Applied Topology Seminar

Models of animal brains are increasingly common and mapped in increasing detail. To simplify analysis of their function, we consider subregions and show that they perform well as classifiers of overall activity, with only a fraction of the neurons. The uniqueness of such ''reliable'' regions seems to be related to the types of connections that pairs of neurons form in them. By focusing on topologically significant structures and reciprocally connected neurons we find even stronger classification results. This is ongoing work across several institutions, including EPFL, the Blue Brain Project, and the University of Aberdeen.

The recent curation of large-scale databases with 3D surface scans of shapes has motivated the development of tools that better detect global patterns in morphological variation. Studies which focus on identifying differences between shapes have been limited to simple pairwise comparisons and rely on pre-specified landmarks (that are often known). In this talk, we present SINATRA: a machine learning pipeline for analyzing collections of shapes without requiring any correspondences. Our method takes in two classes of shapes and highlights the physical features that best describe the variation between them. We develop a rigorous simulation framework to assess our approach, which themselves are a novel contribution to 3D image and shape analyses. Lastly, as case studies with real data, we use SINATRA to (1) analyze mandibular molars from four different suborders of primates and (2) facilitate the visual identification of structural signatures differentiating between the trajectories of two protein ensembles resulting from molecular dynamics simulations. Together, these results highlight a promising future for interpretable machine learning to facilitate the non-trivial task of pattern recognition in evolutionary and structural biology with substantially increased resolution.

A common problem in data science is "use this function defined over this small set to generate predictions over that larger set." Extrapolation, interpolation, statistical inference and forecasting all reduce to this problem. The Kan extension is a powerful tool in category theory that generalizes this notion. In this work we explore several applications of Kan extensions to data science. We begin by deriving simple classification and clustering algorithms as Kan extensions and experimenting with these algorithms on real data. Next, we build more complex and resilient algorithms from these simple parts.

Discrete structures have a long history of use in applied mathematics. Graphs and hypergraphs provide models of social networks, biological systems, academic collaborations, and much more. Network science, and more recently hypernetwork science, have been used to great effect in analyzing these types of discrete structures. Separately, the field of applied topology has gathered many successes through the development of persistent homology, mapper, sheaves, and other concepts. Recent work by our group has focused on the convergence of these two areas, developing and applying topological concepts to study discrete structures that model real data.

This talk will survey our body of work in this area showing our work in both the theoretical and applied spaces. Theory topics will include an introduction to hypernetwork science and its relation to traditional network science, topological interpretations of graphs and hypergraphs, and dynamics of topology and network structures. I will show examples of how we are applying each of these concepts to real data sets.

Directed graphs are a model for various phenomena in the
sciences. In topological data analysis particularly the advent of
applying topological tools to networks of brain neurons has spawned
interest in constructing topological spaces out of digraphs, developing
computational tools for obtaining topological information, and using
these to understand networks. At the end of the day, (homological)
computations of the spaces reveal something about the geometric
realisation, thereby losing the directionality information.

However, digraphs can also be associated with path algebras. We can now
consider applying Hochschild homology to extract information, hopefully
obtaining something more refined in terms of the combinatorics of the
directed edges and paths in the digraph. Unfortunately, Hochschild
homology tends to vanish beyond degree 1. We can overcome this by
considering different higher paths of simplices, and thus introduce
Hochschild homology of digraphs in higher degrees. Moreover, this
procedure gives an implementable persistence pipeline for network
analysis. This is a joint work with Luigi Caputi.

In this talk, we will present a new class of invariants of multi-parameter persistence modules : \emph{projected barcodes}. Relying on Grothendieck's six operations for sheaves, projected barcodes are defined as derived pushforwards of persistence modules onto $\R$ (which can be seen as sheaves on a vector space in a precise sense). We will prove that the well-known fibered barcode is a particular instance of projected barcodes. Moreover, our construction is able to distinguish persistence modules that have the same fibered barcodes but are not isomorphic. We will present a systematic study of the stability of projected barcodes. Given F a subset of the 1-Lipschitz functions, this leads us to define a new class of well-behaved distances between persistence modules, the  F-Integral Sheaf Metrics (F-ISM), as the supremum over p in F of the bottleneck distance of the projected barcodes by p of two persistence modules. 

In the case where M is the collection in all degrees of the sublevel-sets persistence modules of a function f : X -> R^n, we prove that the projected barcode of M by a linear map p : R^n \to R is nothing but the collection of sublevel-sets barcodes of the post-composition of f by p. In particular, it can be computed using already existing softwares, without having to compute entirely M. We also provide an explicit formula for the gradient with respect to p of the bottleneck distance between projected barcodes, allowing to use a gradient ascent scheme of approximation for the linear ISM. This is joint work with François Petit.

Persistent homology is an important methodology from topological data analysis which adapts theory from algebraic topology to data settings and has been successfully implemented in many applications. It produces a statistical summary in the form of a persistence diagram, which captures the shape and size of the data. Despite its widespread use, persistent homology is simply impossible to implement when a dataset is very large. In this talk, I will address the problem of finding a representative persistence diagram for prohibitively large datasets. We adapt the classical statistical method of bootstrapping, namely, drawing and studying smaller multiple subsamples from the large dataset. We show that the mean of the persistence diagrams of subsamples—taken as a mean persistence measure computed from the subsamples—is a valid approximation of the true persistent homology of the larger dataset. We give the rate of convergence of the mean persistence diagram to the true persistence diagram in terms of the number of subsamples and size of each subsample. Given the complex algebraic and geometric nature of persistent homology, we adapt the convexity and stability properties in the space of persistence diagrams together with random set theory to achieve our theoretical results for the general setting of point cloud data. We demonstrate our approach on simulated and real data, including an application of shape clustering on complex large-scale point cloud data.

This is joint work with Yueqi Cao (Imperial College London).

The statistical analysis of data which lies in a non-Euclidean space has become increasingly common over the last decade, starting from the point of view of shape analysis, but also being driven by a number of novel application areas. However, while there are a number of interesting avenues this analysis has taken, particularly around positive definite matrix data and data which lies in function spaces, it has increasingly raised more questions than answers. In this talk, I'll introduce some non-Euclidean data from applications in brain imaging and in linguistics, but spend considerable time asking questions, where I hope the interaction of statistics and topological data analysis (understood broadly) could potentially start to bring understanding into the applications themselves.