Maximum likelihood estimation for tensor normal models via castling transforms

TL;DR

This paper investigates the conditions under which maximum likelihood estimation for tensor normal models is feasible, focusing on sample size thresholds and using invariant theory and castling transforms.

Contribution

It provides a complete characterization of sample size thresholds for existence and uniqueness of MLEs in tensor normal models, extending to real and complex cases.

Findings

Sample size thresholds for MLE existence and uniqueness are established.

Boundedness of the likelihood function implies the existence of MLEs.

Results apply to both real and complex tensor normal models.

Abstract

In this paper, we study sample size thresholds for maximum likelihood estimation for tensor normal models. Given the model parameters and the number of samples, we determine whether, almost surely, (1) the likelihood function is bounded from above, (2) maximum likelihood estimates (MLEs) exist, and (3) MLEs exist uniquely. We obtain a complete answer for both real and complex models. One consequence of our results is that almost sure boundedness of the log-likelihood function guarantees almost sure existence of an MLE. Our techniques are based on invariant theory and castling transforms.