# Multi-Linear Kernel Regression and Imputation in Data Manifolds

arXiv: https://arxiv.org/abs/2304.03041 · HTML: https://mlsanity.com/paper/2304.03041/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 47 references that are in the map.
- First posted on arXiv 6 Apr 2023.

## Builds on (most cited first, 10 of 10)

- [A Deep Cascade of Convolutional Neural Networks for Dynamic MR Image Reconstruction](https://mlsanity.com/paper/1704.02422/) (2017)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Time-Dependent Deep Image Prior for Dynamic MRI](https://mlsanity.com/paper/1910.01684/) (2019)
- [Dynamic imaging using a deep generative SToRM (Gen-SToRM) model](https://mlsanity.com/paper/2102.00034/) (2021)
- [FRIST - Flipping and Rotation Invariant Sparsifying Transform Learning and Applications](https://mlsanity.com/paper/1511.06359/) (2015)
- [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)
- [Low-rank and Adaptive Sparse Signal (LASSI) Models for Highly Accelerated Dynamic Imaging](https://mlsanity.com/paper/1611.04069/) (2016)
- [Learning Tensors in Reproducing Kernel Hilbert Spaces with Multilinear Spectral Penalties](https://mlsanity.com/paper/1310.4977/) (2013)
- [Bi-Linear Modeling of Data Manifolds for Dynamic-MRI Recovery](https://mlsanity.com/paper/1812.10617/) (2018)

## Abstract (arXiv)

This paper introduces an efficient multi-linear nonparametric (kernel-based) approximation framework for data regression and imputation, and its application to dynamic magnetic-resonance imaging (dMRI). Data features are assumed to reside in or close to a smooth manifold embedded in a reproducing kernel Hilbert space. Landmark points are identified to describe concisely the point cloud of features by linear approximating patches which mimic the concept of tangent spaces to smooth manifolds. The multi-linear model effects dimensionality reduction, enables efficient computations, and extracts data patterns and their geometry without any training data or additional information. Numerical tests on dMRI data under severe under-sampling demonstrate remarkable improvements in efficiency and accuracy of the proposed approach over its predecessors, popular data modeling methods, as well as recent tensor-based and deep-image-prior schemes.