Tuesday, 6 April, 11:15
Topological analysis of spike train data
The methods of topological data analysis have found in neuroscience a particularly fertile field of application. Their ability to capture and quantify information about shape and connections makes them relevant to study, for example, the geometry and topology of spaces of neural representations of stimuli. I will describe some successful topological methods for the analysis of neural data, focusing on the information they can extract from spike train data, i.e. lists of firing times of different neurons. As a concrete example, I will present results on the spaces of neural responses elicited by particular visual stimuli.
Tuesday, 30 March, 11:15
Elizabeth Gross (University of Hawaiʻi at Mānoa)
Mixed volumes of steady-state systems
The steady-state degree of a chemical reaction network is the number of complex solutions to the steady-state system for generic parameters. In general, the steady-state degree may be difficult to compute, but it can be bounded above by the mixed volume of the system. In this presentation, using tools from combinatorial polyhedral geometry, we compute the mixed volume for three infinite families of networks, each generated by joining smaller networks to create larger ones. Each of these examples illustrate a different relationship between the steady-state degree and the mixed volume of the steady-state system. This is joint work with Cvetelina Hill.
Tuesday, 23 March, 16:15
Peter Bubenik (University of Florida)
Summaries and Distances for Topological Data Analysis
Topological data analysis starts with encoding data as a diagram of topological spaces. To this diagram we apply standard topological, algebraic, and combinatorial tools to produce a summary of our data. Next, we would like to be able perform a quantitative analysis. I will show how a family of distances, called Wasserstein distances, arise in a natural way, and how an extension of these ideas produces a graded version of this summary and also a summary for which we have not only distances but also angles (in fact, a Hilbert space) allowing us to apply statistics and machine learning. Key mathematical ingredients will include optimal transport, Mobius inversion, and Morse theory.
Tuesday, 16 March, 11:15
Carlos Améndola (TU Munich / Ulm University)
Likelihood Geometry of Correlation Models
Correlation matrices are standardized covariance matrices. They form an affine space of symmetric matrices defined by setting the diagonal entries to one. In this talk we will consider the fascinating geometry of maximum likelihood estimation for this model and linear submodels that encode additional symmetries. We also consider the problem of minimizing two closely related functions of the covariance matrix, the Stein's loss and the symmetrized Stein's loss, which lead naturally to the algebraic statistical concepts of dual ML degree and SSL degree. I will also present exciting open problems in this direction. This is joint work with Piotr Zwiernik.
Tuesday, 9 March, 11:15
Aida Maraj (MPI MiS Leipzig)
Asymptotic Behaviors of Hierarchical Models
A hierarchical model is realizable by a simplicial complex that describes the dependency relationships among random variables and the number of states for each random variable. Diaconis and Sturmfels have constructed toric ideals that produce Markov bases for these models. This talk concerns quantitative properties for families of ideals arising from hierarchical models with the same dependency relations and varying number of states. We introduce and study invariant filtrations of such ideals related by an action of the symmetric group, and their equivariant Hilbert series. Conditions that guarantee this multivariate series is a rational function will be presented. The key is to construct finite automata that recognize languages corresponding to invariant filtrations. Lastly, we show that one can similarly prove the rationality of an equivariant Hilbert series for some filtrations of algebras. This is joint work with Uwe Nagel.
Tuesday, 2 March, 11:15
Mathias Drton (TU Munich)
Parameter Identification in Linear Causal Models with Latent Variables
Linear causal models relate random variables of interest via a linear equation system that features stochastic noise. These models, also known as structural equation models, are naturally represented by directed graphs whose edges indicate non-zero coefficients in the linear equations. In this talk I will report on progress on combinatorial conditions for parameter identifiability in models with latent (i.e., unobserved) variables. Identifiability holds if the coefficients associated with the edges of the graph can be uniquely recovered from the covariance matrix they define.
Tuesday, 23 February, 11:15
Beatriz Pascual Escudero (Universidad Carlos III de Madrid)
Using Algebraic Geometry to detect robustness in Reaction Networks
An interesting feature of some biological systems is their capacity to preserve certain features against changes in the environmental conditions. In particular, we are concerned with the preservation of the concentration of specific species interacting in a system, motivated by examples in Reaction Networks: a biological system has absolute concentration robustness (ACR) for some molecular species if the concentration of this species does not vary among the different equilibria that the network admits. In particular, this concentration is independent of the initial conditions. While some classes of networks with ACR have been described, as well as some techniques to check ACR for a given network, finding networks with this property is a difficult task in general.
