26,867 research outputs found
General CMB and Primordial Bispectrum Estimation I: Mode Expansion, Map-Making and Measures of f_NL
We present a detailed implementation of two bispectrum estimation methods
which can be applied to general non-separable primordial and CMB bispectra. The
method exploits bispectrum mode decompositions on the domain of allowed
wavenumber or multipole values. Concrete mode examples constructed from
symmetrised tetrahedral polynomials are given, demonstrating rapid convergence
for known bispectra. We use these modes to generate simulated CMB maps of high
resolution (l > 2000) given an arbitrary primordial power spectrum and
bispectrum or an arbitrary late-time CMB angular power spectrum and bispectrum.
By extracting coefficients for the same separable basis functions from an
observational map, we are able to present an efficient and general f_NL
estimator for a given theoretical model. The estimator has two versions
comparing theoretical and observed coefficients at either primordial or late
times, thus encompassing a wider range of models, including secondary
anisotropies, lensing and cosmic strings. We provide examples and validation of
both f_NL estimation methods by direct comparison with simulations in a
WMAP-realistic context. In addition, we show how the full bispectrum can be
extracted from observational maps using these mode expansions, irrespective of
the theoretical model under study. We also propose a universal definition of
the bispectrum parameter F_NL for more consistent comparison between
theoretical models. We obtain WMAP5 estimates of f_NL for the equilateral model
from both our primordial and late-time estimators which are consistent with
each other, as well as with results already published in the literature. These
general bispectrum estimation methods should prove useful for the analysis of
nonGaussianity in the Planck satellite data, as well as in other contexts.Comment: 41 pages, 17 figure
Learning Geometric Concepts with Nasty Noise
We study the efficient learnability of geometric concept classes -
specifically, low-degree polynomial threshold functions (PTFs) and
intersections of halfspaces - when a fraction of the data is adversarially
corrupted. We give the first polynomial-time PAC learning algorithms for these
concept classes with dimension-independent error guarantees in the presence of
nasty noise under the Gaussian distribution. In the nasty noise model, an
omniscient adversary can arbitrarily corrupt a small fraction of both the
unlabeled data points and their labels. This model generalizes well-studied
noise models, including the malicious noise model and the agnostic (adversarial
label noise) model. Prior to our work, the only concept class for which
efficient malicious learning algorithms were known was the class of
origin-centered halfspaces.
Specifically, our robust learning algorithm for low-degree PTFs succeeds
under a number of tame distributions -- including the Gaussian distribution
and, more generally, any log-concave distribution with (approximately) known
low-degree moments. For LTFs under the Gaussian distribution, we give a
polynomial-time algorithm that achieves error , where
is the noise rate. At the core of our PAC learning results is an efficient
algorithm to approximate the low-degree Chow-parameters of any bounded function
in the presence of nasty noise. To achieve this, we employ an iterative
spectral method for outlier detection and removal, inspired by recent work in
robust unsupervised learning. Our aforementioned algorithm succeeds for a range
of distributions satisfying mild concentration bounds and moment assumptions.
The correctness of our robust learning algorithm for intersections of
halfspaces makes essential use of a novel robust inverse independence lemma
that may be of broader interest
Grid-free compressive beamforming
The direction-of-arrival (DOA) estimation problem involves the localization
of a few sources from a limited number of observations on an array of sensors,
thus it can be formulated as a sparse signal reconstruction problem and solved
efficiently with compressive sensing (CS) to achieve high-resolution imaging.
On a discrete angular grid, the CS reconstruction degrades due to basis
mismatch when the DOAs do not coincide with the angular directions on the grid.
To overcome this limitation, a continuous formulation of the DOA problem is
employed and an optimization procedure is introduced, which promotes sparsity
on a continuous optimization variable. The DOA estimation problem with
infinitely many unknowns, i.e., source locations and amplitudes, is solved over
a few optimization variables with semidefinite programming. The grid-free CS
reconstruction provides high-resolution imaging even with non-uniform arrays,
single-snapshot data and under noisy conditions as demonstrated on experimental
towed array data.Comment: 14 pages, 8 figures, journal pape
Robust Padé approximation via SVD
Padé approximation is considered from the point of view of robust methods of numerical linear algebra, in particular the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors; a Matlab code is provided. The success of this algorithm suggests that there might be variants of Padé approximation that would be pointwise convergent as the degrees of the numerator and denominator increase to infinity, unlike traditional Padé approximants, which converge only in measure or capacity
A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding
Set-membership estimation is usually formulated in the context of set-valued
calculus and no probabilistic calculations are necessary. In this paper, we
show that set-membership estimation can be equivalently formulated in the
probabilistic setting by employing sets of probability measures. Inference in
set-membership estimation is thus carried out by computing expectations with
respect to the updated set of probability measures P as in the probabilistic
case. In particular, it is shown that inference can be performed by solving a
particular semi-infinite linear programming problem, which is a special case of
the truncated moment problem in which only the zero-th order moment is known
(i.e., the support). By writing the dual of the above semi-infinite linear
programming problem, it is shown that, if the nonlinearities in the measurement
and process equations are polynomial and if the bounding sets for initial
state, process and measurement noises are described by polynomial inequalities,
then an approximation of this semi-infinite linear programming problem can
efficiently be obtained by using the theory of sum-of-squares polynomial
optimization. We then derive a smart greedy procedure to compute a polytopic
outer-approximation of the true membership-set, by computing the minimum-volume
polytope that outer-bounds the set that includes all the means computed with
respect to P
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