75,518 research outputs found
A Quasi-Random Approach to Matrix Spectral Analysis
Inspired by the quantum computing algorithms for Linear Algebra problems
[HHL,TaShma] we study how the simulation on a classical computer of this type
of "Phase Estimation algorithms" performs when we apply it to solve the
Eigen-Problem of Hermitian matrices. The result is a completely new, efficient
and stable, parallel algorithm to compute an approximate spectral decomposition
of any Hermitian matrix. The algorithm can be implemented by Boolean circuits
in parallel time with a total cost of Boolean
operations. This Boolean complexity matches the best known rigorous parallel time algorithms, but unlike those algorithms our algorithm is
(logarithmically) stable, so further improvements may lead to practical
implementations.
All previous efficient and rigorous approaches to solve the Eigen-Problem use
randomization to avoid bad condition as we do too. Our algorithm makes further
use of randomization in a completely new way, taking random powers of a unitary
matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian
perturbation and a random polynomial power are sufficient to ensure almost
pairwise independence of the phases is the main technical
contribution of this work. This randomization enables us, given a Hermitian
matrix with well separated eigenvalues, to sample a random eigenvalue and
produce an approximate eigenvector in parallel time and
Boolean complexity. We conjecture that further improvements of
our method can provide a stable solution to the full approximate spectral
decomposition problem with complexity similar to the complexity (up to a
logarithmic factor) of sampling a single eigenvector.Comment: Replacing previous version: parallel algorithm runs in total
complexity and not . However, the depth of the
implementing circuit is : hence comparable to fastest
eigen-decomposition algorithms know
Diffusion in sparse networks: linear to semi-linear crossover
We consider random networks whose dynamics is described by a rate equation,
with transition rates that form a symmetric matrix. The long time
evolution of the system is characterized by a diffusion coefficient . In one
dimension it is well known that can display an abrupt percolation-like
transition from diffusion () to sub-diffusion (D=0). A question arises
whether such a transition happens in higher dimensions. Numerically can be
evaluated using a resistor network calculation, or optionally it can be deduced
from the spectral properties of the system. Contrary to a recent expectation
that is based on a renormalization-group analysis, we deduce that is
finite; suggest an "effective-range-hopping" procedure to evaluate it; and
contrast the results with the linear estimate. The same approach is useful for
the analysis of networks that are described by quasi-one-dimensional sparse
banded matrices.Comment: 13 pages, 4 figures, proofed as publishe
Weighted random--geometric and random--rectangular graphs: Spectral and eigenfunction properties of the adjacency matrix
Within a random-matrix-theory approach, we use the nearest-neighbor energy
level spacing distribution and the entropic eigenfunction localization
length to study spectral and eigenfunction properties (of adjacency
matrices) of weighted random--geometric and random--rectangular graphs. A
random--geometric graph (RGG) considers a set of vertices uniformly and
independently distributed on the unit square, while for a random--rectangular
graph (RRG) the embedding geometry is a rectangle. The RRG model depends on
three parameters: The rectangle side lengths and , the connection
radius , and the number of vertices . We then study in detail the case
which corresponds to weighted RGGs and explore weighted RRGs
characterized by , i.e.~two-dimensional geometries, but also approach
the limit of quasi-one-dimensional wires when . In general we look for
the scaling properties of and as a function of , and .
We find that the ratio , with , fixes the
properties of both RGGs and RRGs. Moreover, when we show that
spectral and eigenfunction properties of weighted RRGs are universal for the
fixed ratio , with .Comment: 8 pages, 6 figure
Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications
In computer vision, many problems such as image segmentation, pixel
labelling, and scene parsing can be formulated as binary quadratic programs
(BQPs). For submodular problems, cuts based methods can be employed to
efficiently solve large-scale problems. However, general nonsubmodular problems
are significantly more challenging to solve. Finding a solution when the
problem is of large size to be of practical interest, however, typically
requires relaxation. Two standard relaxation methods are widely used for
solving general BQPs--spectral methods and semidefinite programming (SDP), each
with their own advantages and disadvantages. Spectral relaxation is simple and
easy to implement, but its bound is loose. Semidefinite relaxation has a
tighter bound, but its computational complexity is high, especially for large
scale problems. In this work, we present a new SDP formulation for BQPs, with
two desirable properties. First, it has a similar relaxation bound to
conventional SDP formulations. Second, compared with conventional SDP methods,
the new SDP formulation leads to a significantly more efficient and scalable
dual optimization approach, which has the same degree of complexity as spectral
methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton
methods, for the dual problem. Both of them are significantly more efficiently
than standard interior-point methods. In practice, the smoothing Newton solver
is faster than the quasi-Newton solver for dense or medium-sized problems,
while the quasi-Newton solver is preferable for large sparse/structured
problems. Our experiments on a few computer vision applications including
clustering, image segmentation, co-segmentation and registration show the
potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern
Analysis and Machine Intelligenc
Optimal Fluctuations and Tail States of non-Hermitian Operators
A statistical field theory is developed to explore the density of states and
spatial profile of `tail states' at the edge of the spectral support of a
general class of disordered non-Hermitian operators. These states, which are
identified with symmetry broken, instanton field configurations of the theory,
are closely related to localized sub-gap states recently identified in
disordered superconductors. By focusing separately on the problems of a quantum
particle propagating in a random imaginary scalar potential, and a random
imaginary vector potential, we discuss the methodology of our approach and the
universality of the results. Finally, we address potential physical
applications of our findings.Comment: 27 pages AMSLaTeX (with LaTeX2e), 12 eps figures (J. Phys. A, to
appear
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