Multilinear Kernel Regression and Imputation via Manifold Learning

TL;DR

This paper presents MultiL-KRIM, a new nonparametric manifold learning framework for data imputation that models data as points near a manifold in a kernel space, improving speed and accuracy over existing methods.

Contribution

It introduces a novel manifold-based regression and imputation method using tangent spaces and multiple kernels, with no training data needed, enhancing robustness and computational efficiency.

Findings

Outperforms existing imputation techniques in speed and accuracy

Effective in recovering time-varying graph signals and dynamic MRI data

Provides a more explainable alternative to deep learning methods

Abstract

This paper introduces a novel nonparametric 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, integrate dimensionality reduction, and streamline 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 and synthetic data demonstrate MultiL-KRIM's remarkable speedups over its predecessors, and outperformance over prevalent "shallow" data-imputation techniques, with a more intuitive and explainable pipeline than deep-image-prior methods.