Equivariant and Scale-Free Tucker Decomposition Models

Analyses of array-valued datasets often involve reduced-rank array approximations, typically obtained via least-squares or truncations of array decompositions. However, least-squares approximations tend to be noisy in high-dimensional settings, and may not be appropriate for arrays that include discrete or ordinal measurements. This article develops methodology to obtain low-rank model-based representations of continuous, discrete and ordinal data arrays. The model is based on a parameterization of the mean array as a multilinear product of a reduced-rank core array and a set of index-specific orthogonal eigenvector matrices. It is shown how orthogonally equivariant parameter estimates can be obtained from Bayesian procedures under invariant prior distributions. Additionally, priors on the core array are developed that act as regularizers, leading to improved inference over the standard least-squares estimator, and providing robustness to misspecification of the array rank. This model-based approach is extended to accommodate discrete or ordinal data arrays using a semiparametric transformation model. The resulting low-rank representation is scale-free, in the sense that it is invariant to monotonic transformations of the data array. In an example analysis of a multivariate discrete network dataset, this scale-free approach provides a more complete description of data patterns.