Reflexive Regular Equivalence for Bipartite Data

Bipartite data is common in data engineering and brings unique challenges, particularly when it comes to clustering tasks that impose strong structural assumptions. This work presents an unsupervised method for assessing similarity in bipartite data. The method is based on regular equivalence in graphs and uses spectral properties of a bipartite adjacency matrix to estimate similarity in both dimensions. The method is reflexive in that similarity in one dimension informs similarity in the other. The method also uses local graph transitivities, a contribution governed by its only free parameter. Reflexive regular equivalence can be used to validate assumptions of co-similarity, which are required but often untested in co-clustering analyses. The method is robust to noise and asymmetric data, making it particularly suited for cluster analysis and recommendation in data of unknown structure. In bipartite data, co-similarity is the notion that similarity in one dimension is matched by similarity in some other dimension. Such data occurs in a many areas: text mining, gene expression networks, consumer co-purchasing data and social affiliation. In bipartite analyses, co-clustering is an increasingly prominent technique in a range of applications [11], but has strong co-similarity assumptions. One example of co-similar structure is in text analysis where similar words appear in similar documents, where there is assumed to be a permutation of the word-document co-occurrence matrix that exposes co-similarity among words and documents [4]. The work here describes a way to assess co-similarity using regular equivalence [8] with a reflexive conception of similarity that accommodates nodes’ (data-points) local structures. This assessment is a kind of pre-condition for co-clustering: if there is little co-similarity, co-clustering will yield a poor clustering solution. Assessing co-similarity will produce similarity in one dimension to expose potential clustering without requiring it across dimensions. This is particularly useful for asymmetric data when a non-clustered dimension informs, but does not reciprocate clustering in the other. Whereas as co-clustering finds clusters across dimensions, our method provides a decoupled solution in each mode. Reflexive regular equivalence is able to quantify how much one dimension informs similarity in the other. Additionally, our results show that by incorporating local structures, it can better overcome noise and accommodate asymmetry. 4 An extended preprint of this paper is available at arxiv.org/abs/1702.04956.