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We study the algebraic complexity of Euclidean distance minimization from ageneric tensor to a variety of rank-one tensors. The Euclidean Distance (ED)degree of the Segre-Veronese variety counts the number of complex criticalpoints of this optimization problem. We regard this invariant as a function ofinner products and conjecture that it achieves its minimal value at Frobeniusinner product. We prove our conjecture in the case of matrices. We discuss theabove optimization problem for other algebraic varieties, classifying allpossible values of the ED degree. Our approach combines tools from SingularityTheory, Morse Theory, and Algebraic Geometry.

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