  Integrating sensory inputs with prior beliefs from past experiences in
unsupervised learning is a common and fundamental characteristic of brain or
artificial neural computation. However, a quantitative role of prior knowledge
in unsupervised learning remains unclear, prohibiting a scientific
understanding of unsupervised learning. Here, we propose a statistical physics
model of unsupervised learning with prior knowledge, revealing that the sensory
inputs drive a series of continuous phase transitions related to spontaneous
intrinsic-symmetry breaking. The intrinsic symmetry includes both reverse
symmetry and permutation symmetry, commonly observed in most artificial neural
networks. Compared to the prior-free scenario, the prior reduces more strongly
the minimal data size triggering the reverse symmetry breaking transition, and
moreover, the prior merges, rather than separates, permutation symmetry
breaking phases. We claim that the prior can be learned from data samples,
which in physics corresponds to a two-parameter Nishimori constraint. This work
thus reveals mechanisms about the influence of the prior on unsupervised
learning.

Disordered Systems and Neural Networks;Statistical Mechanics;Machine Learning