# Multilinear Kernel Regression and Imputation via Manifold Learning

arXiv: https://arxiv.org/abs/2402.03648 · HTML: https://mlsanity.com/paper/2402.03648/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 122 references that are in the map.
- First posted on arXiv 6 Feb 2024.

## Builds on (most cited first, 30 of 31)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview](https://mlsanity.com/paper/1809.09573/) (2018)
- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Neural Collaborative Filtering vs. Matrix Factorization Revisited](https://mlsanity.com/paper/2005.09683/) (2020)
- [Deep Learning Methods for Parallel Magnetic Resonance Image Reconstruction](https://mlsanity.com/paper/1904.01112/) (2019)
- [Time-Dependent Deep Image Prior for Dynamic MRI](https://mlsanity.com/paper/1910.01684/) (2019)
- [Complete Dictionary Recovery over the Sphere I: Overview and the Geometric Picture](https://mlsanity.com/paper/1511.03607/) (2015)
- [Low-Rank Matrix Completion: A Contemporary Survey](https://mlsanity.com/paper/1907.11705/) (2019)
- [Compressive sensing with un-trained neural networks: Gradient descent finds the smoothest approximation](https://mlsanity.com/paper/2005.03991/) (2020)
- [Nonconvex Low-Rank Tensor Completion from Noisy Data](https://mlsanity.com/paper/1911.04436/) (2019)
- [Matrix Completion on Graphs](https://mlsanity.com/paper/1408.1717/) (2014)
- [Signal Recovery on Graphs: Variation Minimization](https://mlsanity.com/paper/1411.7414/) (2014)
- [Complete Dictionary Recovery over the Sphere II: Recovery by Riemannian Trust-region Method](https://mlsanity.com/paper/1511.04777/) (2015)
- [Graph Signal Processing for Geometric Data and Beyond: Theory and Applications](https://mlsanity.com/paper/2008.01918/) (2020)
- [Reconstruction of Time-varying Graph Signals via Sobolev Smoothness](https://mlsanity.com/paper/2207.06439/) (2022)
- [Dynamic imaging using a deep generative SToRM (Gen-SToRM) model](https://mlsanity.com/paper/2102.00034/) (2021)
- [Untrained Graph Neural Networks for Denoising](https://mlsanity.com/paper/2109.11700/) (2021)
- [Kernel-based Reconstruction of Space-time Functions on Dynamic Graphs](https://mlsanity.com/paper/1612.03615/) (2016)
- [Kernel-based Graph Learning from Smooth Signals: A Functional Viewpoint](https://mlsanity.com/paper/2008.10065/) (2020)
- [Grid-Graph Signal Processing (Grid-GSP): A Graph Signal Processing Framework for the Power Grid](https://mlsanity.com/paper/2103.06068/) (2021)
- [Nonparametric Basis Pursuit via Sparse Kernel-based Learning](https://mlsanity.com/paper/1302.5449/) (2013)
- [Parallel Selective Algorithms for Big Data Optimization](https://mlsanity.com/paper/1402.5521/) (2014)
- [Fej'er-monotone hybrid steepest descent method for affinely constrained and composite convex minimization tasks](https://mlsanity.com/paper/1608.02500/) (2016)
- [A Comparative Study of Pairwise Learning Methods based on Kernel Ridge Regression](https://mlsanity.com/paper/1803.01575/) (2018)
- [Bi-Linear Modeling of Data Manifolds for Dynamic-MRI Recovery](https://mlsanity.com/paper/1812.10617/) (2018)

## Built on by (most cited first, 3 of 3)

- [Imputation of Time-varying Edge Flows in Graphs by Multilinear Kernel Regression and Manifold Learning](https://mlsanity.com/paper/2409.05135/) (2024)
- [Kernel Regression of Multi-Way Data via Tensor Trains with Hadamard Overparametrization: The Dynamic Graph Flow Case](https://mlsanity.com/paper/2509.22197/) (2025)
- [Kernel Regression with Tensor Trains and Hadamard Overparameterization](https://mlsanity.com/paper/2607.17390/) (2026)

## Abstract (arXiv)

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.