Grassmann and Flag Varieties in Linear Algebra, Optimization, and Statistics: An Algebraic Perspective

TL;DR

This paper explores the algebraic complexity of optimization problems on Grassmann and flag varieties, revealing the number of complex critical points as a key measure, with implications for linear algebra and statistics.

Contribution

It provides an algebraic perspective on the complexity of optimization problems over Grassmann and flag varieties, including explicit computations of critical points.

Findings

Computed algebraic complexity for various optimization problems

Connected algebraic geometry with applications in linear algebra and statistics

Provided realizations of manifolds as algebraic varieties

Abstract

Grassmann and flag varieties lead many lives in pure and applied mathematics. Here we focus on the algebraic complexity of solving various problems in linear algebra and statistics as optimization problems over these varieties. The measure of the algebraic complexity is the amount of complex critical points of the corresponding optimization problem. After an exposition of different realizations of these manifolds as algebraic varieties we present a sample of optimization problems over them and we compute their algebraic complexity.