| Summary: | In 1943, McCulloch and Pitts introduced a discrete recurrent neural network as a model for computation in brains. The work inspired breakthroughs such as the first computer design and the theory of finite automata. We focus on learning in Hopfield networks, a special case with symmetric weights and fixed-point attractor dynamics. Specifically, we explore minimum energy flow (MEF) as a scalable convex objective for determining network parameters. We catalog various properties of MEF, such as biological plausibility, and then compare to classical approaches in the theory of learning. Trained Hopfield networks can perform unsupervised clustering and define novel error-correcting coding schemes. They also efficiently find hidden structures (cliques) in graph theory. We extend this known connection from graphs to hypergraphs and discover <i>n</i>-node networks with robust storage of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mrow><mi mathvariant="sans-serif">Ω</mi><mo>(</mo><msup><mi>n</mi><mrow><mn>1</mn><mo>−</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></mrow></msup></semantics></math></inline-formula> memories for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϵ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. In the case of graphs, we also determine a critical ratio of training samples at which networks generalize completely.
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