Partitioned K-nearest neighbor local depth for scalable comparison-based learning

Abstract

A triplet comparison oracle on a set $S$ takes an object $x\in S$ and for any pair $\{y,z\}\subset S\setminus \{x\}$ declares which of $y$ and $z$ is more similar to $x$. Partitioned Local Depth (PaLD) supplies a principled non-parametric partitioning of $S$ under such triplet comparisons but needs $O(n^2\log n)$ oracle calls and $O(n^3)$ post-processing steps. We introduce Partitioned Nearest Neighbors Local Depth (PaNNLD), a computationally tractable variant of PaLD leveraging the $K$-nearest neighbors digraph on $S$. PaNNLD needs only $O(nK\log n)$ oracle calls, by replacing an oracle call by a coin flip when neither $y$ nor $z$ is adjacent to $x$ in the undirected version of the $K$-nearest neighbors digraph. By averaging over randomizations, PaNNLD subsequently requires (at best) only $O(n{K}^2)$ post-processing steps. Concentration of measure shows that the probability of randomization-induced error $\delta$ in PaNNLD is no more than $2e^{-{\delta }^2{K}^2}$.

27 pages, 2 figures