60,622 research outputs found
Rates of convergence for empirical spectral measures: a soft approach
Understanding the limiting behavior of eigenvalues of random matrices is the
central problem of random matrix theory. Classical limit results are known for
many models, and there has been significant recent progress in obtaining more
quantitative, non-asymptotic results. In this paper, we describe a systematic
approach to bounding rates of convergence and proving tail inequalities for the
empirical spectral measures of a wide variety of random matrix ensembles. We
illustrate the approach by proving asymptotically almost sure rates of
convergence of the empirical spectral measure in the following ensembles:
Wigner matrices, Wishart matrices, Haar-distributed matrices from the compact
classical groups, powers of Haar matrices, randomized sums and random
compressions of Hermitian matrices, a random matrix model for the Hamiltonians
of quantum spin glasses, and finally the complex Ginibre ensemble. Many of the
results appeared previously and are being collected and described here as
illustrations of the general method; however, some details (particularly in the
Wigner and Wishart cases) are new.
Our approach makes use of techniques from probability in Banach spaces, in
particular concentration of measure and bounds for suprema of stochastic
processes, in combination with more classical tools from matrix analysis,
approximation theory, and Fourier analysis. It is highly flexible, as evidenced
by the broad list of examples. It is moreover based largely on "soft" methods,
and involves little hard analysis
Fast Matrix Multiplication Without Tears: A Constraint Programming Approach
It is known that the multiplication of an matrix with an matrix can be performed using fewer multiplications than what the
naive approach suggests. The most famous instance of this is Strassen's
algorithm for multiplying two matrices in 7 instead of 8
multiplications. This gives rise to the constraint satisfaction problem of fast
matrix multiplication, where a set of multiplication terms must be
chosen and combined such that they satisfy correctness constraints on the
output matrix. Despite its highly combinatorial nature, this problem has not
been exhaustively examined from that perspective, as evidenced for example by
the recent deep reinforcement learning approach of AlphaTensor. In this work,
we propose a simple yet novel Constraint Programming approach to find
non-commutative algorithms for fast matrix multiplication or provide proof of
infeasibility otherwise. We propose a set of symmetry-breaking constraints and
valid inequalities that are particularly helpful in proving infeasibility. On
the feasible side, we find that exploiting solver performance variability in
conjunction with a sparsity-based problem decomposition enables finding
solutions for larger (feasible) instances of fast matrix multiplication. Our
experimental results using CP Optimizer demonstrate that we can find fast
matrix multiplication algorithms for matrices up to in a short
amount of time
New Orders Among Hilbert Space Operators
This article introduces several new relations among related Hilbert space
operators. In particular, we prove some L\"{o}wener partial orderings among and many other related forms, as a
new discussion in this field; where and are the
real and imaginary parts of the operator . Our approach will be based on
proving the positivity of some new matrix operators, where several new forms
for positive matrix operators will be presented as a key tool in obtaining the
other ordering results. As an application, we present some results treating
numerical radius inequalities in a way that extends some known results in this
direction, in addition to some results about the singular values
A Characterization of Lyapunov Inequalities for Stability of Switched Systems
We study stability criteria for discrete-time switched systems and provide a
meta-theorem that characterizes all Lyapunov theorems of a certain canonical
type. For this purpose, we investigate the structure of sets of LMIs that
provide a sufficient condition for stability. Various such conditions have been
proposed in the literature in the past fifteen years. We prove in this note
that a family of languagetheoretic conditions recently provided by the authors
encapsulates all the possible LMI conditions, thus putting a conclusion to this
research effort. As a corollary, we show that it is PSPACE-complete to
recognize whether a particular set of LMIs implies stability of a switched
system. Finally, we provide a geometric interpretation of these conditions, in
terms of existence of an invariant set.Comment: arXiv admin note: text overlap with arXiv:1201.322
Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization
The purpose of this note is to survey a methodology to solve systems of
polynomial equations and inequalities. The techniques we discuss use the
algebra of multivariate polynomials with coefficients over a field to create
large-scale linear algebra or semidefinite programming relaxations of many
kinds of feasibility or optimization questions. We are particularly interested
in problems arising in combinatorial optimization.Comment: 28 pages, survey pape
Joint Spectral Radius and Path-Complete Graph Lyapunov Functions
We introduce the framework of path-complete graph Lyapunov functions for
approximation of the joint spectral radius. The approach is based on the
analysis of the underlying switched system via inequalities imposed among
multiple Lyapunov functions associated to a labeled directed graph. Inspired by
concepts in automata theory and symbolic dynamics, we define a class of graphs
called path-complete graphs, and show that any such graph gives rise to a
method for proving stability of the switched system. This enables us to derive
several asymptotically tight hierarchies of semidefinite programming
relaxations that unify and generalize many existing techniques such as common
quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov
functions. We compare the quality of approximation obtained by certain classes
of path-complete graphs including a family of dual graphs and all path-complete
graphs with two nodes on an alphabet of two matrices. We provide approximation
guarantees for several families of path-complete graphs, such as the De Bruijn
graphs, establishing as a byproduct a constructive converse Lyapunov theorem
for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has
gone through two major rounds of revision. In particular, a section on the
performance of our algorithm on application-motivated problems has been added
and a more comprehensive literature review is presente
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