多重线性核回归与基于流形学习的插补
Multilinear Kernel Regression and Imputation via Manifold Learning
AI总结：
本文提出MultiL-KRIM，一种基于流形切空间与多重核的非参数数据插补框架，在时变图信号恢复和动态磁共振成像重建中显著加速并优于浅层方法。
摘要：
本文提出了一种新颖的非参数数据插补框架，称为基于流形假设的多重线性核回归与插补（MultiL-KRIM）。受流形学习的启发，MultiL-KRIM将数据特征建模为位于或接近一个用户未知的光滑流形上的点云，该流形嵌入在再生核希尔伯特空间中。与典型的流形学习方法不同，后者通过基于图拉普拉斯矩阵的正则化器来寻求低维模式，MultiL-KRIM则基于流形切空间的直观概念，并直接将点云邻居（回归器）之间的协作纳入损失函数的数据建模项中。允许多个核函数以提供鲁棒性和丰富的逼近性质，而多个矩阵因子则提供低秩建模、整合降维，并在无需训练数据的情况下简化计算。两个重要的应用领域展示了MultiL-KRIM的功能：时变图信号（TVGS）恢复，以及高度加速的动态磁共振成像（dMRI）数据的重建。在真实和合成数据上的大量数值测试表明，MultiL-KRIM相比其前身具有显著的加速优势，并且优于流行的“浅层”数据插补技术，其流程比深度图像先验方法更直观、更可解释。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
