Regarding the temporal requirements of a hierarchical Willshaw network Jo˜aoSacramento a,∗ , Francisco Burnay b , Andreas Wichert a a INESC-ID Lisboa and Instituto Superior T´ ecnico, Technical University of Lisbon, Av. Prof. Dr. An´ ıbal Cavaco Silva, 2744-016 Porto Salvo, Portugal b Instituto de Plasmas e Fus˜ ao Nuclear, Instituto Superior T´ ecnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Abstract In a recent communication, Sacramento & Wichert (2011) proposed a hierarchical re- trieval prescription for Willshaw-type associative networks. Through simulation it was shown that one could make use of low resolution descriptor patterns to decrease the total time requirements of recalling a learnt association. However, such method introduced a dependence on a set of new parameters which define the structure of the hierarchy. In this work we compute the expected retrieval time for the random neural activity regime which maximises the capacity of the Willshaw model and we study the task of finding the optimal hierarchy parametrisation with respect to the derived temporal expectation. Still in regard to this performance measure, we investigate some asymptotic properties of the algorithm. Keywords: Associative memory, Willshaw model, Retrieval time, Hierarchical neural network, Sparse coding 1. Introduction In the strictest technical sense, an associative memory model is designed to solve a variation of the classical nearest neighbour determination problem. Instead of finding a solution for the original labelled classification task formulation (Fix & Hodges, 1951; Cover & Hart, 1967; Minsky & Papert, 1969), an associative memory is a system that stores information about a finite set of M associations of the form S := {(x μ → y μ ): µ =1,...,M }, (1) with most memory models assuming the patterns are binary vectors, i.e., x ∈{0, 1} m and y ∈{0, 1} n . Given a possibly corrupt or incomplete pattern ˜ x ∈{0, 1} m , the system * Corresponding author. Tel.: +351 21 423 32 31; fax: +351 21 314 48 43. Email addresses: joao.sacramento@ist.utl.pt (Jo˜aoSacramento), francisco.burnay@ist.utl.pt (Francisco Burnay), andreas.wichert@ist.utl.pt (Andreas Wichert) Preprint submitted to Neural Networks July 17, 2011