16 research outputs found
Beyond pairwise clustering
We consider the problem of clustering in domains where the affinity relations are not dyadic (pairwise), but rather triadic, tetradic or higher. The problem is an instance of the hypergraph partitioning problem. We propose a two-step algorithm for solving this problem. In the first step we use a novel scheme to approximate the hypergraph using a weighted graph. In the second step a spectral partitioning algorithm is used to partition the vertices of this graph. The algorithm is capable of handling hyperedges of all orders including order two, thus incorporating information of all orders simultaneously. We present a theoretical analysis that relates our algorithm to an existing hypergraph partitioning algorithm and explain the reasons for its superior performance. We report the performance of our algorithm on a variety of computer vision problems and compare it to several existing hypergraph partitioning algorithms
Recognizing Treelike k-Dissimilarities
A k-dissimilarity D on a finite set X, |X| >= k, is a map from the set of
size k subsets of X to the real numbers. Such maps naturally arise from
edge-weighted trees T with leaf-set X: Given a subset Y of X of size k, D(Y) is
defined to be the total length of the smallest subtree of T with leaf-set Y .
In case k = 2, it is well-known that 2-dissimilarities arising in this way can
be characterized by the so-called "4-point condition". However, in case k > 2
Pachter and Speyer recently posed the following question: Given an arbitrary
k-dissimilarity, how do we test whether this map comes from a tree? In this
paper, we provide an answer to this question, showing that for k >= 3 a
k-dissimilarity on a set X arises from a tree if and only if its restriction to
every 2k-element subset of X arises from some tree, and that 2k is the least
possible subset size to ensure that this is the case. As a corollary, we show
that there exists a polynomial-time algorithm to determine when a
k-dissimilarity arises from a tree. We also give a 6-point condition for
determining when a 3-dissimilarity arises from a tree, that is similar to the
aforementioned 4-point condition.Comment: 18 pages, 4 figure
Lassoing and corraling rooted phylogenetic trees
The construction of a dendogram on a set of individuals is a key component of
a genomewide association study. However even with modern sequencing
technologies the distances on the individuals required for the construction of
such a structure may not always be reliable making it tempting to exclude them
from an analysis. This, in turn, results in an input set for dendogram
construction that consists of only partial distance information which raises
the following fundamental question. For what subset of its leaf set can we
reconstruct uniquely the dendogram from the distances that it induces on that
subset. By formalizing a dendogram in terms of an edge-weighted, rooted
phylogenetic tree on a pre-given finite set X with |X|>2 whose edge-weighting
is equidistant and a set of partial distances on X in terms of a set L of
2-subsets of X, we investigate this problem in terms of when such a tree is
lassoed, that is, uniquely determined by the elements in L. For this we
consider four different formalizations of the idea of "uniquely determining"
giving rise to four distinct types of lassos. We present characterizations for
all of them in terms of the child-edge graphs of the interior vertices of such
a tree. Our characterizations imply in particular that in case the tree in
question is binary then all four types of lasso must coincide
The Mathematics of Ivo Rosenberg
International audienceThis paper is dedicated to the memory of the distinguished scholar and friend Professor I.G .Rosenberg. We survey some of his most well known and not so known results, as well as present some new ones related to the study of maximal partial clones and their intersections
On representations of permutations groups as isometry groups of n-semimetric spaces
We prove that every finite permutation group can be represented as the isometry group of some n-semimetric space. We show that if a finite permutation group can be realized as the isometry group of some n-semimetric space then this permutation group can be represented as the isometry group of some (n+1)-semimetric space. The notion of the semimetric rank of a permutation group is introduced
Узагальнений біотопний простір
У статті введено узагальнення біотопного простору
If d is super-metric, then d/(1+d) is super-metric
If a function d is metric, a well-known result is that d/(1 + d) is also metric. We consider m-ary analogs of the binary notion of semimetric, called hemi-metrics and super-metrics. The metrics are totally symmetric maps from Xm+1 into R≥0. It is shown that, if d is supermetric, then d/(1 + d) is also super-metri
Three-way symbolic tree-maps and ultrametrics
Three-way dissimilarities are a generalization of (two-way) dissimilarities which can be used to indicate the lack of homogeneity or resemblance between any three objects. Such maps have applications in cluster analysis and have been used in areas such as psychology and phylogenetics, where three-way data tables can arise. Special examples of such dissimilarities are three-way tree-metrics and ultrametrics, which arise from leaf-labelled trees with edges labelled by positive real numbers. Here we consider three-way maps which arise from leaf-labelled trees where instead the interior vertices are labelled by an arbitrary set of values. For unrooted trees, we call such maps three-way symbolic tree-maps; for rooted trees, we call them three-way symbolic ultrametrics since they can be considered as a generalization of the (two-way) symbolic ultrametrics of Bocker and Dress. We show that, as with two- and three-way tree-metrics and ultrametrics, three-way symbolic tree-maps and ultrametrics can be characterized via certain k-point conditions. In the unrooted case, our characterization is mathematically equivalent to one presented by Gurvich for a certain class of edge-labelled hypergraphs. We also show that it can be decided whether or not an arbitrary three-way symbolic map is a tree-map or a symbolic ultrametric using a triplet-based approach that relies on the so-called BUILD algorithm for deciding when a set of 3-leaved trees or triplets can be displayed by a single tree. We envisage that our results will be useful in developing new approaches and algorithms for understanding 3-way data, especially within the area of phylogenetics