Multilinear Kernel Regression and Imputation via Manifold Learning

TL;DR

MultiL-KRIM introduces a manifold-aware, nonparametric imputation framework that operates in an RKHS by leveraging tangent-space–like local patches and collaborative filtering among neighboring data points.Data are modeled via a multilinear kernel factorization with multiple kernels and Q-layer structure, enabling low-rank, geometry-informed representation without training data.The framework is validated on time-varying graph-signal recovery and highly accelerated dynamic MRI, achieving faster computations and improved accuracy over shallow methods while remaining more interpretable than deep priors.

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.

Figures (7)

• Figure 1: Pipeline of MultiL-KRIM
• Figure 2: The manifold-learning (ManL) modeling assumption: points $\Set{ \varphi (\mathbfit{l}_{k}) }_{k=1}^{N_{\mathit{l}}}$ lie into or close to an unknown-to-the-user smooth manifold $\mathscr{M}$ which is embedded into an ambient RKHS $\mathscr{H}$. The "collaborative-filtering" modeling assumption: only a few points, $\Set{ \varphi (\mathbfit{l}_{k_n}) }_{n=1}^3$ here, collaborate affinely to approximate $\varphi (\check{\bm{\mu}}_t)$. The small number of "collaborating neighbors" implies a low-dimensional structure. All affine combinations of $\Set{ \varphi (\mathbfit{l}_{k_n})}_{n=1}^3$ define the approximating "linear patch" (affine hull, gray-colored plane), which mimics the concept of a tangent space to $\mathscr{M}$.
• Figure 3: Performance of MultiL-KRIM vs. other methods on different datasets (D1, D2, D3) and sampling patterns (P1, P2).
• Figure 4: (a) Domain of size $I_1 \times I_2 \times I_3$ where the dMRI data are observed and collected, with $I_3$ denoting the number of time frames. The marked $\upsilon \times I_2 \times I_3$ box shows the typical location of the faithful "navigator/pilot" data, corresponding, usually, to the "low-frequency" area of the domain. The "white dots" indicate the locations where data are collected, with the black-colored majority of the (k,t)-space to correspond to locations where there are missing data. After "flattening" the 3D (k,t)-space into a 2D one (see \ref{['sec:problem.formulation']}), these "white dots" define the index set $\Omega$, and thus the sampling operator $\mathscr{S}_{\Omega}$. (b) The $(I_1 \times I_2 \times I_3)$-sized image domain. The image-domain data are obtained by applying the two dimensional (2D) inverse DFT $\mathscr{F}^{-1}$ to the (k,t)-space data zhi2000principles. (c) 1D Cartesian and (d) radial sampling trajectories in k-space. The white-colored lines indicate locations where data are collected.
• Figure 5: Normalized RMSE values (the lower the better) w.r.t. the ground truth vs. (sampling-time) acceleration factors. An acceleration factor "$\beta$x" means that only $1/\beta$ of the full-scan $I_1 \times I_2$ k-space is sampled (see \ref{['fig:cartesian.sampling', 'fig:radial.sampling']}).