  Permutation of any two hidden units yields invariant properties in typical
deep generative neural networks. This permutation symmetry plays an important
role in understanding the computation performance of a broad class of neural
networks with two or more hidden units. However, a theoretical study of the
permutation symmetry is still lacking. Here, we propose a minimal model with
only two hidden units in a restricted Boltzmann machine, which aims to address
how the permutation symmetry affects the critical learning data size at which
the concept-formation (or spontaneous symmetry breaking in physics language)
starts, and moreover semi-rigorously prove a conjecture that the critical data
size is independent of the number of hidden units once this number is finite.
Remarkably, we find that the embedded correlation between two receptive fields
of hidden units reduces the critical data size. In particular, the
weakly-correlated receptive fields have the benefit of significantly reducing
the minimal data size that triggers the transition, given less noisy data.
Inspired by the theory, we also propose an efficient fully-distributed
algorithm to infer the receptive fields of hidden units. Furthermore, our
minimal model reveals that the permutation symmetry can also be spontaneously
broken following the spontaneous symmetry breaking. Overall, our results
demonstrate that the unsupervised learning is a progressive combination of
spontaneous symmetry breaking and permutation symmetry breaking which are both
spontaneous processes driven by data streams (observations). All these effects
can be analytically probed based on the minimal model, providing theoretical
insights towards understanding unsupervised learning in a more general context.
