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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 (SSB) 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 SSB. Overall, our results demonstrate that the unsupervised learning is a progressive combination of SSB 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.

Sparse superposition codes were originally proposed as a capacity-achieving communication scheme over the gaussian channel, whose coding matrices were made of i.i.d. gaussian entries [1]. We extend this coding scheme to more generic ensembles of rotational invariant coding matrices with arbitrary spectrum, which include the gaussian ensemble as a special case. We further introduce and analyse a decoder based on vector approximate message-passing (VAMP) [2]. Our main findings, based on both a standard replica symmetric potential theory and state evolution analysis, are the superiority of certain structured ensembles of coding matrices (such as partial row-orthogonal) when compared to i.i.d. matrices, as well as a spectrum-independent upper bound on VAMP's threshold. Most importantly, we derive a simple "spectral criterion" for the scheme to be at the same time capacity-achieving while having the best possible algorithmic threshold, in the "large section size" asymptotic limit. Our results therefore provide practical design principles for the coding matrices in this promising communication scheme.

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