A mathematical analysis of hierarchical Hopfield models

Markus Heydenreich; Christian Hirsch; Matthias Löwe

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arXiv · 2604.25470

A mathematical analysis of hierarchical Hopfield models

Markus Heydenreich·Christian Hirsch·Matthias Löwe

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Abstract

The central question that we address is: How can structured information be stored in a hierarchical Hopfield model involving hidden layers? To this end, we develop a formalism of strokes and concepts that allows us to appropriately structure information: initial features are first classified into strokes, which in a second step are aggregated into concepts. We rigorously derive criteria under which concepts can be retrieved from noisy input data. A remarkable effect is that we do not require a perfect retrieval at the level of strokes, as the second-layer retrieval procedure compensates for first-layer errors. We treat separately the cases of fixed and variable-sized concepts.

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Markus Heydenreich, Christian Hirsch, Matthias Löwe. 2026-04-28. A mathematical analysis of hierarchical Hopfield models. https://arxiv.org/abs/2604.25470

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