71 research outputs found
Ramified rectilinear polygons: coordinatization by dendrons
Simple rectilinear polygons (i.e. rectilinear polygons without holes or
cutpoints) can be regarded as finite rectangular cell complexes coordinatized
by two finite dendrons. The intrinsic -metric is thus inherited from the
product of the two finite dendrons via an isometric embedding. The rectangular
cell complexes that share this same embedding property are called ramified
rectilinear polygons. The links of vertices in these cell complexes may be
arbitrary bipartite graphs, in contrast to simple rectilinear polygons where
the links of points are either 4-cycles or paths of length at most 3. Ramified
rectilinear polygons are particular instances of rectangular complexes obtained
from cube-free median graphs, or equivalently simply connected rectangular
complexes with triangle-free links. The underlying graphs of finite ramified
rectilinear polygons can be recognized among graphs in linear time by a
Lexicographic Breadth-First-Search. Whereas the symmetry of a simple
rectilinear polygon is very restricted (with automorphism group being a
subgroup of the dihedral group ), ramified rectilinear polygons are
universal: every finite group is the automorphism group of some ramified
rectilinear polygon.Comment: 27 pages, 6 figure
On partitioning multivariate self-affine time series
Given a multivariate time series, possibly of high dimension, with unknown and time-varying joint distribution, it is of interest to be able to completely partition the time series into disjoint, contiguous subseries, each of which has different distributional or pattern attributes from the preceding and succeeding subseries. An additional feature of many time series is that they display self-affinity, so that subseries at one time scale are similar to subseries at another after application of an affine transformation. Such qualities are observed in time series from many disciplines, including biology, medicine, economics, finance, and computer science. This paper defines the relevant multiobjective combinatorial optimization problem with limited assumptions as a biobjective one, and a specialized evolutionary algorithm is presented which finds optimal self-affine time series partitionings with a minimum of choice parameters. The algorithm not only finds partitionings for all possible numbers of partitions given data constraints, but also for self-affinities between these partitionings and some fine-grained partitioning. The resulting set of Pareto-efficient solution sets provides a rich representation of the self-affine properties of a multivariate time series at different locations and time scales
Trees, Tight-Spans and Point Configuration
Tight-spans of metrics were first introduced by Isbell in 1964 and
rediscovered and studied by others, most notably by Dress, who gave them this
name. Subsequently, it was found that tight-spans could be defined for more
general maps, such as directed metrics and distances, and more recently for
diversities. In this paper, we show that all of these tight-spans as well as
some related constructions can be defined in terms of point configurations.
This provides a useful way in which to study these objects in a unified and
systematic way. We also show that by using point configurations we can recover
results concerning one-dimensional tight-spans for all of the maps we consider,
as well as extend these and other results to more general maps such as
symmetric and unsymmetric maps.Comment: 21 pages, 2 figure
Computing the blocks of a quasi-median graph
Quasi-median graphs are a tool commonly used by evolutionary biologists to
visualise the evolution of molecular sequences. As with any graph, a
quasi-median graph can contain cut vertices, that is, vertices whose removal
disconnect the graph. These vertices induce a decomposition of the graph into
blocks, that is, maximal subgraphs which do not contain any cut vertices. Here
we show that the special structure of quasi-median graphs can be used to
compute their blocks without having to compute the whole graph. In particular
we present an algorithm that, for a collection of aligned sequences of
length , can compute the blocks of the associated quasi-median graph
together with the information required to correctly connect these blocks
together in run time , independent of the size of the
sequence alphabet. Our primary motivation for presenting this algorithm is the
fact that the quasi-median graph associated to a sequence alignment must
contain all most parsimonious trees for the alignment, and therefore
precomputing the blocks of the graph has the potential to help speed up any
method for computing such trees.Comment: 17 pages, 2 figure
Learning Probabilistic Logic Programs in Continuous Domains
The field of statistical relational learning aims at unifying logic and
probability to reason and learn from data. Perhaps the most successful paradigm
in the field is probabilistic logic programming: the enabling of stochastic
primitives in logic programming, which is now increasingly seen to provide a
declarative background to complex machine learning applications. While many
systems offer inference capabilities, the more significant challenge is that of
learning meaningful and interpretable symbolic representations from data. In
that regard, inductive logic programming and related techniques have paved much
of the way for the last few decades.
Unfortunately, a major limitation of this exciting landscape is that much of
the work is limited to finite-domain discrete probability distributions.
Recently, a handful of systems have been extended to represent and perform
inference with continuous distributions. The problem, of course, is that
classical solutions for inference are either restricted to well-known
parametric families (e.g., Gaussians) or resort to sampling strategies that
provide correct answers only in the limit. When it comes to learning, moreover,
inducing representations remains entirely open, other than "data-fitting"
solutions that force-fit points to aforementioned parametric families.
In this paper, we take the first steps towards inducing probabilistic logic
programs for continuous and mixed discrete-continuous data, without being
pigeon-holed to a fixed set of distribution families. Our key insight is to
leverage techniques from piecewise polynomial function approximation theory,
yielding a principled way to learn and compositionally construct density
functions. We test the framework and discuss the learned representations.Comment: Accepted at the 2018 KR Workshop on Hybrid Reasoning and Learnin
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