45,812 research outputs found
Statistical Compressed Sensing of Gaussian Mixture Models
A novel framework of compressed sensing, namely statistical compressed
sensing (SCS), that aims at efficiently sampling a collection of signals that
follow a statistical distribution, and achieving accurate reconstruction on
average, is introduced. SCS based on Gaussian models is investigated in depth.
For signals that follow a single Gaussian model, with Gaussian or Bernoulli
sensing matrices of O(k) measurements, considerably smaller than the O(k
log(N/k)) required by conventional CS based on sparse models, where N is the
signal dimension, and with an optimal decoder implemented via linear filtering,
significantly faster than the pursuit decoders applied in conventional CS, the
error of SCS is shown tightly upper bounded by a constant times the best k-term
approximation error, with overwhelming probability. The failure probability is
also significantly smaller than that of conventional sparsity-oriented CS.
Stronger yet simpler results further show that for any sensing matrix, the
error of Gaussian SCS is upper bounded by a constant times the best k-term
approximation with probability one, and the bound constant can be efficiently
calculated. For Gaussian mixture models (GMMs), that assume multiple Gaussian
distributions and that each signal follows one of them with an unknown index, a
piecewise linear estimator is introduced to decode SCS. The accuracy of model
selection, at the heart of the piecewise linear decoder, is analyzed in terms
of the properties of the Gaussian distributions and the number of sensing
measurements. A maximum a posteriori expectation-maximization algorithm that
iteratively estimates the Gaussian models parameters, the signals model
selection, and decodes the signals, is presented for GMM-based SCS. In real
image sensing applications, GMM-based SCS is shown to lead to improved results
compared to conventional CS, at a considerably lower computational cost
Constructing smooth potentials of mean force, radial, distribution functions and probability densities from sampled data
In this paper a method of obtaining smooth analytical estimates of
probability densities, radial distribution functions and potentials of mean
force from sampled data in a statistically controlled fashion is presented. The
approach is general and can be applied to any density of a single random
variable. The method outlined here avoids the use of histograms, which require
the specification of a physical parameter (bin size) and tend to give noisy
results. The technique is an extension of the Berg-Harris method [B.A. Berg and
R.C. Harris, Comp. Phys. Comm. 179, 443 (2008)], which is typically inaccurate
for radial distribution functions and potentials of mean force due to a
non-uniform Jacobian factor. In addition, the standard method often requires a
large number of Fourier modes to represent radial distribution functions, which
tends to lead to oscillatory fits. It is shown that the issues of poor sampling
due to a Jacobian factor can be resolved using a biased resampling scheme,
while the requirement of a large number of Fourier modes is mitigated through
an automated piecewise construction approach. The method is demonstrated by
analyzing the radial distribution functions in an energy-discretized water
model. In addition, the fitting procedure is illustrated on three more
applications for which the original Berg-Harris method is not suitable, namely,
a random variable with a discontinuous probability density, a density with long
tails, and the distribution of the first arrival times of a diffusing particle
to a sphere, which has both long tails and short-time structure. In all cases,
the resampled, piecewise analytical fit outperforms the histogram and the
original Berg-Harris method.Comment: 14 pages, 15 figures. To appear in J. Chem. Phy
PieceTimer: A Holistic Timing Analysis Framework Considering Setup/Hold Time Interdependency Using A Piecewise Model
In static timing analysis, clock-to-q delays of flip-flops are considered as
constants. Setup times and hold times are characterized separately and also
used as constants. The characterized delays, setup times and hold times, are
ap- plied in timing analysis independently to verify the perfor- mance of
circuits. In reality, however, clock-to-q delays of flip-flops depend on both
setup and hold times. Instead of being constants, these delays change with
respect to different setup/hold time combinations. Consequently, the simple ab-
straction of setup/hold times and constant clock-to-q delays introduces
inaccuracy in timing analysis. In this paper, we propose a holistic method to
consider the relation between clock-to-q delays and setup/hold time
combinations with a piecewise linear model. The result is more accurate than
that of traditional timing analysis, and the incorporation of the
interdependency between clock-to-q delays, setup times and hold times may also
improve circuit performance.Comment: IEEE/ACM International Conference on Computer-Aided Design (ICCAD),
November 201
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