Optimizing the Induced Correlation in Omnibus Joint Graph Embeddings

TL;DR

This paper introduces an automated method to optimize the correlation structure in joint graph embeddings, improving inference quality by addressing correlation issues in the Omnibus framework.

Contribution

It presents the first automated approach to construct Omnibus matrices that optimize correlation, including theoretical bounds and an algorithm called corr2Omni.

Findings

The classical Omnibus construction induces maximal flat correlation.

The corr2Omni algorithm effectively estimates optimal Omnibus weights.

Enhanced inference accuracy demonstrated in both simulated and real data.

Abstract

Theoretical and empirical evidence suggests that joint graph embedding algorithms induce correlation across the networks in the embedding space. In the Omnibus joint graph embedding framework, previous results explicitly delineated the dual effects of the algorithm-induced and model-inherent correlations on the correlation across the embedded networks. Accounting for and mitigating the algorithm-induced correlation is key to subsequent inference, as sub-optimal Omnibus matrix constructions have been demonstrated to lead to loss in inference fidelity. This work presents the first efforts to automate the Omnibus construction in order to address two key questions in this joint embedding framework: the correlation-to-OMNI problem and the flat correlation problem. In the flat correlation problem, we seek to understand the minimum algorithm-induced flat correlation (i.e., the same across all graph pairs) produced by a generalized Omnibus embedding. Working in a subspace of the fully general Omnibus matrices, we prove both a lower bound for this flat correlation and that the classical Omnibus construction induces the maximal flat correlation. In the correlation-to-OMNI problem, we present an algorithm -- named corr2Omni -- that, from a given matrix of estimated pairwise graph correlations, estimates the matrix of generalized Omnibus weights that induces optimal correlation in the embedding space. Moreover, in both simulated and real data settings, we demonstrate the increased effectiveness of our corr2Omni algorithm versus the classical Omnibus construction.