1,698 research outputs found
Eigenvalue Invariance of Inhomogeneous Matrix Products in Distributed Algorithms
This letter establishes a general theorem concerning the eigenvalue invariance of certain inhomogeneous matrix products with respect to changes of individual multiplicands’ orderings. Instead of detailed entries, it is the zero-nonzero structure that matters in determining such eigenvalue invariance. The theorem is then applied in analyzing the convergence rate of a distributed algorithm for solving linear equations over networks modeled by undirected graphs
THE SMALLEST SINGULAR VALUE OF INHOMOGENEOUS SQUARE RANDOM MATRICES
We show that for an random matrix with independent uniformly
anti-concentrated entries, such that , the
smallest singular value of satisfies This extends earlier results
of Rudelson and Vershynin, and Rebrova and Tikhomirov by removing the
assumption of mean zero and identical distribution of the entries across the
matrix, as well as the recent result of Livshyts, where the matrix was required
to have i.i.d. rows. Our model covers "inhomogeneus" matrices allowing
different variances of the entries, as long as the sum of the second moments is
of order .
In the past advances, the assumption of i.i.d. rows was required due to lack
of Littlewood--Offord--type inequalities for weighted sums of non-i.i.d. random
variables. Here, we overcome this problem by introducing the Randomized Least
Common Denominator (RLCD) which allows to study anti-concentration properties
of weighted sums of independent but not identically distributed variables. We
construct efficient nets on the sphere with lattice structure, and show that
the lattice points typically have large RLCD. This allows us to derive strong
anti-concentration properties for the distance between a fixed column of
and the linear span of the remaining columns, and prove the main result
Dynamical Localization of Quantum Walks in Random Environments
The dynamics of a one dimensional quantum walker on the lattice with two
internal degrees of freedom, the coin states, is considered. The discrete time
unitary dynamics is determined by the repeated action of a coin operator in
U(2) on the internal degrees of freedom followed by a one step shift to the
right or left, conditioned on the state of the coin. For a fixed coin operator,
the dynamics is known to be ballistic. We prove that when the coin operator
depends on the position of the walker and is given by a certain i.i.d. random
process, the phenomenon of Anderson localization takes place in its dynamical
form. When the coin operator depends on the time variable only and is
determined by an i.i.d. random process, the averaged motion is known to be
diffusive and we compute the diffusion constants for all moments of the
position
Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
This article reviews recent developments in the theoretical understanding and
the numerical implementation of variational renormalization group methods using
matrix product states and projected entangled pair states.Comment: Review from 200
Polynomial-Time Algorithms for Quadratic Isomorphism of Polynomials: The Regular Case
Let and be
two sets of nonlinear polynomials over
( being a field). We consider the computational problem of finding
-- if any -- an invertible transformation on the variables mapping
to . The corresponding equivalence problem is known as {\tt
Isomorphism of Polynomials with one Secret} ({\tt IP1S}) and is a fundamental
problem in multivariate cryptography. The main result is a randomized
polynomial-time algorithm for solving {\tt IP1S} for quadratic instances, a
particular case of importance in cryptography and somewhat justifying {\it a
posteriori} the fact that {\it Graph Isomorphism} reduces to only cubic
instances of {\tt IP1S} (Agrawal and Saxena). To this end, we show that {\tt
IP1S} for quadratic polynomials can be reduced to a variant of the classical
module isomorphism problem in representation theory, which involves to test the
orthogonal simultaneous conjugacy of symmetric matrices. We show that we can
essentially {\it linearize} the problem by reducing quadratic-{\tt IP1S} to
test the orthogonal simultaneous similarity of symmetric matrices; this latter
problem was shown by Chistov, Ivanyos and Karpinski to be equivalent to finding
an invertible matrix in the linear space of matrices over and to compute the square root in a matrix
algebra. While computing square roots of matrices can be done efficiently using
numerical methods, it seems difficult to control the bit complexity of such
methods. However, we present exact and polynomial-time algorithms for computing
the square root in for various fields (including
finite fields). We then consider \\#{\tt IP1S}, the counting version of {\tt
IP1S} for quadratic instances. In particular, we provide a (complete)
characterization of the automorphism group of homogeneous quadratic
polynomials. Finally, we also consider the more general {\it Isomorphism of
Polynomials} ({\tt IP}) problem where we allow an invertible linear
transformation on the variables \emph{and} on the set of polynomials. A
randomized polynomial-time algorithm for solving {\tt IP} when
is presented. From an algorithmic point
of view, the problem boils down to factoring the determinant of a linear matrix
(\emph{i.e.}\ a matrix whose components are linear polynomials). This extends
to {\tt IP} a result of Kayal obtained for {\tt PolyProj}.Comment: Published in Journal of Complexity, Elsevier, 2015, pp.3
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