4,950 research outputs found
On a Generalized Matching Problem Arising in Estimating the Eigenvalue Variation of Two Matrices
It is shown that if G is a graph having vertices P1, P2, ..., Pn, Q1, Q2, ..., Qn and satisfying some conditions, then there is a permutation σ of {1, 2, ..., n} such that there is a path, for i = 1, 2, ..., n connecting Pi with Qσ(i) having a length at most {n/2}. This is used to prove a theorem having applications in eigenvalue variation estimation
On the Structure, Covering, and Learning of Poisson Multinomial Distributions
An -Poisson Multinomial Distribution (PMD) is the distribution of the
sum of independent random vectors supported on the set of standard basis vectors in . We prove
a structural characterization of these distributions, showing that, for all
, any -Poisson multinomial random vector is
-close, in total variation distance, to the sum of a discretized
multidimensional Gaussian and an independent -Poisson multinomial random vector. Our structural characterization extends
the multi-dimensional CLT of Valiant and Valiant, by simultaneously applying to
all approximation requirements . In particular, it overcomes
factors depending on and, importantly, the minimum eigenvalue of the
PMD's covariance matrix from the distance to a multidimensional Gaussian random
variable.
We use our structural characterization to obtain an -cover, in
total variation distance, of the set of all -PMDs, significantly
improving the cover size of Daskalakis and Papadimitriou, and obtaining the
same qualitative dependence of the cover size on and as the
cover of Daskalakis and Papadimitriou. We further exploit this structure
to show that -PMDs can be learned to within in total
variation distance from samples, which is
near-optimal in terms of dependence on and independent of . In
particular, our result generalizes the single-dimensional result of Daskalakis,
Diakonikolas, and Servedio for Poisson Binomials to arbitrary dimension.Comment: 49 pages, extended abstract appeared in FOCS 201
A Size-Free CLT for Poisson Multinomials and its Applications
An -Poisson Multinomial Distribution (PMD) is the distribution of the
sum of independent random vectors supported on the set of standard basis vectors in . We show
that any -PMD is -close in total
variation distance to the (appropriately discretized) multi-dimensional
Gaussian with the same first two moments, removing the dependence on from
the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is
obtained by bootstrapping the Valiant-Valiant CLT itself through the structural
characterization of PMDs shown in recent work by Daskalakis, Kamath, and
Tzamos. In turn, our stronger CLT can be leveraged to obtain an efficient PTAS
for approximate Nash equilibria in anonymous games, significantly improving the
state of the art, and matching qualitatively the running time dependence on
and of the best known algorithm for two-strategy anonymous
games. Our new CLT also enables the construction of covers for the set of
-PMDs, which are proper and whose size is shown to be essentially
optimal. Our cover construction combines our CLT with the Shapley-Folkman
theorem and recent sparsification results for Laplacian matrices by Batson,
Spielman, and Srivastava. Our cover size lower bound is based on an algebraic
geometric construction. Finally, leveraging the structural properties of the
Fourier spectrum of PMDs we show that these distributions can be learned from
samples in -time, removing
the quasi-polynomial dependence of the running time on from the
algorithm of Daskalakis, Kamath, and Tzamos.Comment: To appear in STOC 201
Random matrices: The Universality phenomenon for Wigner ensembles
In this paper, we survey some recent progress on rigorously establishing the
universality of various spectral statistics of Wigner Hermitian random matrix
ensembles, focusing on the Four Moment Theorem and its refinements and
applications, including the universality of the sine kernel and the Central
limit theorem of several spectral parameters.
We also take the opportunity here to issue some errata for some of our
previous papers in this area.Comment: 58 page
RascalC: A Jackknife Approach to Estimating Single and Multi-Tracer Galaxy Covariance Matrices
To make use of clustering statistics from large cosmological surveys,
accurate and precise covariance matrices are needed. We present a new code to
estimate large scale galaxy two-point correlation function (2PCF) covariances
in arbitrary survey geometries that, due to new sampling techniques, runs times faster than previous codes, computing finely-binned covariance
matrices with negligible noise in less than 100 CPU-hours. As in previous
works, non-Gaussianity is approximated via a small rescaling of shot-noise in
the theoretical model, calibrated by comparing jackknife survey covariances to
an associated jackknife model. The flexible code, RascalC, has been publicly
released, and automatically takes care of all necessary pre- and
post-processing, requiring only a single input dataset (without a prior 2PCF
model). Deviations between large scale model covariances from a mock survey and
those from a large suite of mocks are found to be be indistinguishable from
noise. In addition, the choice of input mock are shown to be irrelevant for
desired noise levels below mocks. Coupled with its generalization
to multi-tracer data-sets, this shows the algorithm to be an excellent tool for
analysis, reducing the need for large numbers of mock simulations to be
computed.Comment: 29 pages, 8 figures. Accepted by MNRAS. Code is available at
http://github.com/oliverphilcox/RascalC with documentation at
http://rascalc.readthedocs.io
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