We analytically study the evolution of the energy function of Dense Associative Memories (DenseAMs) as they train on randomly sampled codewords of random expander low-density parity-check codes (a.k.a. solutions of random XORSAT problems). We identify two distinct regimes and a transition between them, reminiscent of grokking in neural networks. A regime of small training set size is where the model memorizes a few training examples but cannot discover the underlying rules of the entire codeword dictionary generative process. The opposite regime, corresponding to a larger training set size, is where the model has uncovered the sparse constraints that generated the entire data set after seeing only polynomially many patterns. We also design a novel class of DenseAMs with a thresholding learning rule, and show that any amount of thresholding enables the model to perform simplified sparse Hamiltonian learning, which exponentially reduces the number of training examples needed to learn the true underlying data set. In this regime, the model learns how to discover $\exp[\mathrm{O}(n)]$ solutions from only $\mathrm{O}(\log n)$ training samples, which is asymptotically optimal for any learning algorithm. When training on codewords of certain random codes, the model can even learn the underlying code when a small O(1) fraction of bits are latent variables. These latent variables are not shown to the model during learning, but will be discovered by sparsifying observed higher-order correlations. For certain random codes, we show that a simple and conservative algorithm learns a minimal latent representation for the data set. This provides a transparent, physics-motivated, and analytically tractable toy model for how deep latent representations emerge in an associative memory training.

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