Covariance Kernels on Unordered Pair Spaces: Theory and Applications to Network-Valued Data

Many scientific problems are fundamentally relational: the quantity of interest is a connection between two objects, while the information that describes similarity is available for the objects themselves. This mismatch is common in brain networks, molecular interactions, and other high-dimensional systems. We develop a covariance framework for unordered relationships that transfers object-level geometry to the relations they form while preserving endpoint identity and invariance to ordering. Building on symmetric pairwise-kernel representations, we develop new theory for the loop-free domains used in undirected networks. We establish spectral interlacing and trace-loss results after self-pairs are removed, connect the relational spectrum to regularization and risk, derive an exact inferential error for spectral truncation, and quantify how perturbations of the underlying geometry propagate to the pair covariance and estimator. Simulations show when structured borrowing improves estimation and when geometric misspecification can erode that benefit. We then apply the framework to autism neuroimaging using resting-state functional-connectivity data from the Autism Brain Imaging Data Exchange (ABIDE). The analysis considers a 116-region brain parcellation, yielding 6,670 unique functional connections, and shows that this large connectome can have a far smaller effective covariance dimension. The application also demonstrates that high explained covariance alone is not sufficient for choosing a low-rank representation when the goal is to preserve inferential accuracy. These results illustrate how biologically meaningful information defined at the brain-region level can organize dependence among connections while retaining the connection itself as the inferential unit. The framework provides a principled foundation for covariance and regularization when the statistical units are unordered relationships.