Related papers: Refinement of Metrics: Erd\H{o}s Number, a Case St…
Searching for high-dimensional vector data with high accuracy is an inevitable search technology for various types of data. Graph-based indexes are known to reduce the query time for high-dimensional data. To further improve the query time…
The algorithmic differentiation (AD) of mathematical functions can be interpreted as a sequence of vertex eliminations in an underlying directed acyclic graph. The problem of determining a minimum-cost elimination ordering, which we call…
We consider the problem of augmenting an n-vertex graph embedded in a metric space, by inserting one additional edge in order to minimize the diameter of the resulting graph. We present exact algorithms for the cases when (i) the input…
In many real-world applications data come as discrete metric spaces sampled around 1-dimensional filamentary structures that can be seen as metric graphs. In this paper we address the metric reconstruction problem of such filamentary…
Data-analysis tasks often involve an iterative process, which requires refining previous solutions. For instance, when analyzing dynamic social networks, we may be interested in monitoring the evolution of a community that was identified at…
The aim of this paper is to present a new estimation procedure that can be applied in many statistical frameworks including density and regression and which leads to both robust and optimal (or nearly optimal) estimators. In density…
Many real-world datasets can be naturally represented as graphs, spanning a wide range of domains. However, the increasing complexity and size of graph datasets present significant challenges for analysis and computation. In response, graph…
A graph G on n vertices is said to be extendable if G can be modified to form a new graph H on more than n vertices, while preserving the degrees of the vertices common to G and H. The added vertices all have the same degree and we define…
We develop an iterative refinement method that improves the accuracy of a user-chosen subset of $k$ eigenvectors ($k\ll n$) of an $n\times n$ real symmetric matrix. Using an orthogonal matrix represented in compact WY form, the method…
Graph sparsification is a powerful tool to approximate an arbitrary graph and has been used in machine learning over homogeneous graphs. In heterogeneous graphs such as knowledge graphs, however, sparsification has not been systematically…
In this paper we study the geometry of graph spaces endowed with a special class of graph edit distances. The focus is on geometrical results useful for statistical pattern recognition. The main result is the Graph Representation Theorem.…
Merge-width is a recently introduced family of graph parameters that unifies treewidth, clique-width, twin-width, and generalised colouring numbers. We prove the equivalence of several alternative definitions of merge-width, thus…
Reeb graphs are structural descriptors that capture shape properties of a topological space from the perspective of a chosen function. In this work we define a combinatorial metric for Reeb graphs of orientable surfaces in terms of the cost…
As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for…
In this thesis I propose an algorithm to heuristically calculate different distance measures on uncertain graphs (i.e. graphs where edges only exist with a certain probability) and apply this to the heuristic calculation of harmonic…
In this work we provide a new technique to design fast approximation algorithms for graph problems where the points of the graph lie in a metric space. Specifically, we present a sampling approach for such metric graphs that, using a…
In recent work we have shown how an accurate reduced model can be utilized to perform mesh refinement in random space. That work relied on the explicit knowledge of an accurate reduced model which is used to monitor the transfer of activity…
We define a new family of similarity and distance measures on graphs, and explore their theoretical properties in comparison to conventional distance metrics. These measures are defined by the solution(s) to an optimization problem which…
How do social networks differ across platforms? How do information networks change over time? Answering questions like these requires us to compare two or more graphs. This task is commonly treated as a measurement problem, but numerical…
This paper considers an extremal version of the Erd\H{o}s distinct distances problem. For a point set $P \subset \mathbb R^d$, let $\Delta(P)$ denote the set of all Euclidean distances determined by $P$. Our main result is the following: if…