Multilinear Kernel Regression and Imputation via Manifold Learning
<|reference_start|>Multilinear Kernel Regression and Imputation via Manifold Learning: 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.<|reference_end|>
@article{nguyen2024multilinear,
    title={Multilinear Kernel Regression and Imputation via Manifold Learning},
    author={Duc Thien Nguyen and Konstantinos Slavakis},
    journal={arXiv preprint arXiv:2402.03648},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.03648},
    primaryClass={eess.SP cs.LG}
}