5,565 research outputs found
Multiclass Data Segmentation using Diffuse Interface Methods on Graphs
We present two graph-based algorithms for multiclass segmentation of
high-dimensional data. The algorithms use a diffuse interface model based on
the Ginzburg-Landau functional, related to total variation compressed sensing
and image processing. A multiclass extension is introduced using the Gibbs
simplex, with the functional's double-well potential modified to handle the
multiclass case. The first algorithm minimizes the functional using a convex
splitting numerical scheme. The second algorithm is a uses a graph adaptation
of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates
between diffusion and thresholding. We demonstrate the performance of both
algorithms experimentally on synthetic data, grayscale and color images, and
several benchmark data sets such as MNIST, COIL and WebKB. We also make use of
fast numerical solvers for finding the eigenvectors and eigenvalues of the
graph Laplacian, and take advantage of the sparsity of the matrix. Experiments
indicate that the results are competitive with or better than the current
state-of-the-art multiclass segmentation algorithms.Comment: 14 page
Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain
The sums of components of the ground states of the O(1) loop model on a
cylinder or of the XXZ quantum spin chain at Delta=-1/2 (of size L) are
expressed in terms of combinatorial numbers. The methods include the
introduction of spectral parameters and the use of integrability, a mapping
from size L to L+1, and knot-theoretic skein relations.Comment: final version to be publishe
Ruelle-Perron-Frobenius spectrum for Anosov maps
We extend a number of results from one dimensional dynamics based on spectral
properties of the Ruelle-Perron-Frobenius transfer operator to Anosov
diffeomorphisms on compact manifolds. This allows to develop a direct operator
approach to study ergodic properties of these maps. In particular, we show that
it is possible to define Banach spaces on which the transfer operator is
quasicompact. (Information on the existence of an SRB measure, its smoothness
properties and statistical properties readily follow from such a result.) In
dimension we show that the transfer operator associated to smooth random
perturbations of the map is close, in a proper sense, to the unperturbed
transfer operator. This allows to obtain easily very strong spectral stability
results, which in turn imply spectral stability results for smooth
deterministic perturbations as well. Finally, we are able to implement an Ulam
type finite rank approximation scheme thus reducing the study of the spectral
properties of the transfer operator to a finite dimensional problem.Comment: 58 pages, LaTe
Structure of the conservation laws in integrable spin chains with short range interactions
We present a detailed analysis of the structure of the conservation laws in
quantum integrable chains of the XYZ-type and in the Hubbard model. With the
use of the boost operator, we establish the general form of the XYZ conserved
charges in terms of simple polynomials in spin variables and derive recursion
relations for the relative coefficients of these polynomials. For two submodels
of the XYZ chain - namely the XXX and XY cases, all the charges can be
calculated in closed form. For the XXX case, a simple description of conserved
charges is found in terms of a Catalan tree. This construction is generalized
for the su(M) invariant integrable chain. We also indicate that a quantum
recursive (ladder) operator can be traced back to the presence of a hamiltonian
mastersymmetry of degree one in the classical continuous version of the model.
We show that in the quantum continuous limits of the XYZ model, the ladder
property of the boost operator disappears. For the Hubbard model we demonstrate
the non-existence of a ladder operator. Nevertheless, the general structure of
the conserved charges is indicated, and the expression for the terms linear in
the model's free parameter for all charges is derived in closed form.Comment: 79 pages in plain TeX plus 4 uuencoded figures; (uses harvmac and
epsf
Network Reconstruction from Intrinsic Noise
This paper considers the problem of inferring an unknown network of dynamical
systems driven by unknown, intrinsic, noise inputs. Equivalently we seek to
identify direct causal dependencies among manifest variables only from
observations of these variables. For linear, time-invariant systems of minimal
order, we characterise under what conditions this problem is well posed. We
first show that if the transfer matrix from the inputs to manifest states is
minimum phase, this problem has a unique solution irrespective of the network
topology. This is equivalent to there being only one valid spectral factor (up
to a choice of signs of the inputs) of the output spectral density.
If the assumption of phase-minimality is relaxed, we show that the problem is
characterised by a single Algebraic Riccati Equation (ARE), of dimension
determined by the number of latent states. The number of solutions to this ARE
is an upper bound on the number of solutions for the network. We give necessary
and sufficient conditions for any two dynamical networks to have equal output
spectral density, which can be used to construct all equivalent networks.
Extensive simulations quantify the number of solutions for a range of problem
sizes. For a slightly simpler case, we also provide an algorithm to construct
all equivalent networks from the output spectral density.Comment: 11 pages, submitted to IEEE Transactions on Automatic Contro
Different approaches to community detection
A precise definition of what constitutes a community in networks has remained
elusive. Consequently, network scientists have compared community detection
algorithms on benchmark networks with a particular form of community structure
and classified them based on the mathematical techniques they employ. However,
this comparison can be misleading because apparent similarities in their
mathematical machinery can disguise different reasons for why we would want to
employ community detection in the first place. Here we provide a focused review
of these different motivations that underpin community detection. This
problem-driven classification is useful in applied network science, where it is
important to select an appropriate algorithm for the given purpose. Moreover,
highlighting the different approaches to community detection also delineates
the many lines of research and points out open directions and avenues for
future research.Comment: 14 pages, 2 figures. Written as a chapter for forthcoming Advances in
network clustering and blockmodeling, and based on an extended version of The
many facets of community detection in complex networks, Appl. Netw. Sci. 2: 4
(2017) by the same author
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