  An ideal cognitively-inspired memory system would compress and organize
incoming items. The Kanerva Machine (Wu et al, 2018) is a Bayesian model that
naturally implements online memory compression. However, the organization of
the Kanerva Machine is limited by its use of a single Gaussian random matrix
for storage. Here we introduce the Product Kanerva Machine, which dynamically
combines many smaller Kanerva Machines. Its hierarchical structure provides a
principled way to abstract invariant features and gives scaling and capacity
advantages over single Kanerva Machines. We show that it can exhibit
unsupervised clustering, find sparse and combinatorial allocation patterns, and
discover spatial tunings that approximately factorize simple images by object.
