18 research outputs found
Diversities and the Geometry of Hypergraphs
The embedding of finite metrics in has become a fundamental tool for
both combinatorial optimization and large-scale data analysis. One important
application is to network flow problems in which there is close relation
between max-flow min-cut theorems and the minimal distortion embeddings of
metrics into . Here we show that this theory can be generalized
considerably to encompass Steiner tree packing problems in both graphs and
hypergraphs. Instead of the theory of metrics and minimal distortion
embeddings, the parallel is the theory of diversities recently introduced by
Bryant and Tupper, and the corresponding theory of diversities and
embeddings which we develop here.Comment: 19 pages, no figures. This version: further small correction
Constant distortion embeddings of Symmetric Diversities
Diversities are like metric spaces, except that every finite subset, instead
of just every pair of points, is assigned a value. Just as there is a theory of
minimal distortion embeddings of finite metric spaces into , there is a
similar, yet undeveloped, theory for embedding finite diversities into the
diversity analogue of spaces. In the metric case, it is well known that
an -point metric space can be embedded into with
distortion. For diversities, the optimal distortion is unknown. Here, we
establish the surprising result that symmetric diversities, those in which the
diversity (value) assigned to a set depends only on its cardinality, can be
embedded in with constant distortion.Comment: 14 pages, 3 figure
Parsimonious Labeling
We propose a new family of discrete energy minimization problems, which we
call parsimonious labeling. Specifically, our energy functional consists of
unary potentials and high-order clique potentials. While the unary potentials
are arbitrary, the clique potentials are proportional to the {\em diversity} of
set of the unique labels assigned to the clique. Intuitively, our energy
functional encourages the labeling to be parsimonious, that is, use as few
labels as possible. This in turn allows us to capture useful cues for important
computer vision applications such as stereo correspondence and image denoising.
Furthermore, we propose an efficient graph-cuts based algorithm for the
parsimonious labeling problem that provides strong theoretical guarantees on
the quality of the solution. Our algorithm consists of three steps. First, we
approximate a given diversity using a mixture of a novel hierarchical
Potts model. Second, we use a divide-and-conquer approach for each mixture
component, where each subproblem is solved using an effficient
-expansion algorithm. This provides us with a small number of putative
labelings, one for each mixture component. Third, we choose the best putative
labeling in terms of the energy value. Using both sythetic and standard real
datasets, we show that our algorithm significantly outperforms other graph-cuts
based approaches
Automorphism groups of universal diversities
We prove that the automorphism group of the Urysohn diversity is a universal
Polish group. Furthermore we show that the automorphism group of the rational
Urysohn diversity has ample generics, a dense conjugacy class and that it
embeds densely into the automorphism group of the (full) Urysohn diversity. It
follows that this latter group also has a dense conjugacy class
Geometry and Topology in Memory and Navigation
Okinawa Institute of Science and Technology Graduate UniversityDoctor of PhilosophyGeometry and topology offer rich mathematical worlds and perspectives with which to study and improve our understanding of cognitive function. Here I present the following examples: (1) a functional role for inhibitory diversity in associative memories with graph- ical relationships; (2) improved memory capacity in an associative memory model with setwise connectivity, with implications for glial and dendritic function; (3) safe and effi- cient group navigation among conspecifics using purely local geometric information; and (4) enhancing geometric and topological methods to probe the relations between neural activity and behaviour. In each work, tools and insights from geometry and topology are used in essential ways to gain improved insights or performance. This thesis contributes to our knowledge of the potential computational affordances of biological mechanisms (such as inhibition and setwise connectivity), while also demonstrating new geometric and topological methods and perspectives with which to deepen our understanding of cognitive tasks and their neural representations.doctoral thesi
Faculty Scholarship Celebration 2021
Program and bibliography for Western Carolina University's annual Faculty Scholarship Celebration