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 $\vec{x}$ 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 $f(\vec{x}) = 1$ 当且仅当离 $\vec{x}$ 最近的锚点标记为 $1$。该模型由 Hajnal、Liu 和 Tur\'an（2022）引入，他们研究了在不同锚点选择（实向量与布尔向量）下表示布尔函数所需锚点数量的界限，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的视角，首次系统研究了最近邻与 $k$-最近邻模型的表示能力。为此，我们在具有多项式最近邻复杂度的布尔函数与那些能够被基于线性不等式的类别（最小加多项式阈值函数）高效表示的布尔函数之间建立了联系，后者先前已在阈值电路的相关研究中被探讨。这扩展了 Hajnal 等人（2022）的一个观察。对于显式的 $n$ 变量函数，在假设 $k \leq n^{1-\epsilon}$ 的条件下，我们获得了其 $k$-最近邻复杂度的指数级下界。此前，对于任何 $k>1$ 的情况，尚无超线性下界被知晓。接下来，我们将最近邻表示与电路之间的关联进一步扩展到 $k$-最近邻情形。结果表明，若要为任意 $k$ 下的显式函数证明其 $k$-最近邻复杂度的超多项式下界，将需要在电路复杂度领域取得突破。此外，我们证明了一个显式函数的最近邻复杂度与 $k$-最近邻复杂度（在 $k$ 无限制时）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 $k$-最近邻强下界以及理解参数 $k$ 作用的问题。最后，我们针对若干显式函数，给出了其最近邻复杂度的新界限。