9,106 research outputs found
A Computation in a Cellular Automaton Collider Rule 110
A cellular automaton collider is a finite state machine build of rings of
one-dimensional cellular automata. We show how a computation can be performed
on the collider by exploiting interactions between gliders (particles,
localisations). The constructions proposed are based on universality of
elementary cellular automaton rule 110, cyclic tag systems, supercolliders, and
computing on rings.Comment: 39 pages, 32 figures, 3 table
Visual Representations: Defining Properties and Deep Approximations
Visual representations are defined in terms of minimal sufficient statistics
of visual data, for a class of tasks, that are also invariant to nuisance
variability. Minimal sufficiency guarantees that we can store a representation
in lieu of raw data with smallest complexity and no performance loss on the
task at hand. Invariance guarantees that the statistic is constant with respect
to uninformative transformations of the data. We derive analytical expressions
for such representations and show they are related to feature descriptors
commonly used in computer vision, as well as to convolutional neural networks.
This link highlights the assumptions and approximations tacitly assumed by
these methods and explains empirical practices such as clamping, pooling and
joint normalization.Comment: UCLA CSD TR140023, Nov. 12, 2014, revised April 13, 2015, November
13, 2015, February 28, 201
Modular circle quotients and PL limit sets
We say that a collection Gamma of geodesics in the hyperbolic plane H^2 is a
modular pattern if Gamma is invariant under the modular group PSL_2(Z), if
there are only finitely many PSL_2(Z)-equivalence classes of geodesics in
Gamma, and if each geodesic in Gamma is stabilized by an infinite order
subgroup of PSL_2(Z). For instance, any finite union of closed geodesics on the
modular orbifold H^2/PSL_2(Z) lifts to a modular pattern. Let S^1 be the ideal
boundary of H^2. Given two points p,q in S^1 we write pq if p and q are the
endpoints of a geodesic in Gamma. (In particular pp.) We show that is an
equivalence relation. We let Q_Gamma=S^1/ be the quotient space. We call
Q_Gamma a modular circle quotient. In this paper we will give a sense of what
modular circle quotients `look like' by realizing them as limit sets of
piecewise-linear group actionsComment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper1.abs.htm
Efficient Clustering on Riemannian Manifolds: A Kernelised Random Projection Approach
Reformulating computer vision problems over Riemannian manifolds has
demonstrated superior performance in various computer vision applications. This
is because visual data often forms a special structure lying on a lower
dimensional space embedded in a higher dimensional space. However, since these
manifolds belong to non-Euclidean topological spaces, exploiting their
structures is computationally expensive, especially when one considers the
clustering analysis of massive amounts of data. To this end, we propose an
efficient framework to address the clustering problem on Riemannian manifolds.
This framework implements random projections for manifold points via kernel
space, which can preserve the geometric structure of the original space, but is
computationally efficient. Here, we introduce three methods that follow our
framework. We then validate our framework on several computer vision
applications by comparing against popular clustering methods on Riemannian
manifolds. Experimental results demonstrate that our framework maintains the
performance of the clustering whilst massively reducing computational
complexity by over two orders of magnitude in some cases
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