Dynamical mean-field theory for associative memory with block-correlated patterns
In associative memories storing correlated patterns, retrieval of individual memories can compete with symmetric mixed states. A dynamical description is needed to determine how the initial state selects between these outcomes. We extend the generating-functional dynamical mean-field theory (DMFT) for independent patterns to independent blocks of correlated binary patterns of fixed finite size. The resulting effective single-neuron process has a self-consistent cross-talk noise covariance and self-feedback coefficients that depend on the block-covariance eigenvalues. For hierarchically correlated patterns with nonmonotonic transfer, increasing the loading can permit retrieval from an imperfect initial state before further loading prevents it. DMFT reproduces the retrieval trajectories and basin geometry found in direct simulations. Near the lower boundary of this re-entrant region, direct networks reveal stationary saddles separating coexisting mixed and target-retrieval states. Parameter sensitivities of DMFT trajectories predict nearby shifts of this boundary. At the tested upper boundary, retrieval remains accessible from initial states with larger target overlap, indicating a change in basin access. Under fixed-point convergence assumptions, the theory recovers the equations of generalized self-consistent signal-to-noise analysis.