Sharp Capacity Thresholds in Linear Associative Memory: From Top-1 Retrieval to Tail-Average Learning

Abstract

How many key-value associations can a $d\times d$ linear memory store? The answer depends not only on the $d^2$ degrees of freedom in the memory matrix, but also on the retrieval criterion. Under isotropic Gaussian embeddings, we prove a sharp threshold for top-1 retrieval, where every signal must beat its largest distractor: the critical value of $d^2/(n\log n)$ is $2$. Above the threshold, we explicitly construct a linear memory that retrieves all $n$ associations with high probability; below it, no data-dependent linear memory can do so. The $\log n$ factor is therefore the unavoidable extreme-value cost of winner-take-all decoding. Without the logarithmic factor---that is, when $n/d^2\to \alpha \in (0,\infty )$---simultaneous top-1 retrieval is impossible. The matched target can nevertheless remain near the top of the ranking. We capture this weaker retrieval goal with the Tail-Average Margin (TAM), which, for list size $k$, compares each signal with the average of its $k$ strongest competitors; a positive TAM margin certifies that the target belongs to the top-$k$ candidate list. When $k/n\to r\in (0,1)$, we learn the memory by empirical risk minimization with a smoothed TAM objective and derive an exact high-dimensional characterization through a two-parameter scalar variational problem. The result gives limiting laws for signal and competitor scores, margins, and percentile ranks. Sending the ridge parameter to zero after the high-dimensional limit yields a closed-form critical load ${\alpha }_c(r)$ separating vanishing from positive average loss. The analysis in this work also develops a coupled leave-one-out method for matrix-valued empirical risk problems in which each sample enters many dependent comparisons, a tool that may be useful beyond associative memory.