Related papers: Probabilistic Fitting of Topological Structure to …
This paper lays the foundations of an approach to applying Gromov's ideas on quantitative topology to topological data analysis. We introduce the "contiguity complex", a simplicial complex of maps between simplicial complexes defined in…
The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct…
We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of,…
Random graph mixture models are now very popular for modeling real data networks. In these setups, parameter estimation procedures usually rely on variational approximations, either combined with the expectation-maximisation (\textsc{em})…
The consistency of the maximum likelihood estimator for mixtures of elliptically-symmetric distributions for estimating its population version is shown, where the underlying distribution $P$ is nonparametric and does not necessarily belong…
Mixture models are often used to identify meaningful subpopulations (i.e., clusters) in observed data such that the subpopulations have a real-world interpretation (e.g., as cell types). However, when used for subpopulation discovery,…
Any symmetric affinity function $w: V\times V \to \mathbb{R}_+$ defined on a discrete set $V$ induces Euclidean space structure on $V$. In particular, an undirected graph specified by an affinity (or adjacency) matrix can be considered as a…
Latent variable models for network data extract a summary of the relational structure underlying an observed network. The simplest possible models subdivide nodes of the network into clusters; the probability of a link between any two nodes…
The topological information of a network can be retrieved equivalently from its complement consisting of the same nodes but complementary edges. Hence the partition of a network into certain substructures based on given criteria should be…
In this paper we parameterize non-negative matrices of sum one and rank at most two. More precisely, we give a family of parameterizations using the least possible number of parameters. We also show how these parameterizations relate to a…
Complex systems, ranging from soft materials to wireless communication, are often organised as random geometric networks in which nodes and edges evenly fill up the volume of some space. Studying such networks is difficult because they…
Simplicial distributions are combinatorial models describing distributions on spaces of measurements and outcomes that generalize non-signaling distributions on contextuality scenarios. This paper studies simplicial distributions on…
Random graphs with a given degree sequence are often constructed using the configuration model, which yields a random multigraph. We may adjust this multigraph by a sequence of switchings, eventually yielding a simple graph. We show that,…
Understanding the structure of relationships between objects in a given database is one of the most important problems in the field of data mining. The structure can be defined for a set of single objects (clustering) or a set of groups of…
We introduce a new `geometric realization' of an (abstract) simplicial complex, inspired by probability theory. This space (and its completion) is a metric space, which has the right (weak) homotopy type, and which can be compared with the…
In many scientific and technological contexts we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal…
We propose models and algorithms for learning about random directions in simplex-valued data. The models are applied to the study of income level proportions and their changes over time in a geostatistical area. There are several notable…
Finite mixture models are statistical models which appear in many problems in statistics and machine learning. In such models it is assumed that data are drawn from random probability measures, called mixture components, which are…
To create heterogeneous, multiscale structures with unprecedented functionalities, recent topology optimization approaches design either fully aperiodic systems or functionally graded structures, which compete in terms of design freedom and…
This paper introduces topological data analysis. Starting from notions of a metric space and some elementary graph theory, we take example sets of data and demonstrate some of their topological properties. We discuss simplicial complexes…