Wasserstein distance to an algebraic variety
Lorenzo Venturello (KTH)
A metric on the finite set {1,…,n} induces a metric on the (n-1)-dimensional probability simplex which is called Wasserstein distance. This distance is well known to statisticians, and possesses properties which make it appealing to several areas of computer science. Moreover, its unit ball is a polytope, whose combinatorial structure encodes information on the metric. We study the problem of computing the Wasserstein distance from a point to a statistical model described by polynomial equations, i.e., an algebraic variety. After recalling the necessary background, I will discuss the combinatorial and algebraic complexity of this problem, with a special focus on models of independent random variables. This talk draws from a joint work with Türkü Özlüm Çelik, Asgar Jamneshan, Guido Montúfar and Bernd Sturmfels, and a joint work with Alexander Heaton and Khazhgali Kozhasov.
Toric Eigenvalue Methods for Solving Sparse Polynomial Systems
Matías Bender (TU Berlin)
In this talk, we introduce a symbolic-numerical algorithm to solve (nearly) degenerate sparse polynomial systems. For that, we consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact toric variety X. By working over the Cox ring of X, our algorithm can handle input systems whose solutions have multiplicities. We investigate the regularity of these systems to provide complexity bounds for our approach, as well as sharper bounds for weighted homogeneous, multihomogeneous and unmixed sparse systems, among others. The talk is based on joint work with Simon Telen.
Families of faces for convex semi-algebraic sets
Daniel Plaumann (TU Dortmund)
Given a convex semi-algebraic set, presented for example as the convex hull of an algebraic variety, by a matrix inequality or in some other way, we seek to subdivide its boundary into well-behaved families of faces. To this end, we give a semi-algebraic definition of a "patch", a notion recently introduced by Ciripoi, Kaihnsa, Löhne, and Sturmfels. The talk will focus on examples, illustrating what we can and cannot hope to achieve, and highlight some key results. (Joint work with Rainer Sinn and Jannik Wesner)
Geometry of sampling
Sandra Di Rocco (KTH)
Sampling an algebraic variety is an essential way of visualising the variety (hence the solution of a system of polynomial equations) and to recover geometrical and topological features (the geometric signature of the solution). Classical invariants from algebraic geometry, like polar classes, provide an explanation and numerical measures for the density of a sample. An introduction to the classical theory as well as recent results will be given during the talk. This is joint work with O. Gäfvert, M. Weinstein and D. Eklund.
Measuring the sensibility of numerical problems, including your favourite one
Carlos Beltrán (Universidad de Cantabria)
I will introduce a general geometric framework to investigate sensibility of numerical solutions. This is to some extent a classical approach, but somehow revisited in a very amenable language. Virtually all numerical problems can be studied from this approach and in particular I will comment on some fundamental (and also on some just beautiful) points about the sensitivity of polynomial solving, eigenvalue computations, tensor decomposition and others.
Invariant determinantal representations and numerical ranges
Cynthia Vinzant (North Carolina State University)
The numerical range of a matrix is a convex body in the complex plane capturing its values as a Hermitian bilinear form on the sphere. By a classical theorem of Kippenhahn, its dual convex body is given by a linear matrix inequality and bounded by the corresponding determinantal plane curve. One can show that the numerical range of a block cyclic weighted shift matrix is invariant under the action of a dihedral group. In this talk, I will talk about the converse, namely that any invariant numerical range comes from such a matrix. This relies on understanding invariance of definite determinantal representations of plane curves. This is based on joint work with Faye Pasley Simon (arXiv:2102.01726).
A normal form method for numerical tensor rank decomposition
Simon Telen (MPI MiS)
We present a new, numerical algorithm for computing the tensor rank decomposition of third order tensors. Suitable assumptions on the rank allow us to use only linear algebra computations, based on normal form methods for polynomial system solving. Experiments show that our method outperforms state of the art implementations both in terms of speed and accuracy. We show how the complexity of our method is governed by the regularity of a homogeneous ideal in a bigraded ring and prove complexity bounds for many formats. This is joint work with Nick Vannieuwenhoven.
