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The Four Point Permutation Test for Latent Block Structure in Incidence Matrices

4 October 2018
R. Darling
Cheyne Homberger
ArXiv (abs)PDFHTML
Abstract

Transactional data may be represented as a bipartite graph G:=(L∪R,E)G:=(L \cup R, E)G:=(L∪R,E), where LLL denotes agents, RRR denotes objects visible to many agents, and an edge in EEE denotes an interaction between an agent and an object. Unsupervised learning seeks to detect block structures in the adjacency matrix ZZZ between LLL and RRR, thus grouping together sets of agents with similar object interactions. New results on quasirandom permutations suggest a non-parametric \textbf{four point test} to measure the amount of block structure in GGG, with respect to vertex orderings on LLL and RRR. Take disjoint 4-edge random samples, order these four edges by left endpoint, and count the relative frequencies of the 4!4!4! possible orderings of the right endpoint. When these orderings are equiprobable, the edge set EEE corresponds to a quasirandom permutation π\piπ of ∣E∣|E|∣E∣ symbols. Total variation distance of the relative frequency vector away from the uniform distribution on 24 permutations measures the amount of block structure. Such a test statistic, based on ⌊∣E∣/4⌋\lfloor |E|/4 \rfloor⌊∣E∣/4⌋ samples, is computable in O(∣E∣/p)O(|E|/p)O(∣E∣/p) time on ppp processors. Possibly block structure may be enhanced by precomputing \textbf{natural orders} on LLL and RRR, related to the second eigenvector of graph Laplacians. In practice this takes O(d∣E∣)O(d |E|)O(d∣E∣) time, where ddd is the graph diameter. Five open problems are described.

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