用张量分解+哈达玛过参数化，高效补全动态图流数据
Kernel Regression of Multi-Way Data via Tensor Trains with Hadamard Overparametrization: The Dynamic Graph Flow Case
- 基于再生核希尔伯特空间的非参数回归，用固定秩张量列车建模
- 在真实图数据上补全缺失边流量，误差低于现有先进方法
- 适合需要可解释性与拓扑先验的动态网络分析场景
提出一种可解释的多维数据补全框架KReTTaH，将补全问题建模为再生核希尔伯特空间中的非参数回归。通过固定张量列车（TT）秩的张量实现参数高效，在低维黎曼流形上学习，并引入哈达玛过参数化促进参数稀疏性。学习过程转化为黎曼流形上的光滑逆问题求解。以动态图流估计为例，该方法可无缝融入基于图结构的拓扑先验。在真实图数据集上的数值实验表明，相较于最先进的张量和神经网络方法，KReTTaH在补全时变边流量方面表现更优。
原文摘要 · Abstract (English)
A regression-based framework for interpretable multi-way data imputation, termed Kernel Regression via Tensor Trains with Hadamard overparametrization (KReTTaH), is introduced. KReTTaH adopts a nonparametric formulation by casting imputation as regression via reproducing kernel Hilbert spaces. Parameter efficiency is achieved through tensors of fixed tensor-train (TT) rank, which reside on low-dimensional Riemannian manifolds, and is further enhanced via Hadamard overparametrization, which promotes sparsity within the TT parameter space. Learning is accomplished by solving a smooth inverse problem posed on the Riemannian manifold of fixed TT-rank tensors. As a representative application, the estimation of dynamic graph flows is considered. In this setting, KReTTaH exhibits flexibility by seamlessly incorporating graph-based (topological) priors via its inverse problem formulation. Numerical tests on real-world graph datasets demonstrate that KReTTaH consistently outperforms state-of-the-art alternatives-including a nonparametric tensor- and a neural-network-based methods-for imputing missing, time-varying edge flows.
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