Imputation of Time-varying Edge Flows in Graphs by Multilinear Kernel Regression and Manifold Learning

TL;DR

This paper introduces MultiL-KRIM, a novel method combining multilinear kernel regression and manifold learning to accurately impute time-varying edge flows in graphs, leveraging graph topology and latent geometries without requiring training data.

Contribution

It extends existing frameworks by integrating simplicial-complex topology, Hodge Laplacians, and manifold learning for efficient, data-free imputation of dynamic graph edge flows.

Findings

MultiL-KRIM outperforms state-of-the-art schemes in real-network tests.

The method effectively captures latent geometries and temporal dynamics.

Dimensionality reduction enables efficient computation without training data.

Abstract

This paper extends the recently developed framework of multilinear kernel regression and imputation via manifold learning (MultiL-KRIM) to impute time-varying edge flows in a graph. MultiL-KRIM uses simplicial-complex arguments and Hodge Laplacians to incorporate the graph topology, and exploits manifold-learning arguments to identify latent geometries within features which are modeled as a point-cloud around a smooth manifold embedded in a reproducing kernel Hilbert space (RKHS). Following the concept of tangent spaces to smooth manifolds, linear approximating patches are used to add a collaborative-filtering flavor to the point-cloud approximations. Together with matrix factorizations, MultiL-KRIM effects dimensionality reduction, and enables efficient computations, without any training data or additional information. Numerical tests on real-network time-varying edge flows demonstrate noticeable improvements of MultiL-KRIM over several state-of-the-art schemes.