Multilinear Kernel Regression and Imputation via Manifold Learning

Abstract

This paper introduces a novel kernel regression framework for data imputation, coined multilinear kernel regression and imputation via the manifold assumption (MultiL-KRIM). Motivated by manifold learning, MultiL-KRIM models data features as a point-cloud located in or close to a user-unknown smooth manifold embedded in a reproducing kernel Hilbert space. Unlike typical manifold-learning routes, which seek low-dimensional patterns via regularizers based on graph-Laplacian matrices, MultiL-KRIM builds instead on the intuitive concept of tangent spaces to manifolds and incorporates collaboration among point-cloud neighbors (regressors) directly into the data-modeling term of the loss function. Multiple kernel functions are allowed to offer robustness and rich approximation properties, while multiple matrix factors offer low-rank modeling, dimensionality reduction and streamlined computations, with no need of training data. Two important application domains showcase the functionality of MultiL-KRIM: time-varying-graph-signal (TVGS) recovery, and reconstruction of highly accelerated dynamic-magnetic-resonance-imaging (dMRI) data. Extensive numerical tests on real TVGS and synthetic dMRI data demonstrate that the “shallow” MultiL-KRIM offers remarkable speedups over its predecessors and outperforms other “shallow” state-of-the-art techniques, with a more intuitive and explainable pipeline than deep-image-prior methods.