Modeling and image reconstruction in the context of magnetic particle imaging
Tobias Kluth (Universität Bremen)
Magnetic particle imaging (MPI) is a tracer-based imaging modality developed to detect the concentration of superparamagnetic iron oxide nanoparticles. It is highly sensitive to the nanoparticle's nonlinear response to an applied magnetic field and allows a rapid data acquisition. The imaging problem is a linear inverse problem given by a Fredholm integral equation of the first kind. The problem of modeling MPI, with respect to finding the correct integral kernel/system function in the forward operator, remains an unsolved problem. Existing model-based reconstruction techniques incorporate particle behavior based on the theory of paramagnetism which have not yet reached the necessary quality of measured system functions. As a result the state of the art reconstruction method still relies on a full calibration of the system function by moving a delta sample through the field-of-view, which is time-consuming, limits the spatial resolution, and does not generalize with respect to particle and scanner parameters. This defines a second important problem in MPI, the calibration problem. One particular challenge of modeling the signal acquisition chain and thus the development of model-based calibration methods is the magnetization dynamic of large ensembles of magnetic nanoparticles. The application MPI motivates the considerations of general and specific problems in various directions such as physical modeling and analysis of the imaging operator, sophisticated reconstruction methods simultaneously determining parameters in the forward operator, learning-based methods for image reconstruction, model-based calibration, time-dependent concentration reconstruction, etc. After an introduction to the imaging modality, we mainly focus on the imaging problem which includes a discussion of the degree of ill-posedness in the equilibrium model and the consideration of nanoparticles' magnetization dynamics to improve model-based reconstruction. In a second part we investigate general deep image prior concepts for inverse problems and their application to the imaging problem in MPI.
Applications of Group Symmetry
Emily King (Colorado State University)
Group actions have implicitly played a role in harmonic analysis since at least the work of Fourier on solutions of the heat equation. Namely, one can generate the classical Fourier basis as an action of the group 𝕋 on L2([0,1]). Hundreds of years later in the 20th century, the two most important transforms in applied harmonic analysis arose, the wavelet transform and the short-time Fourier transform, which were created using projective unitary representations of the affine group and (a group related to the) the Weyl-Heisenberg group, respectively. Even in the modern era of data-driven approaches, group symmetries still play an important role, from generating transformations like diffusion wavelets and the scattering transform to characterizing overtrained networks in neural collapse. The talk will start as a colloquium-style talk on applications of group symmetry and finish with some new results on the relationship between group symmetry and optimality.
Likelihood geometry of correlation models
Piotr Zwiernik (Universitat Pompeu Fabra)
Correlation matrices are standardized covariance matrices. They form an affine space of symmetric matrices defined by setting the diagonal entries to one. We study the 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. Unlike the Gaussian log-likelihood these two functions are convex and hence admit a unique positive definite optimum. (Joint work with Carlos Améndola)
Stabilizing Invertible Neural Networks Using Mixture Models
Sebastian Neumayer (TU Berlin)
to be added...
Disguised toric dynamical systems
Miruna-Ştefana Sorea (SISSA)
We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric dynamical systems, by Craciun, Dickenstein, Shiu and Sturmfels. These systems are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and global stability, persistence, and permanence. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all dynamical properties mentioned above extend naturally. We show that, for some families of reaction networks, this new class is much larger than the class of toric systems. For example, for some networks we may even go from an empty locus of toric systems in parameter space to a positive-measure locus of disguised toric systems. We focus on the characterization of the disguised toric locus by means of real algebraic geometry. Joint work with Laura Brustenga i Moncusi and Gheorghe Craciun.
Curve recovery from moments and applications
Paul Catala (Universität Osnabrück)
We introduce a novel algorithm for recovering a Radon measure from a finite number of trigonometric moments. Our approach is not restricted to the discrete setting, and is able to recover finite approximations of infinitely supported measures. The crucial step consists in the approximate joint-diagonalization of a few non-commuting matrices, which we perform using a quasi-Newton algorithm. Experiments show that our method performs well in regimes where no recovery guarantees are known. Our algorithm broadens the scope of Lasserre's hierarchies to solve optimization problems in spaces of measures. We illustrate its applicability in optimal transport problems. This is joint work with Jean-François Cardoso, Vincent Duval and Gabriel Peyré.
Exploiting Sparsity for Large-Scale Polynomial Optimization
Jie Wang (LAAS-CNRS)
Over the past twenty years, the moment-SOS hierarchy has become a popular tool to handle polynomial optimization as it possesses nice theoretical properties. From the view of applications, the bottleneck of the moment-SOS hierarchy is that the involved sequence of SDP relaxations becomes intractable very quickly as the problem size increases. In this talk, I will introduce the moment-SOS hierarchy and show how to exploit certain sparsity patterns arising from the problem data to develop sparsity-adapted moment-SOS hierarchies. As a consequence, we are able to solve instances of large-scale (commutative and noncommutative) polynomial optimization problems involving up to tens of thousands of variables and constraints.