Motivated by this problem, we will explore local and global notions of robustness on the set of (real positive) solutions of a system of polynomial equations. The goal is to provide a practical test on necessary conditions for ACR, using algebraic-geometric techniques. This is based on joint work with E. Feliu.
Tuesday, 16 February, 11:15
Rickard Brüel Gabrielsson (MIT)
Topology and Machine Learning
We will talk about connections between topology and machine learning. Statistics and local geometry has for a long time served as the underpinning of machine learning, but recent developments allow us to incorporate global geometry and topology with great results. First, topological features such as persistent homology and clustering methods can help us understand what neural networks learn and construct topology inspired learning theory. Next, many of these features are differentiable which allows us to create a general purpose machine learning layer that allows anyone to easily incorporate topology into any machine learning pipeline. Lastly, we will talk about universal function approximation on graphs and the promise of unsupervised learning and pre-training.
Tuesday, 9 February, 15:45
Rehka Thomas (University of Washington)
Multiview Chirality, Two Cameras and a Cubic Surface
The set of images captured by an arrangement of pinhole cameras is usually modeled by an algebraic variety called the multiview variety. The true set is in fact a semialgebraic subset of this variety, arising from the physical restriction that cameras can only image points in front of them. I will discuss this set for multiple cameras. For a pair of cameras, the minimal problem in this semialgebraic setting is given by 5 point pairs, which even in general position, can fail to have a "chiral" 3-dimensional reconstruction. We will see that the combinatorics and arithmetic information of this minimal case is carried by a cubic surface with 27 real lines.
Joint work with Sameer Agarwal, Andrew Pryhuber and Rainer Sinn
Tuesday, 26 January, 11:15
Kaie Kubjas (Aalto University, Helsinki)
Uniqueness of nonnegative matrix factorizations
Nonnegative matrix factorizations are often encountered in data mining applications where they are used to explain datasets by a small number of parts. For many of these applications it is desirable that there exists a unique nonnegative matrix factorization up to trivial modifications given by scalings and permutations. This means that model parameters are uniquely identifiable from the data. Different sufficient conditions for the uniqueness of nonnegative matrix factorizations have been well studied, however, a little is known about necessary conditions. We will give so far the strongest necessary condition for the uniqueness of a nonnegative factorization. The talk is based on the joint work with Robert Krone.
Tuesday, 19 January, 11:15
Points, polytopes, polynomials
Ehrhart theory is concerned with the enumeration of lattice points in polytopes. Given a polytope with vertices in the integer lattice, the function counting the lattice points in its integer dilates agrees with a polynomial, the Ehrhart polynomial of the polytope. In this talk, I will give a brief overview of some of my research interests in Ehrhart theory, in particular, geometric models for enumerative problems, structural results for Ehrhart polynomials as well as ties to valuation theory.
Slides (pdf 911 kB)
Tuesday, 17 November, 11:15
Nonlinear Algebra of Data Science and AI
Nonlinear algebra is the study of systems of multivariate polynomial equations and inequalities. At the heart of this lies algebraic geometry, but the field is driven by many other areas of mathematics such as combinatorics, convex and discrete geometry, multilinear algebra, tropical geometry, or representation theory. Systems of polynomial (in)equalities appear vastly throughout the sciences and engineering. In this talk, I will focus on applications in data science and artificial intelligence. My research focuses on uncovering geometric structures in applied problems and on finding connections between seemingly unrelated areas of mathematics or science in general. I will demonstrate this by describing the underlying (algebraic) geometry of machine learning with neural networks and of 3D reconstruction in computer vision. I will also outline surprising connections between statistics, invariant theory, geometric modeling, intersection theory, and physics.
Slides (pdf 4.3 MB)
Tuesday, 10 November, 11:15
Some of my interests and (old) results
I work in complexity theory and recently I have studied efficient approximability of NP-hard optimization problems. In particular I have been interested in constraint satisfaction problems (CSPs). In a CSP you are given a large number of constraints each only depending on a constant number of variables and the goal is to find an assignment to satisfy as many constraints as possible. Central examples are Max-3Sat and sets of linear equations over a finite field where each equation only depends on three variables. I will also briefly mention some problems in post-quantum cryptography.
slides (pdf 264 kB)
- Tuesday, 3 November, 11:15
Wasserstein distance in algebraic statistics
Any metric on the set of states of a discrete random variable induces a metric called Wasserstein distance on the probability simplex. The unit ball of this norm is a polytope, well known in discrete and tropical geometry. Given any data distribution, we seek to minimize its Wasserstein distance to an algebraic model, i.e., a model described by polynomials. The solution to this optimization problem is a piecewise algebraic function of the data. After a gentle introduction on the topic, I will comment on the combinatorial and algebraic complexity of the problem. This talk is based on joint work with Türkü Özlüm Ç̧̧elik, Asgar Jamneshan, Guido Montúfar and Bernd Sturmfels.
slides (pdf 1.2 MB)
- Tuesday, 27 October, *10:15*
¡ PhD Day !
