Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may apply, and a standard test-bed for studying this prospect is matrix factorization (matrix completion via linear neural networks).

A Bayesian approach to partitioning distance matrices is presented. It is inspired by the Translation-invariant Wishart-Dirichlet process (TIWD) in [1] and shares a number of advantageous properties like the fully probabilistic nature of the inference model, automatic selection of the number of clusters and applicability in semi-supervised settings. In addition, our method (which we call fastTIWD) overcomes the main shortcoming of the original TIWD, namely its high computational costs.

Bayesian inference has theoretical attractions as a principled framework for reasoning about beliefs. However, the motivations of Bayesian inference which claim it to be the only 'rational' kind of reasoning do not apply in practice. They create a binary split in which all approximate inference is equally 'irrational'. Instead, we should ask ourselves how to define a spectrum of more- and less-rational reasoning that explains why we might prefer one Bayesian approximation to another. I explore approximate inference in Bayesian neural networks and consider the unintended interactions between the probabilistic model, approximating distribution, optimization algorithm, and dataset. The complexity of these interactions highlights the difficulty of any strategy for evaluating Bayesian approximations which focuses entirely on the method, outside the context of specific datasets and decision-problems. For given applications, the expected utility of the approximate posterior can measure inference quality. To assess a model's ability to incorporate different parts of the Bayesian framework we can identify desirable characteristic behaviours of Bayesian reasoning and pick decision-problems that make heavy use of those behaviours. Here, we use continual learning (testing the ability to update sequentially) and active learning (testing the ability to represent credence). But existing continual and active learning set-ups pose challenges that have nothing to do with posterior quality which can distort their ability to evaluate Bayesian approximations. These unrelated challenges can be removed or reduced, allowing better evaluation of approximate inference methods.