Wojciech Chachólski (KTH)
Geometry, Homology and Data
One of the key steps towards a successful analysis is to provide suitable representations of data by objects amenable for statistical and ML methods.
Since geometrical properties are often not amenable for such tools, one of the most important contribution of Topological data analysis has been to provide strategies and algorithms of transforming geometrical information into objects for which statistical and ML methods can be applied to. During the last decade there has been an explosion of applications in which such representations of data played a significant role. In my talks I will present one such strategy based on hierarchical stabilisation process leading to invariants called stable ranks.
I will use classical Wisconsin Breast Cancer data as one of examples when homological invariants can give interesting information.
Anne Estrade (Université Paris Cité)
The geometry of Gaussian fields
Érika Roldán (MPI Leipzig)
Topology and Geometry of Random Cubical Complexes
Rasmus Waagepetersen (Aalborg University)
Cox processes – mixed models for point processes
Cox process is the point process counterpart of a latent variable model. It arises by adding latent random effects to a Poisson point process model where these random effects constitute a random field and are used to model unobserved sources of variation influencing the occurrence of points. We will review a range of Cox process models including log Gaussian Cox processes and shot-noise Cox processes. We consider moment properties and methodology for statistical inference including estimating functions. We also consider extensions to multivariate point patterns.
Péter Juhász (Aarhus University)
Topological data analysis of higher-order networks
Preferential attachment is a popular mechanism for generating scale-free networks. While it offers a compelling narrative, the underlying reinforced processes make it difficult to rigorously establish subtle properties. Recently, age-dependent random connection models were proposed as an alternative that are capable of generating similar networks with a mechanism that is amenable to a more refined analysis. In this poster, we analyze the asymptotic behavior of higher-order topological characteristics such as higher-order degree distributions and Betti numbers in large domains.
Nikolaj Lundbye (Aarhus University)
Extremal lifetimes in the sparse regime
The persistence diagram is one of the main object of study in topological data analysis. Given a point cloud, the persistence diagram represents the collection of birth and death times of topological features such as loops or cavities. In particular, the points in the persistence diagram with a large lifetime are of special interest. We study the extremal lifetimes when the point cloud is given by a Poisson point process in a sparse regime, i.e., where the expected number of neighbors of a typical vertex tends to 0. We derive a Poisson approximation result for the maximal lifetimes.
Karthik Viswanathan (University of Amsterdam)
Information maximizing persistent homology for inference
A way to summarize a complex data set is to represent it via a filtered simplicial complex and compute the corresponding persistent homology. It is possible that we lose information about the dataset while we construct the persistence diagram. In this poster, I'll quantify the information content of the persistent diagram using Fisher Information. This algorithm may be useful for integrating topological information into statistical inference. We illustrate the pipeline using examples such as Gaussian Random Fields with a defined power spectrum. In certain cases, we show that the persistence diagrams can provide optimal summaries for inference, in the sense that the Cramer-Rao bound is saturated. The possibility of using Fisher Information as a loss function for unsupervised learning tasks to learn an "optimal" filtration is explored in this poster.
Jacky Yip (University of Wisconsin-Madison)
TDA for cosmology: parameter constraints and estimations from the large-scale structure
Persistent homology naturally describes the multi-scale characteristics of the large-scale structure in our universe. We apply the tool to mock galaxy catalogues to construct a simple and interpretable summary statistic. With the Fisher matrix formalism, constraints on cosmological parameters are found to be generally tighter than those from state-of-the-art momentum-space statistics conventionally used by the community. For parameter estimation, we explore the use of a neural network model trained on persistence images and find that it outperforms a traditional Bayesian inference method, demonstrating that TDA and machine learning can be combined for cosmology.
Navyanth Kusampudi (Max-Planck-Institut für Eisenforschung)
Application of topological data analysis and sliding window embedding to analyze time series data from bird songs
Jose Licon-Salaiz (University of Hamburg)
A topological analysis of pattern formation in shallow cumulus convection
Turbulent convection in the planetary boundary layer is an archetypal case of pattern emergence and self-organization in nonlinear dynamics. Given the spatiotemporal complexity inherent to these patterns, it is difficult to give a precise quantitative characterization of their structure and interactions with the environment. This is even more difficult in a moist atmosphere, as the appearance of a cloud layer introduces nonlinear feedbacks to the system. Yet having a quantitative understanding of these patterns could greatly improve their representation in large-scale climate models, for which clouds constitute the largest source of uncertainty. Previous work has shown that topological invariants computed from two-dimensional cross sections of vertical wind velocity in a dry boundary layer carry important information about the underlying dynamics. Here we extend those methods to a moist boundary layer, and explore relationships between topological invariants obtained from model variables and the processes of cloud formation and evolution.