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.

该论文介绍了一种新颖的非参数框架用于数据插值，称为多线性核回归和流形假设下的插值（MultiL-KRIM）。

多线性核回归与流形学习中的插补

Multilinear Kernel Regression and Imputation via Manifold Learning

Conditional kernel mean embeddings form an attractive nonparametric framework
for representing conditional means of functions, describing the observation
processes for many complex models. However, the recovery of the original
underlying function of interest whose conditional mean was observed is a
challenging inference task. We formalize deconditional kernel mean embeddings
as a solution to this inverse problem, and show that it can be naturally viewed
as a nonparametric Bayes' rule. Critically, we introduce the notion of task
transformed Gaussian processes and establish deconditional kernel means as
their posterior predictive mean. This connection provides Bayesian
interpretations and uncertainty estimates for deconditional kernel mean
embeddings, explains their regularization hyperparameters, and reveals a
marginal likelihood for kernel hyperparameter learning. These revelations
further enable practical applications such as likelihood-free inference and
learning sparse representations for big data.

本研究提出了非参数贝叶斯规则的去条件化核均值嵌入，并且展示了其与任务转换高斯过程的后验预测均值的联系。这提供了贝叶斯解释和不确定性估计，解释了它们的正则化超参数，揭示了内核超参数学习的边缘似然函数。这些发现进一步促进了无似然推断和大数据的稀疏表征等实际应用。

贝叶斯去条件核均值嵌入

Bayesian Deconditional Kernel Mean Embeddings

We propose a nonparametric framework for the analysis of networks, based on a
natural limit object termed a graphon. We prove consistency of graphon
estimation under general conditions, giving rates which include the important
practical setting of sparse networks. Our results cover dense and sparse
stochastic blockmodels with a growing number of classes, under model
misspecification. We use profile likelihood methods, and connect our results to
approximation theory, nonparametric function estimation, and the theory of
graph limits.

本文提出了一种非参数分析网络的框架，基于一种自然的极限对象 —— 图源。我们证明了在一般条件下，图源估计的一致性，包括稀疏网络等重要的实际情况。我们使用档案似然方法，并将我们的结果与逼近理论、非参数函数估计和图限理论联系起来。