Nearest Neighbor Complexity and Boolean Circuits

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.