Tensors in statistics and rigidity theory

TL;DR

This paper surveys the role of tensors in algebraic statistics and rigidity theory, highlighting algebraic techniques for analyzing statistical models and discussing secant varieties and their connections to rigidity theory.

Contribution

It provides an overview of algebraic and geometric methods for studying tensors in statistics and introduces new research directions, including tensor decompositions and their applications.

Findings

Decomposition of conditional independence varieties reveals new dependencies.

Application of tensor methods extends classical CI axioms.

Connection between secant varieties and rigidity theory is outlined.

Abstract

This is a short report on the discussions of appearance of tensors in algebraic statistics and rigidity theory, during the semester ``AGATES: Algebraic Geometry with Applications to TEnsors and Secants". We briefly survey some of the existing results in the literature and further research directions. We first provide an overview of algebraic and geometric techniques in the study of conditional independence (CI) statistical models. We study different families of algebraic varieties arising in statistics. This includes the determinantal varieties related to CI statements with hidden random variables. Such statements correspond to determinantal conditions on the tensor of joint probabilities of events involving the observed random variables. We show how to compute the irreducible decompositions of the corresponding CI varieties, which leads to finding further conditional dependencies (or independencies) among the involved random variables. As an example, we show how these methods can be applied to extend the classical intersection axiom for CI statements. We then give a brief overview about secant varieties and their appearance in the study of mixture models. We focus on examples and briefly mention the connection to rigidity theory which will appear in the forthcoming paper \cite{rigidity}.