The average number of critical rank-one approximations to a tensor

Abstract

Motivated by the many potential applications of low-rank multiway tensor approximations, we set out to count the rank-one tensors that are critical points of the distance function to a general tensor v. As this count depends on v, we average over v drawn from a Gaussian distribution, and find a formula that relates this average to problems in random matrix theory.

The Euclidean Distance Degree of an Algebraic Variety

“…In this section, we present a brief account of recent work on multidimensional tensors of rank one [14]. For these, the ED degree is computed in [15], and the average ED degree is computed in [10]. Our discussion includes partially symmetric tensors, and it represents a step towards extending the Eckart-Young theorem from matrices to tensors.…”

“…The case of multidimensional tensors, while of equally fundamental importance, is much more involved. In Section 8, following [10,14,15], we give an account of recent results on the ordinary and average ED degree of the variety of rank one tensors.…”

The Expected Number of Eigenvalues of a Real Gaussian Tensor

“…In [8] Draisma and Horobet point out that application-oriented algorithms to compute the best rank-one approximations, or, more generally, the best low-rank approximations, are mostly of a local nature. Therefore, they face difficulties when close to nonminimal critical points of the distance function that one wants to minimize.…”

The number of real eigenvectors of a real polynomial

“…Again for a sample of 100000 forms f , we have estimated the probabilities of the aleatory variables X f = (0, 2, 4) for d = 4, Y f = (1, 3, 5) for d = 5 and respectively X yfx−xfy = (0, 2, 4), Y yfx−xfy = (1, 3, 5) with respect to f and yf x − xf y and then relative expected values and we expect that E(X f ) ≈ √ d and E(X yfx−xfy ) ≈ √ 3d − 2 and the same for E(Y f ) and E(Y yfx−xfy ) (see Example 1.6 in [8] and Example 4.8 in [9]):…”