Statistical testing of epiphenomena for multi-index data

As large tensor-variate data increasingly become the norm in applied machine learning and statistics, complex analysis methods similarly increase in prevalence. Such a trend offers the opportunity to understand more intricate features of the data that, ostensibly, could not be studied with simpler datasets or simpler methodologies. While promising, these advances are also perilous: these novel analysis techniques do not always consider the possibility that their results are in fact an expected consequence of some simpler, already-known feature of simpler data (for example, treating the tensor like a matrix or a univariate quantity) or simpler statistic (for example, the mean and covariance of one of the tensor modes). I will present two works that address this growing problem, the first of which uses Kronecker algebra to derive a tensor-variate maximum entropy distribution that shares modal moments with the real data. This distribution of surrogate data forms the basis of a statistical hypothesis test, and I use this method to answer a question of epiphenomenal tensor structure in populations of neural recordings in the motor and prefrontal cortex. In the second part, I will discuss how to extend this maximum entropy formulation to arbitrary constraints using deep neural network architectures in the flavor of implicit generative modeling, and I will use this method in a texture synthesis application.