3,436 research outputs found
A Bayesian alternative to mutual information for the hierarchical clustering of dependent random variables
The use of mutual information as a similarity measure in agglomerative
hierarchical clustering (AHC) raises an important issue: some correction needs
to be applied for the dimensionality of variables. In this work, we formulate
the decision of merging dependent multivariate normal variables in an AHC
procedure as a Bayesian model comparison. We found that the Bayesian
formulation naturally shrinks the empirical covariance matrix towards a matrix
set a priori (e.g., the identity), provides an automated stopping rule, and
corrects for dimensionality using a term that scales up the measure as a
function of the dimensionality of the variables. Also, the resulting log Bayes
factor is asymptotically proportional to the plug-in estimate of mutual
information, with an additive correction for dimensionality in agreement with
the Bayesian information criterion. We investigated the behavior of these
Bayesian alternatives (in exact and asymptotic forms) to mutual information on
simulated and real data. An encouraging result was first derived on
simulations: the hierarchical clustering based on the log Bayes factor
outperformed off-the-shelf clustering techniques as well as raw and normalized
mutual information in terms of classification accuracy. On a toy example, we
found that the Bayesian approaches led to results that were similar to those of
mutual information clustering techniques, with the advantage of an automated
thresholding. On real functional magnetic resonance imaging (fMRI) datasets
measuring brain activity, it identified clusters consistent with the
established outcome of standard procedures. On this application, normalized
mutual information had a highly atypical behavior, in the sense that it
systematically favored very large clusters. These initial experiments suggest
that the proposed Bayesian alternatives to mutual information are a useful new
tool for hierarchical clustering
Time Evolution within a Comoving Window: Scaling of signal fronts and magnetization plateaus after a local quench in quantum spin chains
We present a modification of Matrix Product State time evolution to simulate
the propagation of signal fronts on infinite one-dimensional systems. We
restrict the calculation to a window moving along with a signal, which by the
Lieb-Robinson bound is contained within a light cone. Signal fronts can be
studied unperturbed and with high precision for much longer times than on
finite systems. Entanglement inside the window is naturally small, greatly
lowering computational effort. We investigate the time evolution of the
transverse field Ising (TFI) model and of the S=1/2 XXZ antiferromagnet in
their symmetry broken phases after several different local quantum quenches.
In both models, we observe distinct magnetization plateaus at the signal
front for very large times, resembling those previously observed for the
particle density of tight binding (TB) fermions. We show that the normalized
difference to the magnetization of the ground state exhibits similar scaling
behaviour as the density of TB fermions. In the XXZ model there is an
additional internal structure of the signal front due to pairing, and wider
plateaus with tight binding scaling exponents for the normalized excess
magnetization. We also observe parameter dependent interaction effects between
individual plateaus, resulting in a slight spatial compression of the plateau
widths.
In the TFI model, we additionally find that for an initial Jordan-Wigner
domain wall state, the complete time evolution of the normalized excess
longitudinal magnetization agrees exactly with the particle density of TB
fermions.Comment: 10 pages with 5 figures. Appendix with 23 pages, 13 figures and 4
tables. Largely extended and improved versio
A new diagrammatic representation for correlation functions in the in-in formalism
In this paper we provide an alternative method to compute correlation
functions in the in-in formalism, with a modified set of Feynman rules to
compute loop corrections. The diagrammatic expansion is based on an iterative
solution of the equation of motion for the quantum operators with only retarded
propagators, which makes each diagram intrinsically local (whereas in the
standard case locality is the result of several cancellations) and endowed with
a straightforward physical interpretation. While the final result is strictly
equivalent, as a bonus the formulation presented here also contains less graphs
than other diagrammatic approaches to in-in correlation functions. Our method
is particularly suitable for applications to cosmology.Comment: 14 pages, matches the published version. includes a modified version
of axodraw.sty that works with the Revtex4 clas
Exact asymmetric Skyrmion in anisotropic ferromagnet and its helimagnetic application
Topological Skyrmions as intricate spin textures were observed experimentally
in helimagnets on 2d plane. Theoretical foundation of such solitonic states to
appear in pure ferromagnetic model, as exact solutions expressed through any
analytic function, was made long ago by Belavin and Polyakov (BP). We propose
an innovative generalization of the BP solution for an anisotropic ferromagnet,
based on a physically motivated geometric (in-)equality, which takes the exact
Skyrmion to a new class of functions beyond analyticity. The possibility of
stabilizing such metastable states in helimagnets is discussed with the
construction of individual Skyrmion and Skyrmion crystal with asymmetry, likely
to be detected in precision experiments.Comment: 12 pages, latex, 3 figures, published in Nucl Phys B (As Frontiers
article
Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the context of adiabatic vacua and the Lewis-Riesenfeld invariant
We use the method of the Lewis-Riesenfeld invariant to analyze the dynamical
properties of the Mukhanov-Sasaki Hamiltonian and, following this approach,
investigate whether we can obtain possible candidates for initial states in the
context of inflation considering a quasi-de Sitter spacetime. Our main interest
lies in the question to which extent these already well-established methods at
the classical and quantum level for finitely many degrees of freedom can be
generalized to field theory. As our results show, a straightforward
generalization does in general not lead to a unitary operator on Fock space
that implements the corresponding time-dependent canonical transformation
associated with the Lewis-Riesenfeld invariant. The action of this operator can
be rewritten as a time-dependent Bogoliubov transformation and we show that its
generalization to Fock space has to be chosen appropriately in order that the
Shale-Stinespring condition is not violated, where we also compare our results
to already existing ones in the literature. Furthermore, our analysis relates
the Ermakov differential equation that plays the role of an auxiliary equation,
whose solution is necessary to construct the Lewis-Riesenfeld invariant, as
well as the corresponding time-dependent canonical transformation to the
defining differential equation for adiabatic vacua. Therefore, a given solution
of the Ermakov equation directly yields a full solution to the differential
equation for adiabatic vacua involving no truncation at some adiabatic order.
As a consequence, we can interpret our result obtained here as a kind of
non-squeezed Bunch-Davies mode, where the term non-squeezed refers to a
possible residual squeezing that can be involved in the unitary operator for
certain choices of the Bogoliubov map.Comment: 40 pages, 5 figures, minor changes: slightly rewrote the
introduction, extended the discussion on the infrared modes, corrected typos
and added reference
SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls
We propose an equivalence of the partition functions of two different 3d
gauge theories. On one side of the correspondence we consider the partition
function of 3d SL(2,R) Chern-Simons theory on a 3-manifold, obtained as a
punctured Riemann surface times an interval. On the other side we have a
partition function of a 3d N=2 superconformal field theory on S^3, which is
realized as a duality domain wall in a 4d gauge theory on S^4. We sketch the
proof of this conjecture using connections with quantum Liouville theory and
quantum Teichmuller theory, and study in detail the example of the
once-punctured torus. Motivated by these results we advocate a direct
Chern-Simons interpretation of the ingredients of (a generalization of) the
Alday-Gaiotto-Tachikawa relation. We also comment on M5-brane realizations as
well as on possible generalizations of our proposals.Comment: 53+1 pages, 14 figures; v2: typos corrected, references adde
Affinity and Fluctuations in a Mesoscopic Noria
We exhibit the invariance of cycle affinities in finite state Markov
processes under various natural probabilistic constructions, for instance under
conditioning and under a new combinatorial construction that we call ``drag and
drop''. We show that cycle affinities have a natural probabilistic meaning
related to first passage non-equilibrium fluctuation relations that we
establish.Comment: 30 pages, 1 figur
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