70,599 research outputs found
Noise correlation-induced splitting of Kramers' escape rate from a metastable state
A correlation between two noise processes driving the thermally activated
particles in a symmetric triple well potential, may cause a symmetry breaking
and a difference in relative stability of the two side wells with respect to
the middle one. This leads to an asymmetric localization of population and
splitting of Kramers' rate of escape from the middle well, ensuring a
preferential distribution of the products in the course of a parallel reaction
Randomized Dimension Reduction on Massive Data
Scalability of statistical estimators is of increasing importance in modern
applications and dimension reduction is often used to extract relevant
information from data. A variety of popular dimension reduction approaches can
be framed as symmetric generalized eigendecomposition problems. In this paper
we outline how taking into account the low rank structure assumption implicit
in these dimension reduction approaches provides both computational and
statistical advantages. We adapt recent randomized low-rank approximation
algorithms to provide efficient solutions to three dimension reduction methods:
Principal Component Analysis (PCA), Sliced Inverse Regression (SIR), and
Localized Sliced Inverse Regression (LSIR). A key observation in this paper is
that randomization serves a dual role, improving both computational and
statistical performance. This point is highlighted in our experiments on real
and simulated data.Comment: 31 pages, 6 figures, Key Words:dimension reduction, generalized
eigendecompositon, low-rank, supervised, inverse regression, random
projections, randomized algorithms, Krylov subspace method
Boolean functions: noise stability, non-interactive correlation distillation, and mutual information
Let be the noise operator acting on Boolean functions , where is the noise parameter. Given
and fixed mean , which Boolean function has the
largest -th moment ? This question has
close connections with noise stability of Boolean functions, the problem of
non-interactive correlation distillation, and Courtade-Kumar's conjecture on
the most informative Boolean function. In this paper, we characterize
maximizers in some extremal settings, such as low noise (
is close to 0), high noise ( is close to 1/2), as well as
when is large. Analogous results are also established in
more general contexts, such as Boolean functions defined on discrete torus
and the problem of noise stability in a tree
model.Comment: Corrections of some inaccuracie
Gaussian Bounds for Noise Correlation of Functions
In this paper we derive tight bounds on the expected value of products of
{\em low influence} functions defined on correlated probability spaces. The
proofs are based on extending Fourier theory to an arbitrary number of
correlated probability spaces, on a generalization of an invariance principle
recently obtained with O'Donnell and Oleszkiewicz for multilinear polynomials
with low influences and bounded degree and on properties of multi-dimensional
Gaussian distributions. The results derived here have a number of applications
to the theory of social choice in economics, to hardness of approximation in
computer science and to additive combinatorics problems.Comment: Typos and references correcte
Nonclassical statistics of intracavity coupled waveguides: the quantum optical dimer
A model is proposed where two nonlinear waveguides are contained
in a cavity suited for second-harmonic generation. The evanescent wave coupling
between the waveguides is considered as weak, and the interplay between this
coupling and the nonlinear interaction within the waveguides gives rise to
quantum violations of the classical limit. These violations are particularly
strong when two instabilities are competing, where twin-beam behavior is found
as almost complete noise suppression in the difference of the fundamental
intensities. Moreover, close to bistable transitions perfect twin-beam
correlations are seen in the sum of the fundamental intensities, and also the
self-pulsing instability as well as the transition from symmetric to asymmetric
states display nonclassical twin-beam correlations of both fundamental and
second-harmonic intensities. The results are based on the full quantum Langevin
equations derived from the Hamiltonian and including cavity damping effects.
The intensity correlations of the output fields are calculated
semi-analytically using a linearized version of the Langevin equations derived
through the positive-P representation. Confirmation of the analytical results
are obtained by numerical simulations of the nonlinear Langevin equations
derived using the truncated Wigner representation.Comment: 15 pages, 8 figures, submitted to Phys. Rev.
On The Probability of a Rational Outcome for Generalized Social Welfare Functions on Three Alternatives
In [G. Kalai, A Fourier-theoretic Perspective on the Condorcet Paradox and
Arrow's Theorem, Adv. in Appl. Math. 29(3) (2002), pp. 412--426], Kalai
investigated the probability of a rational outcome for a generalized social
welfare function (GSWF) on three alternatives, when the individual preferences
are uniform and independent. In this paper we generalize Kalai's results to a
broader class of distributions of the individual preferences, and obtain new
lower bounds on the probability of a rational outcome in several classes of
GSWFs. In particular, we show that if the GSWF is monotone and balanced and the
distribution of the preferences is uniform, then the probability of a rational
outcome is at least 3/4, proving a conjecture raised by Kalai. The tools used
in the paper are analytic: the Fourier-Walsh expansion of Boolean functions on
the discrete cube, properties of the Bonamie-Beckner noise operator, and the
FKG inequality.Comment: 23 pages, 1 figur
Intrinsically-generated fluctuating activity in excitatory-inhibitory networks
Recurrent networks of non-linear units display a variety of dynamical regimes
depending on the structure of their synaptic connectivity. A particularly
remarkable phenomenon is the appearance of strongly fluctuating, chaotic
activity in networks of deterministic, but randomly connected rate units. How
this type of intrinsi- cally generated fluctuations appears in more realistic
networks of spiking neurons has been a long standing question. To ease the
comparison between rate and spiking networks, recent works investigated the
dynami- cal regimes of randomly-connected rate networks with segregated
excitatory and inhibitory populations, and firing rates constrained to be
positive. These works derived general dynamical mean field (DMF) equations
describing the fluctuating dynamics, but solved these equations only in the
case of purely inhibitory networks. Using a simplified excitatory-inhibitory
architecture in which DMF equations are more easily tractable, here we show
that the presence of excitation qualitatively modifies the fluctuating activity
compared to purely inhibitory networks. In presence of excitation,
intrinsically generated fluctuations induce a strong increase in mean firing
rates, a phenomenon that is much weaker in purely inhibitory networks.
Excitation moreover induces two different fluctuating regimes: for moderate
overall coupling, recurrent inhibition is sufficient to stabilize fluctuations,
for strong coupling, firing rates are stabilized solely by the upper bound
imposed on activity, even if inhibition is stronger than excitation. These
results extend to more general network architectures, and to rate networks
receiving noisy inputs mimicking spiking activity. Finally, we show that
signatures of the second dynamical regime appear in networks of
integrate-and-fire neurons
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