| This workshop will look at how to employ data in
estimating or inferring topological properties of bodies when the
data dimension is high. Introductory talks in topology and statistics
will provide students a common base, and then several invited speakers
will examine geometric or topological properties in high dimensions
that can be exploited statistically or computationally. Beyond its
mathematical interest, the workshop will also highlight applications
to object recognition, machine learning and genomics.
||Dans cet atelier, on essayera de voir comment estimer
ou déduire des propriétés topologiques des
corps à partir de l'échantillon de données,
lorsque la taille de cet échantillon est importante. La première
partie comprendra des exposés d'introduction à la
topologie et aux statistiques, afin de fournir aux étudiants
une base commune. Dans un second temps, plusieurs conférenciers
invités examineront quelles sont les propriétés
géométriques ou topologiques en dimensions élevées
qui peuvent être exploitées statistiquement ou numériquement.
Au-delà de son intérêt mathématique,
l'atelier mettra aussi en évidence des applications à
la reconnaissance d'objets, l'intelligence artificielle et la génomique.
Conférenciers invites/ Invited Speakers
Gunnar Carlsson (Stanford U.)
Persistent Homology (slides from talk .pdf
Alexander Gorban (University of Leicester)
How to discover a geometry and topology in a finite dataset by means
of elastic nets .pdf (.ppt format)
Peter Kim (Guelph U.)
Nonparametrics in High Dimensions
Concernant l'inscription et l'appui potentiel aux étudiants
pour les frais de voyage, veuillez contacter
Regarding registration and potential student travel support, contact
Andre Dabrowski (email@example.com),
Les étudiants sont encouragés à présenter
de courts exposés.
Students are encouraged to present short communications.
Paul-Eugène Parent and Barry Jessup
Elements of topology.
Assuming only an undergraduate knowledge of mathematics, we will review
some basic notions in manifolds and algebraic topology. We will also
introduce the concept of an (algebraic) invariant for such objects,
one in particular being the homology. We will recall the construction
of a combinatorial tool to compute this invariant, namely a simplicial
talk .pdf format)
André Dabrowski (Ottawa)
Aspects of statistics.
This session will employ the basic elements from introductory undergraduate
courses on probability theory and statistics as a springboard to the
discussion of more comprehensive results such as Donskers theorem
(Functional Central Limit Theorem) and empirical processes.
Gunnar Carlsson (Stanford)
Algebraic topology is a mathematical formalism which makes precise mathematics
out of certain kinds of intuitive concepts concerning geometrical objects.
These concepts come under the heading of "connectivity information",
i.e. they include the possibility of decomposing the object into disjoint
pieces, the study of holes in the space, the nature of closed loops,
and so on. Until recently, these methods have been restricted
to situations where the space is given in closed form, and where by
hand calculation is feasible. In recent years, methods have been developed
which permit the automatic computation of some of this information in
situations where we are not given complete information about the space,
but only sets sampled from the space. These ideas can be used to study
high dimensional data sets qualitatively in situations where actual
visualization is not possible. We will present this work, and show examples
using some real data sets. We will also show how the ideas can be extended
to study qualitative information which is not a priori topological,
such as the presence of corners and edges, and apply the results to
(slides from talk .pdf format)
Alexander Gorban (Leicester)
How to discover a geometry and topology in a finite dataset by means
of elastic nets
Principal manifolds were introduced in 1989 as lines or surfaces passing
through "the middle" of the data distribution. This intuitive
notion, corresponding to the human brain generalization ability, was
supported by a mathematical notion of selfconsistency: every point of
the principal manifold is a conditional mean of all points that are
projected into this point. Most scientific and industrial applications
of principal manifold methodology were implemented using the SOM (self-organizing
maps) approach, coming from the theory of neural networks. In the lecture,
algorithms for fast construction of approximate principal manifolds
with various topology are presented. These algorithms are based on analogy
of principal manifold and elastic membrane and corresponding variational
The relation between the classical statistics and the data modelling
approaches is discussed. In Introduction, brief review of clustering
algorithms and SOM construction is presented. Further steps, principal
graphs construction and a graph grammar extraction are outlined
Peter Kim (Guelph)
Nonparametrics in high dimensions
This series of talks will investigate the interplay between geometry
and statistics. We will begin with a look at a parametric problem on
the sphere with the data being the directed unit normals of the elliptic
planes of long period cometary orbits. It is the belief by astronomers
that the intrinsic distribution of the directed normals is the spherical
uniform distribution. Nevertheless conventional statistical tests always
reject uniformity if applied directly. Part of the difficulty comes
from the fact that there is considerable selection bias in the observed
directed normals. One can model this selection bias by using properties
of spherical geometry and once this selection bias has been accounted
for, one can no longer reject spherical uniformity of the directed unit
normals of long period cometary orbits.
The second topic discusses a deconvolution problem on the space of 3x3
rotation matrices. The technique involves using the irreducible representations
of rotation matrices. The main result is to show that one can obtain
minimax deconvolution density estimators on the space of 3x3 rotation
matrices. This represents a sufficiently rich example so that one can
extend the theory to compact Lie groups. Some applications are discussed
which include rotational matching in bioinformatics, texture analysis
in physical chemistry, encryption in quantum computing as well as an
application to persistent homology.
The third topic will then be a general approach to what may be termed
as a statistical inverse problem on a Riemannian manifold. Here one
is interested in recovering a transformation of a density function on
a Riemannian manifold. Both rate and sharp minimaxity will be discussed
along with additional examples.
Maia Lesosky (Guelph)
Introduction to Quantum Computing.
Quantum computing has been generating intense interest lately in a large
number of fields. I will introduce the concept of quantum error correction,
particularly a method known as the noiseless subsystems method. Along
the way I will discuss some interesting extensions of classical probability
theory. In addition we will see a nice geometrical interpretation of
one of the key players in quantum computing, the density matrix.
Peter Bubenik (Lausanne)
Persistent homology and directional statistics.
We combine statistical and topological approaches to study data sampled
from densities on spheres. In particular we define a persistent homology
for densities, and calculate barcode and function descriptors for the
homology of various densities on spheres. We use the theory of spacings
to compare different ways of combining statistics and topology to study
very large data sets sampled from the circle.
Ulrich Fahrenberg (Aalborg)
Back to Top
Parallel composition of automata.
We show how parallel composition of higher-dimensional automata (HDA)
can be expressed categorically in the spirit of Winskel & Nielsen.
Employing the notion of computation path introduced by van Glabbeek,
we define a new notion of bisimulation of HDA using open maps. We derive
a connection between computation paths and carrier sequences of dipaths
and show that bisimilarity of HDA can be decided by the use of geometric
techniques. For a mathematical audience, we concentrate more on the
topological aspects, and include material on how equivalence of computation
paths is related to dihomotopy of dipaths