Robust retrieval in a hierarchical associative memory

We study a hierarchical associative-memory model inspired by hierarchical Hopfield networks with hidden layers. While such architectures are often viewed as biologically plausible realisations of dense associative memories, it remains less clear which mathematical advantages are provided by the hierarchical organisation itself. To address this question, we introduce a sparse stroke–concept model in which elementary feature patterns (strokes) are first stored and subsequently combined into higher-level memories (concepts). Retrieval proceeds in two stages: a feature cue activates candidate strokes, and a second-layer decoding rule identifies the most likely concept. Our main result shows that successful concept retrieval does not require exact recovery of all constituent strokes. Instead, it is sufficient to control aggregate false negatives and false positives at the intermediate stroke layer. We derive deterministic and probabilistic retrieval criteria that quantify the interplay between stroke-layer errors and overlaps among competing concepts. In particular, the second retrieval stage can compensate for a substantial amount of first-layer noise. We analyse both fixed-size and variable-size concept ensembles. For variable concept sizes, we introduce suitable penalised and normalised scoring rules and establish corresponding retrieval guarantees. The results provide a rigorous mathematical description of how hierarchical organisation can improve retrieval robustness, even in situations where it does not alter the classical Hopfield-type storage-capacity scaling.