227 research outputs found
ESPRIT for multidimensional general grids
We present a new method for complex frequency estimation in several
variables, extending the classical (1d) ESPRIT-algorithm. We also consider how
to work with data sampled on non-standard domains (i.e going beyond
multi-rectangles)
Local conservation laws in spin-1/2 XY chains with open boundary conditions
We revisit the conserved quantities of the spin-1/2 XY model with open
boundary conditions. In the absence of a transverse field, we find new families
of local charges and show that half of the seeming conservation laws are
conserved only if the number of sites is odd. In even chains the set of
noninteracting charges is abelian, like in the periodic case when the number of
sites is odd. In odd chains the set is doubled and becomes non-abelian, like in
even periodic chains. The dependence of the charges on the parity of the
chain's size undermines the common belief that the thermodynamic limit of
diagonal ensembles exists. We consider also the transverse-field Ising chain,
where the situation is more ordinary. The generalization to the XY model in a
transverse field is not straightforward and we propose a general framework to
carry out similar calculations. We conjecture the form of the bulk part of the
local charges and discuss the emergence of quasilocal conserved quantities. We
provide evidence that in a region of the parameter space there is a reduction
of the number of quasilocal conservation laws invariant under chain inversion.
As a by-product, we study a class of block-Toeplitz-plus-Hankel operators and
identify the conditions that their symbols satisfy in order to commute with a
given block-Toeplitz.Comment: 49 pages, 5 figures, 3 tables; published versio
Bogoliubov modes of a dipolar condensate in a cylindrical trap
The calculation of properties of Bose-Einstein condensates with dipolar
interactions has proven a computationally intensive problem due to the long
range nature of the interactions, limiting the scope of applications. In
particular, the lowest lying Bogoliubov excitations in three dimensional
harmonic trap with cylindrical symmetry were so far computed in an indirect
way, by Fourier analysis of time dependent perturbations, or by approximate
variational methods. We have developed a very fast and accurate numerical
algorithm based on the Hankel transform for calculating properties of dipolar
Bose-Einstein condensates in cylindrically symmetric traps. As an application,
we are able to compute many excitation modes by directly solving the
Bogoliubov-De Gennes equations. We explore the behavior of the excited modes in
different trap geometries. We use these results to calculate the quantum
depletion of the condensate by a combination of a computation of the exact
modes and the use of a local density approximation
Random Matrix Theory and Entanglement in Quantum Spin Chains
We compute the entropy of entanglement in the ground states of a general
class of quantum spin-chain Hamiltonians - those that are related to quadratic
forms of Fermi operators - between the first N spins and the rest of the system
in the limit of infinite total chain length. We show that the entropy can be
expressed in terms of averages over the classical compact groups and establish
an explicit correspondence between the symmetries of a given Hamiltonian and
those characterizing the Haar measure of the associated group. These averages
are either Toeplitz determinants or determinants of combinations of Toeplitz
and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture
are used to compute the leading order asymptotics of the entropy as N -->
infinity . This is shown to grow logarithmically with N. The constant of
proportionality is determined explicitly, as is the next (constant) term in the
asymptotic expansion. The logarithmic growth of the entropy was previously
predicted on the basis of numerical computations and conformal-field-theoretic
calculations. In these calculations the constant of proportionality was
determined in terms of the central charge of the Virasoro algebra. Our results
therefore lead to an explicit formula for this charge. We also show that the
entropy is related to solutions of ordinary differential equations of
Painlev\'e type. In some cases these solutions can be evaluated to all orders
using recurrence relations.Comment: 39 pages, 1 table, no figures. Revised version: minor correction
A Method of Moments for Mixture Models and Hidden Markov Models
Mixture models are a fundamental tool in applied statistics and machine
learning for treating data taken from multiple subpopulations. The current
practice for estimating the parameters of such models relies on local search
heuristics (e.g., the EM algorithm) which are prone to failure, and existing
consistent methods are unfavorable due to their high computational and sample
complexity which typically scale exponentially with the number of mixture
components. This work develops an efficient method of moments approach to
parameter estimation for a broad class of high-dimensional mixture models with
many components, including multi-view mixtures of Gaussians (such as mixtures
of axis-aligned Gaussians) and hidden Markov models. The new method leads to
rigorous unsupervised learning results for mixture models that were not
achieved by previous works; and, because of its simplicity, it offers a viable
alternative to EM for practical deployment
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