On the Information Capacity of Nearest Neighbor Representations
This work addresses the theoretical foundations of associative computation models, which could impact neuromorphic computing and AI architectures, though it is incremental in focusing on symmetric Boolean functions.
The paper tackles the problem of representing Boolean functions using a brain-inspired associative computation model based on nearest neighbor anchors, and presents constructions that achieve optimal nearest neighbor complexity and resolution for symmetric Boolean functions.
The $\textit{von Neumann Computer Architecture}$ has a distinction between computation and memory. In contrast, the brain has an integrated architecture where computation and memory are indistinguishable. Motivated by the architecture of the brain, we propose a model of $\textit{associative computation}$ where memory is defined by a set of vectors in $\mathbb{R}^n$ (that we call $\textit{anchors}$), computation is performed by convergence from an input vector to a nearest neighbor anchor, and the output is a label associated with an anchor. Specifically, in this paper, we study the representation of Boolean functions in the associative computation model, where the inputs are binary vectors and the corresponding outputs are the labels ($0$ or $1$) of the nearest neighbor anchors. The information capacity of a Boolean function in this model is associated with two quantities: $\textit{(i)}$ the number of anchors (called $\textit{Nearest Neighbor (NN) Complexity}$) and $\textit{(ii)}$ the maximal number of bits representing entries of anchors (called $\textit{Resolution}$). We study symmetric Boolean functions and present constructions that have optimal NN complexity and resolution.