523 research outputs found
Localized bases for kernel spaces on the unit sphere
Approximation/interpolation from spaces of positive definite or conditionally
positive definite kernels is an increasingly popular tool for the analysis and
synthesis of scattered data, and is central to many meshless methods. For a set
of scattered sites, the standard basis for such a space utilizes
\emph{globally} supported kernels; computing with it is prohibitively expensive
for large . Easily computable, well-localized bases, with "small-footprint"
basis elements - i.e., elements using only a small number of kernels -- have
been unavailable. Working on \sphere, with focus on the restricted surface
spline kernels (e.g. the thin-plate splines restricted to the sphere), we
construct easily computable, spatially well-localized, small-footprint, robust
bases for the associated kernel spaces. Our theory predicts that each element
of the local basis is constructed by using a combination of only
kernels, which makes the construction computationally
cheap. We prove that the new basis is stable and satisfies polynomial
decay estimates that are stationary with respect to the density of the data
sites, and we present a quasi-interpolation scheme that provides optimal
approximation orders. Although our focus is on , much of the
theory applies to other manifolds - , the rotation group, and so
on. Finally, we construct algorithms to implement these schemes and use them to
conduct numerical experiments, which validate our theory for interpolation
problems on involving over one hundred fifty thousand data
sites.Comment: This article supersedes arXiv:1111.1013 "Better bases for kernel
spaces," which proved existence of better bases for various kernel spaces.
This article treats a smaller class of kernels, but presents an algorithm for
constructing better bases and demonstrates its effectiveness with more
elaborate examples. A quasi-interpolation scheme is introduced that provides
optimal linear convergence rate
On Recursive Edit Distance Kernels with Application to Time Series Classification
This paper proposes some extensions to the work on kernels dedicated to
string or time series global alignment based on the aggregation of scores
obtained by local alignments. The extensions we propose allow to construct,
from classical recursive definition of elastic distances, recursive edit
distance (or time-warp) kernels that are positive definite if some sufficient
conditions are satisfied. The sufficient conditions we end-up with are original
and weaker than those proposed in earlier works, although a recursive
regularizing term is required to get the proof of the positive definiteness as
a direct consequence of the Haussler's convolution theorem. The classification
experiment we conducted on three classical time warp distances (two of which
being metrics), using Support Vector Machine classifier, leads to conclude
that, when the pairwise distance matrix obtained from the training data is
\textit{far} from definiteness, the positive definite recursive elastic kernels
outperform in general the distance substituting kernels for the classical
elastic distances we have tested.Comment: 14 page
Sliced Wasserstein Kernel for Persistence Diagrams
Persistence diagrams (PDs) play a key role in topological data analysis
(TDA), in which they are routinely used to describe topological properties of
complicated shapes. PDs enjoy strong stability properties and have proven their
utility in various learning contexts. They do not, however, live in a space
naturally endowed with a Hilbert structure and are usually compared with
specific distances, such as the bottleneck distance. To incorporate PDs in a
learning pipeline, several kernels have been proposed for PDs with a strong
emphasis on the stability of the RKHS distance w.r.t. perturbations of the PDs.
In this article, we use the Sliced Wasserstein approximation SW of the
Wasserstein distance to define a new kernel for PDs, which is not only provably
stable but also provably discriminative (depending on the number of points in
the PDs) w.r.t. the Wasserstein distance between PDs. We also demonstrate
its practicality, by developing an approximation technique to reduce kernel
computation time, and show that our proposal compares favorably to existing
kernels for PDs on several benchmarks.Comment: Minor modification
Highly Localized RBF Lagrange Functions for Finite Difference Methods on Spheres
The aim of this paper is to show how rapidly decaying RBF Lagrange functions
on the spheres can be used to create effective, stable finite difference
methods based on radial basis functions (RBF-FD). For certain classes of PDEs
this approach leads to precise convergence estimates for stencils which grow
moderately with increasing discretization fineness
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