239 research outputs found
On the Schoenberg Transformations in Data Analysis: Theory and Illustrations
The class of Schoenberg transformations, embedding Euclidean distances into
higher dimensional Euclidean spaces, is presented, and derived from theorems on
positive definite and conditionally negative definite matrices. Original
results on the arc lengths, angles and curvature of the transformations are
proposed, and visualized on artificial data sets by classical multidimensional
scaling. A simple distance-based discriminant algorithm illustrates the theory,
intimately connected to the Gaussian kernels of Machine Learning
Geometrical Insights for Implicit Generative Modeling
Learning algorithms for implicit generative models can optimize a variety of
criteria that measure how the data distribution differs from the implicit model
distribution, including the Wasserstein distance, the Energy distance, and the
Maximum Mean Discrepancy criterion. A careful look at the geometries induced by
these distances on the space of probability measures reveals interesting
differences. In particular, we can establish surprising approximate global
convergence guarantees for the -Wasserstein distance,even when the
parametric generator has a nonconvex parametrization.Comment: this version fixes a typo in a definitio
Positive definite metric spaces
Magnitude is a numerical invariant of finite metric spaces, recently
introduced by T. Leinster, which is analogous in precise senses to the
cardinality of finite sets or the Euler characteristic of topological spaces.
It has been extended to infinite metric spaces in several a priori distinct
ways. This paper develops the theory of a class of metric spaces, positive
definite metric spaces, for which magnitude is more tractable than in general.
Positive definiteness is a generalization of the classical property of negative
type for a metric space, which is known to hold for many interesting classes of
spaces. It is proved that all the proposed definitions of magnitude coincide
for compact positive definite metric spaces and further results are proved
about the behavior of magnitude as a function of such spaces. Finally, some
facts about the magnitude of compact subsets of l_p^n for p \le 2 are proved,
generalizing results of Leinster for p=1,2, using properties of these spaces
which are somewhat stronger than positive definiteness.Comment: v5: Corrected some misstatements in the last few paragraphs. Updated
reference
Re-ranking Permutation-Based Candidate Sets with the n-Simplex Projection
In the realm of metric search, the permutation-based approaches have shown very good performance in indexing and supporting approximate search on large databases. These methods embed the metric objects into a permutation space where candidate results to a given query can be efficiently identified. Typically, to achieve high effectiveness, the permutation-based result set is refined by directly comparing each candidate object to the query one. Therefore, one drawback of these approaches is that the original dataset needs to be stored and then accessed during the refining step. We propose a refining approach based on a metric embedding, called n-Simplex projection, that can be used on metric spaces meeting the n-point property. The n-Simplex projection provides upper- and lower-bounds of the actual distance, derived using the distances between the data objects and a finite set of pivots. We propose to reuse the distances computed for building the data permutations to derive these bounds and we show how to use them to improve the permutation-based results. Our approach is particularly advantageous for all the cases in which the traditional refining step is too costly, e.g. very large dataset or very expensive metric function
Solving parabolic equations on the unit sphere via Laplace transforms and radial basis functions
We propose a method to construct numerical solutions of parabolic equations
on the unit sphere. The time discretization uses Laplace transforms and
quadrature. The spatial approximation of the solution employs radial basis
functions restricted to the sphere. The method allows us to construct high
accuracy numerical solutions in parallel. We establish error estimates
for smooth and nonsmooth initial data, and describe some numerical experiments.Comment: 26 pages, 1 figur
Splines and Wavelets on Geophysically Relevant Manifolds
Analysis on the unit sphere found many applications in
seismology, weather prediction, astrophysics, signal analysis, crystallography,
computer vision, computerized tomography, neuroscience, and statistics.
In the last two decades, the importance of these and other applications
triggered the development of various tools such as splines and wavelet bases
suitable for the unit spheres , and the
rotation group . Present paper is a summary of some of results of the
author and his collaborators on generalized (average) variational splines and
localized frames (wavelets) on compact Riemannian manifolds. The results are
illustrated by applications to Radon-type transforms on and
.Comment: The final publication is available at http://www.springerlink.co
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