On the minimal algebraic complexity of the rank-one approximation problem for general inner products

TL;DR

This paper investigates the algebraic complexity of approximating tensors by rank-one tensors under various inner products, identifying the Frobenius inner product as a local minimum and conjecturing it as a global minimum, with proofs in specific cases.

Contribution

The paper introduces a conjecture that the Frobenius inner product minimizes the ED degree globally, proves it for matrices and certain tensors, and classifies ED degrees for various algebraic varieties.

Findings

Frobenius inner product is a local minimum of the ED degree.

Conjecture: Frobenius inner product is the global minimum.

Classified ED degrees for different algebraic varieties.

Abstract

We study the algebraic complexity of Euclidean distance minimization from a generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED) degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products. We prove that Frobenius inner product is a local minimum of the ED degree, and conjecture that it is a global minimum. We prove our conjecture in the case of matrices and symmetric binary and $3\times 3\times 3$ tensors. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry.