119 research outputs found
An Efficient Parallel Algorithm for Spectral Sparsification of Laplacian and SDDM Matrix Polynomials
For "large" class of continuous probability density functions
(p.d.f.), we demonstrate that for every there is mixture of
discrete Binomial distributions (MDBD) with
distinct Binomial distributions that -approximates a
discretized p.d.f. for all , where
. Also, we give two efficient parallel
algorithms to find such MDBD.
Moreover, we propose a sequential algorithm that on input MDBD with
for that induces a discretized p.d.f. ,
that is either Laplacian or SDDM matrix and parameter ,
outputs in time a spectral
sparsifier of a matrix-polynomial, where
notation hides factors.
This improves the Cheng et al.'s [CCLPT15] algorithm whose run time is
.
Furthermore, our algorithm is parallelizable and runs in work
and depth . Our main algorithmic contribution is to
propose the first efficient parallel algorithm that on input continuous p.d.f.
, matrix as above, outputs a spectral sparsifier of
matrix-polynomial whose coefficients approximate component-wise the discretized
p.d.f. .
Our results yield the first efficient and parallel algorithm that runs in
nearly linear work and poly-logarithmic depth and analyzes the long term
behaviour of Markov chains in non-trivial settings. In addition, we strengthen
the Spielman and Peng's [PS14] parallel SDD solver
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Spectral sparsification
We survey recent literature focused on the following spectral sparsification question: Given an integer n and
> 0, does there exist a function N(n; ) such that for every collection of C1; : : : ;Cm of n n real symmetric
positive semidefinite matrices whose sum is the identity, there exists a weighted subset of size N(n; ) whose
sum has eigenvalues lying between 1 and 1 + ?
We present the algorithms for solving this problem given in [4, 8, 10]. These algorithms obtain N(n; ) =
O(n= 2), which is optimal up to constant factors, through use of the barrier method, a proof technique
involving potential functions which control the locations of the eigenvalues of a matrix under certain matrix
updates.
We then survey the applications of this sparsification result and its proof techniques to graph sparsification
[4, 10], low-rank matrix approximation [8], and estimating the covariance of certain distributions of random
matrices [32, 26]. We end our survey by examining a multivariate generalization of the barrier method used
in Marcus, Spielman, and Srivastava’s recent proof [19] of the Kadison-Singer conjecture
Quantum Speedup for Graph Sparsification, Cut Approximation and Laplacian Solving
Graph sparsification underlies a large number of algorithms, ranging from
approximation algorithms for cut problems to solvers for linear systems in the
graph Laplacian. In its strongest form, "spectral sparsification" reduces the
number of edges to near-linear in the number of nodes, while approximately
preserving the cut and spectral structure of the graph. In this work we
demonstrate a polynomial quantum speedup for spectral sparsification and many
of its applications. In particular, we give a quantum algorithm that, given a
weighted graph with nodes and edges, outputs a classical description of
an -spectral sparsifier in sublinear time
. This contrasts with the optimal classical
complexity . We also prove that our quantum algorithm is optimal
up to polylog-factors. The algorithm builds on a string of existing results on
sparsification, graph spanners, quantum algorithms for shortest paths, and
efficient constructions for -wise independent random strings. Our algorithm
implies a quantum speedup for solving Laplacian systems and for approximating a
range of cut problems such as min cut and sparsest cut.Comment: v2: several small improvements to the text. An extended abstract will
appear in FOCS'20; v3: corrected a minor mistake in Appendix
Space-Time Sampling for Network Observability
Designing sparse sampling strategies is one of the important components in
having resilient estimation and control in networked systems as they make
network design problems more cost-effective due to their reduced sampling
requirements and less fragile to where and when samples are collected. It is
shown that under what conditions taking coarse samples from a network will
contain the same amount of information as a more finer set of samples. Our goal
is to estimate initial condition of linear time-invariant networks using a set
of noisy measurements. The observability condition is reformulated as the frame
condition, where one can easily trace location and time stamps of each sample.
We compare estimation quality of various sampling strategies using estimation
measures, which depend on spectrum of the corresponding frame operators. Using
properties of the minimal polynomial of the state matrix, deterministic and
randomized methods are suggested to construct observability frames. Intrinsic
tradeoffs assert that collecting samples from fewer subsystems dictates taking
more samples (in average) per subsystem. Three scalable algorithms are
developed to generate sparse space-time sampling strategies with explicit error
bounds.Comment: Submitted to IEEE TAC (Revised Version
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
Singular Value Approximation and Sparsifying Random Walks on Directed Graphs
In this paper, we introduce a new, spectral notion of approximation between
directed graphs, which we call singular value (SV) approximation.
SV-approximation is stronger than previous notions of spectral approximation
considered in the literature, including spectral approximation of Laplacians
for undirected graphs (Spielman Teng STOC 2004), standard approximation for
directed graphs (Cohen et. al. STOC 2017), and unit-circle approximation for
directed graphs (Ahmadinejad et. al. FOCS 2020). Further, SV approximation
enjoys several useful properties not possessed by previous notions of
approximation, e.g., it is preserved under products of random-walk matrices and
bounded matrices.
We provide a nearly linear-time algorithm for SV-sparsifying (and hence
UC-sparsifying) Eulerian directed graphs, as well as -step random walks
on such graphs, for any . Combined with the Eulerian
scaling algorithms of (Cohen et. al. FOCS 2018), given an arbitrary (not
necessarily Eulerian) directed graph and a set of vertices, we can
approximate the stationary probability mass of the cut in an
-step random walk to within a multiplicative error of
and an additive error of in nearly
linear time. As a starting point for these results, we provide a simple
black-box reduction from SV-sparsifying Eulerian directed graphs to
SV-sparsifying undirected graphs; such a directed-to-undirected reduction was
not known for previous notions of spectral approximation.Comment: FOCS 202
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