École Polytechnique Fédérale de Lausanne (EPFL)

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.

Motivated by the desire to automate classification of neuron morphologies, we designed a topological signature, the Topological Morphology Descriptor (TMD),  that assigns a so-called “barcode" to any geometric tree (i.e, any finite binary tree embedded in R^3). We showed that the TMD effectively determines  reliable clusterings of random and neuronal trees. Moreover, using the TMD we performed an objective, stable classification of pyramidal cells in the rat neocortex, based only on the shape of their dendrites.

We have also reverse-engineered the TMD, in order to digitally synthesize dendrites, to compensate for the relatively small number of available biological reconstructions. The algorithm we developed, called Topological Neuron Synthesis (TNS), stochastically generates a geometric tree from a barcode, in a biologically grounded manner. The synthesized neurons are statistically indistinguishable from real neurons of the same type, in terms of morpho-electrical properties and  connectivity. We synthesized networks of structurally altered neurons, revealing principles linking branching properties to the structure of large-scale networks.  We have also successfully applied these classification and synthesis techniques to microglia and astrocytes, two other types of cells that populate the brain.

In this talk I will provide an overview of the TMD and the TNS and then describe the results of our theoretical and computational analysis of their behavior and properties.

This talk is based on work in collaborations led by Lida Kanari at the Blue Brain Project.

The well known node2vec algorithm has been used to explore network structures and represent the nodes of a graph in a vector space in a way that reflects the structure of the graph. Random walks in node2vec have been used to study the local structure through pairwise interactions. Our motivation for this project comes from a desire to understand higher-order relationships by a similar approach. To this end, we propose an extension of node2vec to a method for representing the k-simplices of a simplicial complex into Euclidean space. 

In this talk I outline a way to do this by performing random walks on simplicial complexes, which have a greater variety of adjacency relations to take into account than in the case of graphs. The walks on simplices are then used to obtain a representation of the simplices. We will show cases in which this method can uncover the roles of higher order simplices in a network and help understand structures in graphs that cannot be seen by using just the random walks on the nodes. 

Computer-based simulation of partial differential equations (PDEs) involves approximating the unknowns and relies on suitable description of geometrical entities such as the computational domain and its properties. The Finite Element Method (FEM) is by large the most popular technique for the computer-based simulation of PDEs and hinges on the assumption that discretized domain and unknown fields are both represented by piecewise polynomials, on tetrahedral or hexahedral partitions. In reality, the simulation of PDEs is a brick within a workflow where, at the beginning, the geometrical entities are created, described and manipulated with a geometry processor, often through Computer-Aided Design systems (CAD), and then used for the simulation of the mechanical behaviour of the designed object. This workflow is often repeated many times as part of a shape optimisation loop. Within this loop, the use of FEM on CAD geometries (which are mainly represented through their boundaries) calls then for (re-) meshing and re-interpolation techniques that often require human intervention and result in inaccurate solutions and lack of robustness of the whole process. In my talk, I will present the mathematical counterpart of this problem, I will discuss the mismatch in the mathematical representations of geometries and PDEs unknowns and introduce a promising framework where geometric objects and PDEs unknowns are represented in a compatible way. Within this framework, the challenges to be addressed in order to construct robust PDE solvers are many and I will discuss some of them. Mathematical results will besupported by numerical validation.

The TMD algorithm (Kanari et al. 2018) computes the barcode of a neuron (tree) with respect to the radial or path distance from the soma (root). We are interested in the inverse problem: how to understand the space of trees that are represented by the same barcode. Our tool to study this spaces is the stochastic TNS algorithm (Kanari et al. 2020) which generates trees from a given barcode in a biologically meaningful way. 

I will present some theoretical results on the space of trees that have the same barcode, as well as the effect of adding noise to the barcode. In addition, I will provide a more combinatorial perspective on the space of barcodes, expressed in terms of the symmetric group. I will illustrate these results with experiments based on the TNS.

This is joint work with L. Kanari and K. Hess. 

In practice, it is standard to initialize Artificial Neural Networks (ANN) with random parameters. We will see that this allows to describe, in the functional space, the limit of the evolution of (fully connected) ANN when their width tends towards infinity. Within this limit, an ANN is initially a Gaussian process and follows, during learning, a gradient descent convoluted by a kernel called the Neural Tangent Kernel. 

This description allows a better understanding of the convergence properties of neural networks, of how they generalize to examples during learning and has 

practical implications on the training of wide ANNs. 

Over the past decade, and particularly over the past five years, research at the interface of topology and neuroscience has grown remarkably fast.  In this talk I will briefly survey a few quite different applications of topology to neuroscience in which members of my lab have been involved over the past four years: the algebraic topology of brain structure and function, topological characterization and classification of neuron morphologies, and (if time allows) topological detection of network dynamics.

Semidefinite convex optimization problems often have low-rank solutions that can be represented with O(p)-storage. However, semidefinite programming methods require us to store the matrix decision variable with size O(p^2), which prevents the application of virtually all convex methods at large scale.

Indeed, storage, not arithmetic computation, is now the obstacle that prevents us from solving large- scale optimization problems. A grand challenge in contemporary optimization is therefore to design storage-optimal algorithms that provably and reliably solve large-scale optimization problems in key scientific and engineering applications. An algorithm is called storage optimal if its working storage is within a constant factor of the memory required to specify a generic problem instance and its solution.

So far, convex methods have completely failed to satisfy storage optimality. As a result, the literature has largely focused on storage optimal non-convex methods to obtain numerical solutions. Unfortunately, these algorithms have been shown to be provably correct only under unverifiable and unrealistic statistical assumptions on the problem template. They can also sacrifice the key benefits of convexity, as they do not use key convex geometric properties in their cost functions.

To this end, my talk introduces a new convex optimization algebra to obtain numerical solutions to semidefinite programs with a low-rank matrix streaming model. This streaming model provides us an opportunity to integrate sketching as a new tool for developing storage optimal convex optimization methods that go beyond semidefinite programming to more general convex templates. The resulting algorithms are expected to achieve unparalleled results for scalable matrix optimization problems in signal processing, machine learning, and computer science.