3,983 research outputs found
Many-to-Many Graph Matching: a Continuous Relaxation Approach
Graphs provide an efficient tool for object representation in various
computer vision applications. Once graph-based representations are constructed,
an important question is how to compare graphs. This problem is often
formulated as a graph matching problem where one seeks a mapping between
vertices of two graphs which optimally aligns their structure. In the classical
formulation of graph matching, only one-to-one correspondences between vertices
are considered. However, in many applications, graphs cannot be matched
perfectly and it is more interesting to consider many-to-many correspondences
where clusters of vertices in one graph are matched to clusters of vertices in
the other graph. In this paper, we formulate the many-to-many graph matching
problem as a discrete optimization problem and propose an approximate algorithm
based on a continuous relaxation of the combinatorial problem. We compare our
method with other existing methods on several benchmark computer vision
datasets.Comment: 1
Tree Edit Distance Learning via Adaptive Symbol Embeddings
Metric learning has the aim to improve classification accuracy by learning a
distance measure which brings data points from the same class closer together
and pushes data points from different classes further apart. Recent research
has demonstrated that metric learning approaches can also be applied to trees,
such as molecular structures, abstract syntax trees of computer programs, or
syntax trees of natural language, by learning the cost function of an edit
distance, i.e. the costs of replacing, deleting, or inserting nodes in a tree.
However, learning such costs directly may yield an edit distance which violates
metric axioms, is challenging to interpret, and may not generalize well. In
this contribution, we propose a novel metric learning approach for trees which
we call embedding edit distance learning (BEDL) and which learns an edit
distance indirectly by embedding the tree nodes as vectors, such that the
Euclidean distance between those vectors supports class discrimination. We
learn such embeddings by reducing the distance to prototypical trees from the
same class and increasing the distance to prototypical trees from different
classes. In our experiments, we show that BEDL improves upon the
state-of-the-art in metric learning for trees on six benchmark data sets,
ranging from computer science over biomedical data to a natural-language
processing data set containing over 300,000 nodes.Comment: Paper at the International Conference of Machine Learning (2018),
2018-07-10 to 2018-07-15 in Stockholm, Swede
Convex Graph Invariant Relaxations For Graph Edit Distance
The edit distance between two graphs is a widely used measure of similarity
that evaluates the smallest number of vertex and edge deletions/insertions
required to transform one graph to another. It is NP-hard to compute in
general, and a large number of heuristics have been proposed for approximating
this quantity. With few exceptions, these methods generally provide upper
bounds on the edit distance between two graphs. In this paper, we propose a new
family of computationally tractable convex relaxations for obtaining lower
bounds on graph edit distance. These relaxations can be tailored to the
structural properties of the particular graphs via convex graph invariants.
Specific examples that we highlight in this paper include constraints on the
graph spectrum as well as (tractable approximations of) the stability number
and the maximum-cut values of graphs. We prove under suitable conditions that
our relaxations are tight (i.e., exactly compute the graph edit distance) when
one of the graphs consists of few eigenvalues. We also validate the utility of
our framework on synthetic problems as well as real applications involving
molecular structure comparison problems in chemistry.Comment: 27 pages, 7 figure
Synthesizing species trees from gene trees using the parameterized and graph-theoretic approaches
Gene trees describe how parts of the species have evolved over time, and it is assumed that gene trees have evolved along the branches of the species tree. However, some of gene trees are often discordant with the corresponding species tree due to the complicated evolution history of genes. To overcome this obstacle, median problems have emerged as a major tool for synthesizing species trees by reconciling discordance in a given collection of gene trees. Given a collection of gene trees and a cost function, the median problem seeks a tree, called median tree, that minimizes the overall cost to the gene trees. Median tree problems are typically NP-hard, and there is an increased interest in making such median tree problems available for large-scale species tree construction.
In this thesis work, we first show that the gene duplication median tree problem satisfied the weaker version of the Pareto property and propose a parameterized algorithm to solve the gene duplication median tree problem. Second, we design two efficient methods to handle the issues of applying the parameterized algorithm to unrooted gene trees which are sampled from the different species. Third, we introduce the graph-theoretic formulation of the Robinson-Foulds median tree problem and a new tree edit operation. Fourth, we propose a new metric between two phylogenetic trees and examine the statistical properties of the metric. Finally, we propose a new clustering criteria in a bipartite network and propose a new NP-hard problem and its ILP formulation
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