Related papers: Towards a theory of statistical tree-shape analysi…
This article presents a new way to understand the descriptive ability of tree shape statistics. Where before tree shape statistics were chosen by their ability to distinguish between macroevolutionary models, the ``resolution'' presented in…
How can one analyze detailed 3D biological objects, such as neurons and botanical trees, that exhibit complex geometrical and topological variation? In this paper, we develop a novel mathematical framework for representing, comparing, and…
Node similarity is a fundamental problem in graph analytics. However, node similarity between nodes in different graphs (inter-graph nodes) has not received a lot of attention yet. The inter-graph node similarity is important in learning a…
Past approaches for statistical shape analysis of objects have focused mainly on objects within the same topological classes, e.g., scalar functions, Euclidean curves, or surfaces, etc. For objects that differ in more complex ways, the…
Evolutionary relationships between species are represented by phylogenetic trees, but these relationships are subject to uncertainty due to the random nature of evolution. A geometry for the space of phylogenetic trees is necessary in order…
In this work we define a metric structure to compare functions defined on different merge trees. The metric introduced possesses some stability properties, which we illustrate within a standard topological data analysis (TDA) framework, and…
Topological structures such as the merge tree provide an abstract and succinct representation of scalar fields. They facilitate effective visualization and interactive exploration of feature-rich data. A merge tree captures the topology of…
Statistical graph models aim at modeling graphs as random realization among a set of possible graphs. One issue is to evaluate whether or not a graph is likely to have been generated by one particular model. In this paper we introduce the…
In this work we define a novel edit distance for trees considered with some abstract weights on the edges. The metric is driven by the idea of considering trees as topological summaries in the context of persistence and topological data…
Comparative analysis of scalar fields is an important problem with various applications including feature-directed visualization and feature tracking in time-varying data. Comparing topological structures that are abstract and succinct…
Associated to any finite metric space are a large number of objects and quantities which provide some degree of structural or geometric information about the space. In this paper we show that in the setting of subsets of weighted Hamming…
An important problem in geometric computing is defining and computing similarity between two geometric shapes, e.g. point sets, curves and surfaces, etc. Important geometric and topological information of many shapes can be captured by…
The classification of shapes is of great interest in diverse areas ranging from medical imaging to computer vision and beyond. While many statistical frameworks have been developed for the classification problem, most are strongly tied to…
It is known that PQ-symmetric maps on the boundary characterize the quasi-isometry type of visual hyperbolic spaces, in particular, of geodesically complete \br-trees. We define a map on pairs of PQ-symmetric ultrametric spaces which…
We develop a new concept of non-positive curvature for metric spaces, based on intersection patterns of closed balls. In contrast to the synthetic approaches of Alexandrov and Buesemann, our concept also applies to metric spaces that might…
Based on solid theoretical foundations, we present strong evidences that a number of real-life networks, taken from different domains like Internet measurements, biological data, web graphs, social and collaboration networks, exhibit…
Computing the similarity between two data points plays a vital role in many machine learning algorithms. Metric learning has the aim of learning a good metric automatically from data. Most existing studies on metric learning for…
Kendall's Shape Theory covers shapes formed by $N$ points in $\mathbb{R}^d$ upon quotienting out the similarity transformations. This theory is based on the geometry and topology of the corresponding configuration space: shape space.…
Kendall's Similarity Shape Theory for constellations of points in the carrier space $\mathbb{R}^n$ was developed for use in Probability and Statistics. It was subsequently shown to reside within (Classical and Quantum) Mechanics'…
In this paper we investigate the geometric properties of quasi-trees, and prove some equivalent criteria. We give a general construction of a tree that approximates the ends of a geodesic space, and use this to prove that every quasi-tree…