Thermal Robustness of Retrieval in Dense Associative Memories: LSE vs LSR Kernels
This work addresses the problem of bridging theoretical capacity proofs with practical thermal noise conditions for researchers in computational neuroscience and machine learning, though it is incremental as it compares existing kernels.
The study investigated the thermal robustness of retrieval in dense associative memories using Monte Carlo simulations for two kernels, finding that the LSE kernel sustains retrieval at high temperatures for low loads, while the LSR kernel shows near-perfect retrieval across most loads with a finite support threshold.
Understanding whether retrieval in dense associative memories survives thermal noise is essential for bridging zero-temperature capacity proofs with the finite-temperature conditions of practical inference and biological computation. We use Monte Carlo simulations to map the retrieval phase boundary of two continuous dense associative memories (DAMs) on the $N$-sphere with an exponential number of stored patterns $M = e^{αN}$: a log-sum-exp (LSE) kernel and a log-sum-ReLU (LSR) kernel. Both kernels share the zero-temperature critical load $α_c(0)=0.5$, but their finite-temperature behavior differs markedly. The LSE kernel sustains retrieval at arbitrarily high temperatures for sufficiently low load, whereas the LSR kernel exhibits a finite support threshold below which retrieval is perfect at any temperature; for typical sharpness values this threshold approaches $α_c$, making retrieval nearly perfect across the entire load range. We also compare the measured equilibrium alignment with analytical Boltzmann predictions within the retrieval basin.