A nearest neighbor representation of a Boolean function $f$ is a set of vectors (anchors) labeled by $0$ or $1$ such that $f(\vec{x}) = 1$ if and only if the closest anchor to $x$ is labeled by $1$. This model was introduced by Hajnal, Liu, and Tur\'an (2022), who studied bounds on the number of anchors required to represent Boolean functions under different choices of anchors (real vs. Boolean vectors) as well as the more expressive model of $k$-nearest neighbors. We initiate the study of the representational power of nearest and $k$-nearest neighbors through Boolean circuit complexity. To this end, we establish a connection between Boolean functions with polynomial nearest neighbor complexity and those that can be efficiently represented by classes based on linear inequalities -- min-plus polynomial threshold functions -- previously studied in relation to threshold circuits. This extends an observation of Hajnal et al. (2022). We obtain exponential lower bounds on the $k$-nearest neighbors complexity of explicit $n$-variate functions, assuming $k \leq n^{1-\epsilon}$. Previously, no superlinear lower bound was known for any $k>1$. Next, we further extend the connection between nearest neighbor representations and circuits to the $k$-nearest neighbors case. As a result, we show that proving superpolynomial lower bounds for the $k$-nearest neighbors complexity of an explicit function for arbitrary $k$ would require a breakthrough in circuit complexity. In addition, we prove an exponential separation between the nearest neighbor and $k$-nearest neighbors complexity (for unrestricted $k$) of an explicit function. These results address questions raised by Hajnal et al. (2022) of proving strong lower bounds for $k$-nearest neighbors and understanding the role of the parameter $k$. Finally, we devise new bounds on the nearest neighbor complexity for several explicit functions.

翻译：布尔函数$f$的最近邻表示是一组标记为0或1的向量（锚点），使得当且仅当距离$x$最近的锚点标记为1时，$f(\vec{x}) = 1$。该模型由Hajnal、Liu和Turán（2022）提出，他们研究了在不同锚点选择（实数向量与布尔向量）下表示布尔函数所需锚点数量的界，以及更具表达力的$k$-最近邻模型。我们通过布尔电路复杂度，首次系统研究最近邻和$k$-最近邻的表示能力。为此，我们建立了具有多项式最近邻复杂度的布尔函数与可被基于线性不等式类（即min-plus多项式阈值函数）高效表示的布尔函数之间的联系，后者此前在阈值电路相关研究中被探讨。该结论扩展了Hajnal等人（2022）的观察结果。我们证明，在假设$k \leq n^{1-\epsilon}$时，显式$n$元函数的$k$-最近邻复杂度具有指数级下界。此前，对于任意$k>1$，均无已知的超线性下界。接着，我们将最近邻表示与电路之间的关联进一步扩展至$k$-最近邻情形。结果表明，若要对任意$k$证明显式函数的$k$-最近邻复杂度的超多项式下界，需在电路复杂度领域取得突破性进展。此外，我们证明了显式函数在最近邻与$k$-最近邻复杂度（$k$无限制）之间存在指数级分离。这些结果回应了Hajnal等人（2022）提出的关于证明$k$-最近邻强下界及理解参数$k$作用的问题。最后，我们为若干显式函数设计了最近邻复杂度的新界限。