Partitioned K-nearest neighbor local depth for scalable comparison-based learning
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A triplet comparison oracle on a set S takes an object x ∈ S and for any pair {y, z}⊂ S ∖{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 logn) 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(n K logn) 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 δ in PaNNLD is no more than 2 e^-δ^2 K^2.
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