Quantify scaling of ill-posedness for rank-2 approximation in n × n × n tensors

Establish quantitative lower and upper bounds that characterize how the measure or probability of n × n × n real tensors lacking a best rank-2 approximation scales as a function of n. In particular, derive bounds on the probability of ill-posedness for rank-2 tensor approximation in n × n × n formats and determine its asymptotic behavior with respect to n.

Background

The authors show that non-closedness of bounded-rank tensor sets and ill-posedness of low-rank approximation persist in higher dimensions. They cite that a positive volume of n × n × n tensors has no best rank-2 approximation, demonstrating pervasive ill-posedness.

Despite this, a quantitative understanding of the prevalence of ill-posedness is missing. The authors explicitly state it remains open to describe how the volume or probability of such ill-posed instances scales with n, motivating the need for precise probabilistic or measure-theoretic bounds.