Maximum likelihood estimation for tensor normal models via castling transforms

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.

arXiv comment22 pages

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• Towards a theory of non-commutative optimization: geodesic first and second order methods for moment maps and polytopes
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• Quantum max-flow in the bridge graph
• Crossed products by compact group actions with the weak tracial Rokhlin property
• Near optimal sample complexity for matrix and tensor normal models via geodesic convexity