Geometric methods in statistical learning theory and applications
Final Report Abstract
Statistical models occur in many contexts and applications such as the analysis of big data sets, machine learning, estimation theory, statistical physics only to name a few. The basic problem is to extract information out of a (typically very large) set of data, for instance to estimate the probability distribution or the expectation value of certain random variables. One of the classical quantities associated to such a data set is the Fisher metric, which is a tensor quantifying the information loss when taking a statistic on the set (i.e., when measuring certain quantities). The typical approach is to start with some conjectured probability distribution, taking a sample of the data and then adjusting the conjectured probability distribution according to some algorithm. Iterating this process, the hope is that the adjusted probability distributions will converge reasonably fast to the actual probability distribution. This is the essential idea of the method of the Fisher gradient flow and of Bayesian statistics. While these approaches work reasonably well in practice as long as the set of data is “reasonably well behaved”, the question of convergence is difficult to formalize in general, as for this, a reasonable geometric structure on the set of probability measures needs to be established. Prior to this project, we had presented a very general geometric model which equally works for finite and infinite data sets. In the first part of the project, we were able to relax some of the assumptions of our previous approaches by replacing the notion of differentiable statistical models by the more general notion of diffeological statistical models. This now allows also to treat statistical models with certain types of singularities, thus defining a notion which includes statistical models with singularities and hence is more feasible in many situations. In this context, a description of Probabilistic morphisms and Bayesian nonparametrics as well as nonparametric estimations using the Diffeological Fisher Metric could be established. In the second part, we noticed an interesting link of our theory of classical statistical models to quantum states. In case of a finite quantum model, we were able to formulate a parametric estimation theory for states on finitedimensional C∗-algebras, which in the classical limit gives a formulation of estimators on finite statistical models. While in general, quantum systems tend to be more involved than classical ones, the transition via classical limits shed some light to the estimator theory of classical systems as well. In particular, the investigation of Jordan algebras, one of the formal tools in quantization, allowed us to interpret the Fisher metric tensor as a classical limit of the Bures-Helstrom metric tensor, corresponding to the embedding of an associative Jordan algebra into the Jordan algebra of Hermitean matrices. We hope that this approach will lead the way to establishing a correspondence between the classical quantum states to arbitrary statistical models even in the infinite dimensional case.
Publications
- Lagrangian submanifolds in a strictly nearly Kähler 6-manifolds, Osaka Journal of Mathematics. Vol. 56 no. 3, 601-629 (2019)
Hông Vân Lê, Lorenz Schwachhöfer
- Differential geometric aspects of parametric estimation theory for states on finite-dimensional C∗-algebras, Entropy Vol. 22(11), 1132 (30 p.) (2020)
Florio M. Ciaglia, Jürgen Jost, Lorenz J. Schwachhöfer
(See online at https://doi.org/10.3390/e22111332) - From the Jordan Product to Riemannian Geometries on Classical and Quantum States, Entropy Vol. 22(6), 637 (27 p.) (2020)
Florio M. Ciaglia, Jürgen Jost, Lorenz J. Schwachhöfer
(See online at https://doi.org/10.3390/e22060637) - Almost formality of manifolds of low dimension, Annali della Scuola Normale Superiore di Pisa, Vol. 22 no. 1, 79-107 (2021)
Domenico Fiorenza, Kotaro Kawai, Hông Vân Lê, Lorenz J. Schwachhöfer
(See online at https://doi.org/10.2422/2036-2145.201905_002)
