Related papers: Towards a theory of statistical tree-shape analysi…
Complex data objects arise in many areas of modern science including evolutionary biology, nueroscience, dynamics of gene expression and medical imaging. Object oriented data analysis (OODA) is the statistical analysis of datasets of…
The manifold of empirical mean values of statistical data ad infinitum has a geometric shape that depends on the probability measure that governs the generating model. Large deviation theory produces entropy functions that depend on both…
In this paper we present with algebraic trees a novel notion of (continuum) trees which generalizes countable graph-theoretic trees to (potentially) uncountable structures. For that purpose we focus on the tree structure given by the branch…
The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension K, where the edges are realized as straight segments of lengths equal (or as close as possible) to the…
Geometric frameworks for analyzing curves are common in applications as they focus on invariant features and provide visually satisfying solutions to standard problems such as computing invariant distances, averaging curves, or registering…
A metric space $\mathcal{T}$ is a \emph{real tree} if for any pair of points $x, y \in \mathcal{T}$ all topological embeddings $\sigma$ of the segment $[0,1]$ into $\mathcal{T}$, such that $\sigma (0)=x$ and $\sigma (1)=y$, have the same…
We introduce a new framework to analyze shape descriptors that capture the geometric features of an ensemble of point clouds. At the core of our approach is the point of view that the data arises as sampled recordings from a metric…
Metric spaces $(X, d)$ are ubiquitous objects in mathematics and computer science that allow for capturing (pairwise) distance relationships $d(x, y)$ between points $x, y \in X$. Because of this, it is natural to ask what useful…
Reachability analysis is a powerful tool when it comes to capturing the behaviour, thus verifying the safety, of autonomous systems. However, general-purpose methods, such as Hamilton-Jacobi approaches, suffer from the curse of…
The unit Euclidean distance degree and the generic Euclidean distance degree are two well-studied invariants of projective varieties. These quantities measure the algebraic complexity of nearest-point problems on a variety, and in many…
Empirically, Deep Learning (DL) has demonstrated unprecedented success in practical applications. However, DL remains by and large a mysterious "black-box", spurring recent theoretical research to build its mathematical foundations. In this…
We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all…
We prove that if X is a complete geodesic metric space with uniformly generated first homology group and $f: X\to R$ is metrically proper on the connected components and bornologous, then X is quasi-isometric to a tree. Using this and…
This work presents novel algorithms for learning Bayesian network structures with bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed-integer linear programming formulations for structure…
CONTEXT. Attack treesare a recommended threat modeling tool, but there is no established method to compare them. OBJECTIVE. We aim to establish a method to compare "real" attack trees, based on both the structure of the tree itself and the…
We propose a new graph metric and study its properties. In contrast to the standard distance in connected graphs, it takes into account all paths between vertices. Formally, it is defined as d(i,j)=q_{ii}+q_{jj}-q_{ij}-q_{ji}, where q_{ij}…
Object oriented data analysis is the statistical analysis of populations of complex objects. In the special case of functional data analysis, these data objects are curves, where standard Euclidean approaches, such as principal component…
To improve our understanding of connected systems, different tools derived from statistics, signal processing, information theory and statistical physics have been developed in the last decade. Here, we will focus on the graph comparison…
The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths),…
We show that an embedding in Euclidean space based on tropical geometry generates stable sufficient statistics for barcodes. In topological data analysis, barcodes are multiscale summaries of algebraic topological characteristics that…