7,880 research outputs found
Fast and accurate con-eigenvalue algorithm for optimal rational approximations
The need to compute small con-eigenvalues and the associated con-eigenvectors
of positive-definite Cauchy matrices naturally arises when constructing
rational approximations with a (near) optimally small error.
Specifically, given a rational function with poles in the unit disk, a
rational approximation with poles in the unit disk may be obtained
from the th con-eigenvector of an Cauchy matrix, where the
associated con-eigenvalue gives the approximation error in the
norm. Unfortunately, standard algorithms do not accurately compute
small con-eigenvalues (and the associated con-eigenvectors) and, in particular,
yield few or no correct digits for con-eigenvalues smaller than the machine
roundoff. We develop a fast and accurate algorithm for computing
con-eigenvalues and con-eigenvectors of positive-definite Cauchy matrices,
yielding even the tiniest con-eigenvalues with high relative accuracy. The
algorithm computes the th con-eigenvalue in operations
and, since the con-eigenvalues of positive-definite Cauchy matrices decay
exponentially fast, we obtain (near) optimal rational approximations in
operations, where is the
approximation error in the norm. We derive error bounds
demonstrating high relative accuracy of the computed con-eigenvalues and the
high accuracy of the unit con-eigenvectors. We also provide examples of using
the algorithm to compute (near) optimal rational approximations of functions
with singularities and sharp transitions, where approximation errors close to
machine precision are obtained. Finally, we present numerical tests on random
(complex-valued) Cauchy matrices to show that the algorithm computes all the
con-eigenvalues and con-eigenvectors with nearly full precision
Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables
Let be a a negative discriminant and let vary over a set of
representatives of the integral equivalence classes of integral binary
quadratic forms of discriminant . We prove an asymptotic formula for for the average over of the number of representations of by an
integral positive definite quaternary quadratic form and obtain results on
averages of Fourier coefficients of linear combinations of Siegel theta series.
We also find an asymptotic bound from below on the number of binary forms of
fixed discriminant which are represented by a given quaternary form. In
particular, we can show that for growing a positive proportion of the
binary quadratic forms of discriminant is represented by the given
quaternary quadratic form.Comment: v5: Some typos correcte
Compressed Sensing over -balls: Minimax Mean Square Error
We consider the compressed sensing problem, where the object x_0 \in \bR^N
is to be recovered from incomplete measurements ; here the
sensing matrix is an random matrix with iid Gaussian entries
and . A popular method of sparsity-promoting reconstruction is
-penalized least-squares reconstruction (aka LASSO, Basis Pursuit).
It is currently popular to consider the strict sparsity model, where the
object is nonzero in only a small fraction of entries. In this paper, we
instead consider the much more broadly applicable -sparsity model,
where is sparse in the sense of having norm bounded by for some fixed .
We study an asymptotic regime in which and both tend to infinity with
limiting ratio , both in the noisy () and
noiseless () cases. Under weak assumptions on , we are able to
precisely evaluate the worst-case asymptotic minimax mean-squared
reconstruction error (AMSE) for penalized least-squares: min over
penalization parameters, max over -sparse objects . We exhibit the
asymptotically least-favorable object (hardest sparse signal to recover) and
the maximin penalization.
Our explicit formulas unexpectedly involve quantities appearing classically
in statistical decision theory. Occurring in the present setting, they reflect
a deeper connection between penalized minimization and scalar soft
thresholding. This connection, which follows from earlier work of the authors
and collaborators on the AMP iterative thresholding algorithm, is carefully
explained.
Our approach also gives precise results under weak- ball coefficient
constraints, as we show here.Comment: 41 pages, 11 pdf figure
On the Power Efficiency of Sensory and Ad Hoc Wireless Networks
We consider the power efficiency of a communications channel, i.e., the maximum bit rate that can be achieved per unit power (energy rate). For additive white Gaussian noise (AWGN) channels, it is well known that power efficiency is attained in the low signal-to-noise ratio (SNR) regime where capacity is proportional to the transmit power. In this paper, we first show that for a random sensory wireless network with n users (nodes) placed in a domain of fixed area, with probability converging to one as n grows, the power efficiency scales at least by a factor of sqrt n. In other words, each user in a wireless channel with n nodes can support the same communication rate as a single-user system, but by expending only 1/(sqrt n) times the energy. Then we look at a random ad hoc network with n relay nodes and r simultaneous transmitter/receiver pairs located in a domain of fixed area. We show that as long as r †sqrt n, we can achieve a power efficiency that scales by a factor of sqrt n. We also give a description of how to achieve these gains
Small deviations for beta ensembles
We establish various small deviation inequalities for the extremal (soft
edge) eigenvalues in the beta-Hermite and beta-Laguerre ensembles. In both
settings, upper bounds on the variance of the largest eigenvalue of the
anticipated order follow immediately
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