- Tuesday, 20 October, 11:15
Algebraic statistics and the maximum likelihood degree
This talk is about statistical models and algebraic varieties. Algebraic Statistics unites these two concepts, turning algebraic structure into statistical insight. We will discuss the basic ideas of this relatively new field and see some examples of the kind of models algebraic statisticians are interested in. We will focus on an important quantity associated to the likelihood estimation problem on these models: the maximum likelihood degree.
- Tuesday, 13 October, 11:15
Convergence of linear neural networks to global minimizers
It is known that gradient flow in linear neural networks using Euclidean loss almost always avoids critical points that have at least one eigendirection with negative curvature. Using algebraic invariants of the gradient flow we try to prove that the set of all critical points with no second-order curvature (zero Hessian) for arbitrary networks is associated to a subset of the invariants of lower dimension. This would mean that these critical points are almost surely avoided. We show that this holds for networks with 3 or less hidden layers and a few other special cases. We show by way of explicit counter-example that it is not true for general deep networks.
slides (pdf 477 kB)
- Tuesday, 6 October, 11:15
Exact semidefinite programming bounds for packing problems
In the first part of the talk, I present how semidefinite programming methods can provide upper bounds for various geometric packing problems, such as kissing numbers, spherical codes, or packings of spheres into a larger sphere. When these bounds are sharp, they give additional information on optimal configurations, that may lead to prove the uniqueness of such packings. For example, we show that the lattice E8 is the unique solution for the kissing number problem on the hemisphere in dimension 8. However, semidefinite programming solvers provide approximate solutions, and some additional work is required to turn them into an exact solution, giving a certificate that the bound is sharp. In the second part of the talk, I explain how, via our rounding procedure, we can obtain an exact rational solution of a semidefinite program from an approximate solution in floating point given by the solver. This is a joined work with David de Laat and Philippe Moustrou.
slides (pdf 6.3 MB)
- Tuesday, 22 September, 11:15
Combinatorics, Algebra, and Intervention
The modern theory of causality is founded in our ability to encode both probabilistic and causal information about a data-generating distribution within the structure of a graph. Doing so in the proper way allows us to derive theorems which can be applied when developing data-driven causal structure learning algorithms, as well as probabilistic and causal inference. Such algorithms work best in the cases where the corresponding theorems point to nice combinatorial structure in the graph. Going deeper, if we also consider a parameterization of the joint distribution, we find that nice properties of the model can be attributed to nice properties of its defining algebraic variety in the ambient parameter space. Classic results of this nature are known for probabilistic graphical models. In this talk, we will see how such results generalize to interventional graphical models, which are now being used to learn causal structure from a mixture of observational and interventional data. Time permitting, we will explore how the proofs of these algebraic results give rise to new methods for modeling causation in context-specific settings.
slides (pdf 277 kB)
- Tuesday, 15 September, 11:15
Dynamics of chemical reaction networks and positivity of polynomials
In biochemical processes, different molecules are consumed and produced as they interact with each other. The evolution of their concentration through time can be modelled with a system of differential equations that, under certain assumptions, is polynomial. In this case, the equilibrium points form an algebraic variety. In this talk I will present how these systems arise, and how deciding whether a polynomial can attain positive or negative values is used for detecting multistationarity and stability in the biochemical system.
slides (pdf 307 kB)
- Tuesday, 8 September, 11:15
An overview of the pipeline of multiparameter persistence
In the last two decades, persistent homology and its generalizations have grown to a substantial branch of mathematical research. It gives rise to various research problems in algebra, category theory, geometry, topology, and other areas. Moreover, having immediate applications in data science, efficient algorithms, computational and probabilistic methods, and concrete implementations deserve the same amount of attention. Multiparameter persistence is the generalization of persistence to multiple independent parameters taken into account at once. While the theory of multiparameter persistence is provably much more complicated than the theory of ordinary persistence, new computational challenges arise as well. In this talk, I give a rough overview of the computational pipeline of multiparameter persistence, mention some important challenges, and outline the projects of my PhD thesis along these lines.
slides (pdf 9.8 MB)