they point to future directions to make these
algorithms useful in even larger-scale machine learning applications.Comment: 60 pages, 15 color figures; updated proof of Frobenius norm bounds,
added comparison to projection-based low-rank approximations, and an analysis
of the power method applied to SPSD sketche
Synchronization of uncoupled oscillators by common gamma impulses: from phase locking to noise-induced synchronization
Nonlinear oscillators can mutually synchronize when they are driven by common
external impulses. Two important scenarios are (i) synchronization resulting
from phase locking of each oscillator to regular periodic impulses and (ii)
noise-induced synchronization caused by Poisson random impulses, but their
difference has not been fully quantified. Here we analyze a pair of uncoupled
oscillators subject to common random impulses with gamma-distributed intervals,
which can be smoothly interpolated between regular periodic and random Poisson
impulses. Their dynamics are charac- terized by phase distributions, frequency
detuning, Lyapunov exponents, and information-theoretic measures, which clearly
reveal the differences between the two synchronization scenarios.Comment: 18 page
Phase Retrieval From Binary Measurements
We consider the problem of signal reconstruction from quadratic measurements
that are encoded as +1 or -1 depending on whether they exceed a predetermined
positive threshold or not. Binary measurements are fast to acquire and
inexpensive in terms of hardware. We formulate the problem of signal
reconstruction using a consistency criterion, wherein one seeks to find a
signal that is in agreement with the measurements. To enforce consistency, we
construct a convex cost using a one-sided quadratic penalty and minimize it
using an iterative accelerated projected gradient-descent (APGD) technique. The
PGD scheme reduces the cost function in each iteration, whereas incorporating
momentum into PGD, notwithstanding the lack of such a descent property,
exhibits faster convergence than PGD empirically. We refer to the resulting
algorithm as binary phase retrieval (BPR). Considering additive white noise
contamination prior to quantization, we also derive the Cramer-Rao Bound (CRB)
for the binary encoding model. Experimental results demonstrate that the BPR
algorithm yields a signal-to- reconstruction error ratio (SRER) of
approximately 25 dB in the absence of noise. In the presence of noise prior to
quantization, the SRER is within 2 to 3 dB of the CRB
Timescale effect estimation in time-series studies of air pollution and health: A Singular Spectrum Analysis approach
A wealth of epidemiological data suggests an association between
mortality/morbidity from pulmonary and cardiovascular adverse events and air
pollution, but uncertainty remains as to the extent implied by those
associations although the abundance of the data. In this paper we describe an
SSA (Singular Spectrum Analysis) based approach in order to decompose the
time-series of particulate matter concentration into a set of exposure
variables, each one representing a different timescale. We implement our
methodology to investigate both acute and long-term effects of
exposure on morbidity from respiratory causes within the urban area of Bari,
Italy.Comment: Published in at http://dx.doi.org/10.1214/07-EJS123 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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