1,320,274 research outputs found
Predictive information in Gaussian processes with application to music analysis
This is the author's accepted manuscript of this article. The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-40020-9.Lecture Notes in Computer ScienceLecture Notes in Computer ScienceWe describe an information-theoretic approach to the analysis of sequential data, which emphasises the predictive aspects of perception, and the dynamic process of forming and modifying expectations about an unfolding stream of data, characterising these using a set of process information measures. After reviewing the theoretical foundations and the definition of the predictive information rate, we describe how this can be computed for Gaussian processes, including how the approach can be adpated to non-stationary processes, using an online Bayesian spectral estimation method to compute the Bayesian surprise. We finish with a sample analysis of a recording of Steve Reich’s Drummin
Chasing Puppies: Mobile Beacon Routing on Closed Curves
We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired
by his and others' work on beacon-based routing. Consider a human and a puppy
on a simple closed curve in the plane. The human can walk along the curve at
bounded speed and change direction as desired. The puppy runs with unbounded
speed along the curve as long as the Euclidean straight-line distance to the
human is decreasing, so that it is always at a point on the curve where the
distance is locally minimal. Assuming that the curve is smooth (with some mild
genericity constraints) or a simple polygon, we prove that the human can always
catch the puppy in finite time.Comment: Full version of a SOCG 2021 paper, 28 pages, 27 figure
Geometric Dequantization
Dequantization is a set of rules which turn quantum mechanics (QM) into
classical mechanics (CM). It is not the WKB limit of QM. In this paper we show
that, by extending time to a 3-dimensional "supertime", we can dequantize the
system in the sense of turning the Feynman path integral version of QM into the
functional counterpart of the Koopman-von Neumann operatorial approach to CM.
Somehow this procedure is the inverse of geometric quantization and we present
it in three different polarizations: the Schroedinger, the momentum and the
coherent states ones.Comment: 50+1 pages, Late
Geometric Transitions
The purpose of this paper is to give, on one hand, a mathematical exposition
of the main topological and geometrical properties of geometric transitions, on
the other hand, a quick outline of their principal applications, both in
mathematics and in physics.Comment: 44 page
Geometric recursion
We propose a general theory to construct functorial assignments for a large class of functors
from a certain category of bordered surfaces to a suitable target category of
topological vector spaces. The construction proceeds by successive excisions of
homotopy classes of embedded pairs of pants, and thus by induction on the Euler
characteristic. We provide sufficient conditions to guarantee the infinite sums
appearing in this construction converge. In particular, we can generate mapping
class group invariant vectors . The initial data
for the recursion encode the cases when is a pair of pants or a torus
with one boundary, as well as the "recursion kernels" used for glueing. We give
this construction the name of Geometric Recursion (GR). As a first application,
we demonstrate that our formalism produce a large class of measurable functions
on the moduli space of bordered Riemann surfaces. Under certain conditions, the
functions produced by the geometric recursion can be integrated with respect to
the Weil--Petersson measure on moduli spaces with fixed boundary lengths, and
we show that the integrals satisfy a topological recursion (TR) generalizing
the one of Eynard and Orantin. We establish a generalization of
Mirzakhani--McShane identities, namely that multiplicative statistics of
hyperbolic lengths of multicurves can be computed by GR, and thus their
integrals satisfy TR. As a corollary, we find an interpretation of the
intersection indices of the Chern character of bundles of conformal blocks in
terms of the aforementioned statistics. The theory has however a wider scope
than functions on Teichm\"uller space, which will be explored in subsequent
papers; one expects that many functorial objects in low-dimensional geometry
could be constructed by variants of our new geometric recursion.Comment: 97 pages, 21 figures. v2: misprint corrected. v3: revised and
abridged version, 66 page
Geometric influences
We present a new definition of influences in product spaces of continuous
distributions. Our definition is geometric, and for monotone sets it is
identical with the measure of the boundary with respect to uniform enlargement.
We prove analogs of the Kahn-Kalai-Linial (KKL) and Talagrand's influence sum
bounds for the new definition. We further prove an analog of a result of
Friedgut showing that sets with small "influence sum" are essentially
determined by a small number of coordinates. In particular, we establish the
following tight analog of the KKL bound: for any set in of
Gaussian measure , there exists a coordinate such that the th
geometric influence of the set is at least , where
is a universal constant. This result is then used to obtain an isoperimetric
inequality for the Gaussian measure on and the class of sets
invariant under transitive permutation group of the coordinates.Comment: Published in at http://dx.doi.org/10.1214/11-AOP643 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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