1,199 research outputs found
The typical cell in anisotropic tessellations
The typical cell is a key concept for stochastic-geometry based modeling in
communication networks, as it provides a rigorous framework for describing
properties of a serving zone associated with a component selected at random in
a large network. We consider a setting where network components are located on
a large street network. While earlier investigations were restricted to street
systems without preferred directions, in this paper we derive the distribution
of the typical cell in Manhattan-type systems characterized by a pattern of
horizontal and vertical streets. We explain how the mathematical description
can be turned into a simulation algorithm and provide numerical results
uncovering novel effects when compared to classical isotropic networks.Comment: 7 pages, 7 figure
Multilevel Artificial Neural Network Training for Spatially Correlated Learning
Multigrid modeling algorithms are a technique used to accelerate relaxation
models running on a hierarchy of similar graphlike structures. We introduce and
demonstrate a new method for training neural networks which uses multilevel
methods. Using an objective function derived from a graph-distance metric, we
perform orthogonally-constrained optimization to find optimal prolongation and
restriction maps between graphs. We compare and contrast several methods for
performing this numerical optimization, and additionally present some new
theoretical results on upper bounds of this type of objective function. Once
calculated, these optimal maps between graphs form the core of Multiscale
Artificial Neural Network (MsANN) training, a new procedure we present which
simultaneously trains a hierarchy of neural network models of varying spatial
resolution. Parameter information is passed between members of this hierarchy
according to standard coarsening and refinement schedules from the multiscale
modelling literature. In our machine learning experiments, these models are
able to learn faster than default training, achieving a comparable level of
error in an order of magnitude fewer training examples.Comment: Manuscript (24 pages) and Supplementary Material (4 pages). Updated
January 2019 to reflect new formulation of MsANN structure and new training
procedur
The typical cell in anisotropic tessellations
The typical cell is a key concept for stochastic-geometry based modeling in communication networks, as it provides a rigorous framework for describing properties of a serving zone associated with a component selected at random in a large network. We consider a setting where network components are located on a large street network. While earlier investigations were restricted to street systems without preferred directions, in this paper we derive the distribution of the typical cell in Manhattan-type systems characterized by a pattern of horizontal and vertical streets. We explain how the mathematical description can be turned into a simulation algorithm and provide numerical results uncovering novel effects when compared to classical isotropic networks
Non-isotropic Persistent Homology: Leveraging the Metric Dependency of PH
Persistent Homology is a widely used topological data analysis tool that
creates a concise description of the topological properties of a point cloud
based on a specified filtration. Most filtrations used for persistent homology
depend (implicitly) on a chosen metric, which is typically agnostically chosen
as the standard Euclidean metric on . Recent work has tried to
uncover the 'true' metric on the point cloud using distance-to-measure
functions, in order to obtain more meaningful persistent homology results. Here
we propose an alternative look at this problem: we posit that information on
the point cloud is lost when restricting persistent homology to a single
(correct) distance function. Instead, we show how by varying the distance
function on the underlying space and analysing the corresponding shifts in the
persistence diagrams, we can extract additional topological and geometrical
information. Finally, we numerically show that non-isotropic persistent
homology can extract information on orientation, orientational variance, and
scaling of randomly generated point clouds with good accuracy and conduct some
experiments on real-world data.Comment: 30 pages, 17 figures, comments welcome
Neighbor selection and hitting probability in small-world graphs
Small-world graphs, which combine randomized and structured elements, are
seen as prevalent in nature. Jon Kleinberg showed that in some graphs of this
type it is possible to route, or navigate, between vertices in few steps even
with very little knowledge of the graph itself. In an attempt to understand how
such graphs arise we introduce a different criterion for graphs to be navigable
in this sense, relating the neighbor selection of a vertex to the hitting
probability of routed walks. In several models starting from both discrete and
continuous settings, this can be shown to lead to graphs with the desired
properties. It also leads directly to an evolutionary model for the creation of
similar graphs by the stepwise rewiring of the edges, and we conjecture,
supported by simulations, that these too are navigable.Comment: Published in at http://dx.doi.org/10.1214/07-AAP499 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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