Maximum Likelihood Estimation for Matrix Normal Models via Quiver Representations

Maximum likelihood estimation for tensor normal models via castling transforms

“…Note that Lemma 2.9 was stated only for . There are many ways to adapt the argument for : for example, one can use [12, Proposition 2.23].…”

“…A special case of tensor normal models is the matrix normal model , where the concentration matrix is a Kronecker product of exactly two matrices. Sample size thresholds for matrix normal models have been investigated in [14, 32, 37, 13, 36, 2, 12]. In particular, a complete answer for matrix normal models was obtained in [12] with techniques from quiver representations.…”

“…Sample size thresholds for matrix normal models have been investigated in [14, 32, 37, 13, 36, 2, 12]. In particular, a complete answer for matrix normal models was obtained in [12] with techniques from quiver representations. We do not use quiver representations in this paper; instead, we use castling transforms and results on stabilisers in general position.…”

“…It was conjectured in [13] and proved in [12] that for matrix normal models (tensor normal models with ), almost sure boundedness of the log-likelihood function implies almost sure existence of an MLE. Theorem 1.1 implies that the same holds for all tensor normal models.…”

Ranks of linear matrix pencils separate simultaneous similarity orbits

“…, m. The classification of matrix tuples up to similarity has been deemed a "hopeless problem" [LB97]. Nevertheless, the study of simultaneous similarity and related group actions on matrix tuples is crucial in multiple areas of mathematics, ranging from operator theory [Fri83,DKS04], invariant and representation theory [Dro80,Pro76] and algebraic geometry [EH88,LBR99] to algebraic statistics [AKRS21,DM21] and computational complexity [GGOW16,DM17,IQS17]. As one would expect, this allows for many perspectives in studying matrix tuples and the transfer of ideas across disciplines can be especially fruitful.…”