The Nearest Neighbor (NN) Representation is an emerging computational model that is inspired by the brain. We study the complexity of representing a neuron (threshold function) using the NN representations. It is known that two anchors (the points to which NN is computed) are sufficient for a NN representation of a threshold function, however, the resolution (the maximum number of bits required for the entries of an anchor) is $O(n\log{n})$. In this work, the trade-off between the number of anchors and the resolution of a NN representation of threshold functions is investigated. We prove that the well-known threshold functions EQUALITY, COMPARISON, and ODD-MAX-BIT, which require 2 or 3 anchors and resolution of $O(n)$, can be represented by polynomially large number of anchors in $n$ and $O(\log{n})$ resolution. We conjecture that for all threshold functions, there are NN representations with polynomially large size and logarithmic resolution in $n$.

翻译：最近邻（NN）表示是一种受大脑启发的新兴计算模型。我们研究了使用NN表示来表征神经元（阈值函数）的复杂度。已知两个锚点（计算NN所依据的点）足以表示一个阈值函数的NN表示，但分辨率（锚点条目所需的最大位数）为$O(n\log{n})$。本文研究了阈值函数的NN表示中锚点数量与分辨率之间的平衡关系。我们证明了著名的阈值函数EQUALITY、COMPARISON和ODD-MAX-BIT（需2或3个锚点且分辨率为$O(n)$）可以通过多项式数量的锚点（关于$n$）和$O(\log{n})$分辨率来表征。我们推测，对于所有阈值函数，存在规模为多项式大小且分辨率关于$n$呈对数级别的NN表示。