Applied Topology Seminar

In topological data analysis, persistence barcodes record the
persistence of homological generators in a one-parameter filtration
built on the data at hand. In contrast, computing the pointwise Euler
characteristic (EC) of the filtration merely records the alternating sum
of the dimensions of each homology vector space.

In this talk, we will show that despite losing the classical
"signal/noise" dichotomy, EC tools are powerful descriptors, especially
when combined with new integral transforms mixing EC techniques with
Lebesgue integration. Our motivation is fourfold: their applicability to
multi-parameter filtrations and time-varying data, their remarkable
performance in supervised and unsupervised tasks at a low computational
cost, their satisfactory properties as integral transforms (e.g.,
regularity and invertibility properties) and the expectation results on
the EC in random settings. Along the way, we will give an insight into
the information these descriptors record.

This talk is based on the work [https://arxiv.org/abs/2111.07829] and
the joint work with Olympio Hacquard [https://arxiv.org/abs/2303.14040].

Various constructions relevant to practical problems such as clustering and graph classification give rise to multiparameter persistence modules (MPPM), that is, linear representations of non-totally ordered sets. Much of the mathematical interest in multiparameter persistence comes from the fact that there exists no tractable classification of MPPM up to isomorphism, meaning that there is a lot of room for devising invariants of MPPM that strike a good balance between discriminating power and complexity of their computation. However, there is no consensus on what type of information we want these invariants to provide us with, and, in particular, there seems to be no good notion of “global” or “high persistence” features of MPPM.

With the goal of substantiating these claims, as well as making them more precise, I will start with an overview of some of the known invariants of MPPM, including joint works with Bauer and Oudot. I will then describe recent work of Bjerkevik, which contains relevant open questions and which will help us make sense of the notion of global feature in multiparameter persistence.

Deep convolutional neural networks such as GoogLeNet and ResNet have become ubiquitous in image classification tasks, whereas
transformer-based language models such as BERT and its variants have found widespread use in natural language processing. In this talk, I
will discuss recent efforts in exploring the topology of artificial neuron activations in deep learning, from images to word embeddings.
First, I will discuss the topology of convolutional neural network activations, which provides semantic insight into how these models
organize hierarchical class knowledge at each layer. Second, I will discuss the topology of word embeddings from transformer-based models.
I will explore the topological changes of word embeddings during the fine-tuning process of various models and discover model confusions in
the embedding spaces. If time permits, I will discuss on-going work in studying the topology of neural activations under adversarial attacks.
 

In this talk, I will present the notion of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto R. Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). I will explain how the fibered barcode is a particular instance of projected barcodes and how the ISM and the SCD provide lower bounds for the convolution distance. 

Furthermore, in the case where the persistence module considered is the sublevel-sets persistence modules of a function f : X -> R^n, we will explain how, under mild conditions, the projected barcode of this module by a linear map u : R^n \to R is the collection of sublevel-sets barcodes of the composition uf . In particular, it can be computed using software dedicated to one-parameter persistence modules. This is joint work with Nicolas Berkouk.

Dimensionality reduction is the machine learning problem of taking a data set whose elements are described with potentially many features (e.g., the pixels in an image), and computing representations which are as economical as possible (i.e., with few coordinates). In this talk, I will present a framework to leverage the topological structure of data (measured via persistent cohomology) and construct low dimensional coordinates in classifying spaces consistent with the underlying data topology.

I will survey on the cluster structure of random geometric graphs in a regime that is less discussed in the literature. The statistics of interest include the number of k-components, the number of components, the number of vertices in the giant component, and the connectivity threshold. We show LLN and normal/Poisson approximation by Stein's method. Based on recent joint works with Mathew Penrose (Bath).

TBC

Multi-parameter persistence is a main research topic in topological data analysis. Major questions involve the computation and the structural properties
of persistence modules. In this context, I will sketch two very recent results:

(1) We define a natural bifiltration called the localized union-of-balls bifiltration that contains filtrations studied in the context of local persistent homology as slices. This bifiltration is not k-critical for any finite k. Still, we show that a representation of it (involving algebraic curves of low degree) can be computed exactly and efficiently. This is joint work with Matthias Soels (TU Graz).

(2) Every persistence modules permits a unique decomposition into indecomposable summands. Intervals are the simplest type of summands, but more complicated indecomposables can appear, and usually do appear in examples. We prove that for homology-dimension 0 and density-Rips bifiltration, at least a quarter of the indecomposables are intervals in expectation for a rather general class of point samples. Moreover, these intervals can be ``peeled off'' the module efficiently. This is joint work with Angel Alonso (TU Graz).

One-dimensional persistent homology encodes geometric information of data by means of a barcode decomposition. Often, one needs to relate the persistence barcodes of two datasets which are intrinsically linked, e.g. consider a sample from a large point cloud. Such connections are encoded through persistence morphisms; as in linear algebra, a (one-dimensional) persistence morphism is fully understood by fixing a basis in the domain and codomain and computing the associated matrix. However, in the literature and existing software, the focus is often restricted to interval decompositions of images, kernels and cokernels. This is the case of the Bauer-Lesnick matching, which is computed using the intervals from the image. Unfortunately, this matching has substantial differences from the structure of the persistence morphism in very simple examples. In this talk I will present an induced block function that is well-behaved in such examples. This block function is computed using the associated matrix of a persistence morphism and is additive with respect to taking direct sums of persistence morphisms. This is joint work with M. Soriano-Trigueros and R. Gonzalez-Díaz from Universidad de Sevilla.

Mapper and Ball Mapper are Topological Data Analysis tools used for exploring high dimensional point clouds and visualizing scalar–valued functions on those point clouds. Inspired by open questions in knot theory, new features are added to Ball Mapper that enable encoding of the structure, internal relations and symmetries of the point cloud. Moreover, the strengths of Mapper and Ball Mapper constructions are combined to create a tool for comparing high dimensional data descriptors of a single dataset. This new hybrid algorithm, Mapper on Ball Mapper, is applicable to high dimensional lens functions. As a proof of concept we include applications to knot and game  theory, as well as material science and cancer research.