用拉普拉斯近似解决张量列车核机器的贝叶斯建模难题，加速系统辨识。
Laplace Approximation For Tensor Train Kernel Machines In System Identification
- 对选定张量核应用拉普拉斯近似估计后验，结合变分推断优化超参数。
- 在逆动力学问题上验证有效，训练速度比交叉验证快65倍。
- 适用于需要快速贝叶斯推理的系统辨识场景，尤其适合高维数据。
为克服高斯过程回归的可扩展性瓶颈，已有多种近似方法被提出。其中基于张量网络的方法能在不引发指数级计算开销的前提下使用指数级基函数。然而，将该模型扩展为全概率形式带来若干设计挑战，特别是针对张量列车（TT）模型，尚不清楚应如何选择需进行贝叶斯处理的TT核。本文提出一种贝叶斯张量列车核机器，采用拉普拉斯近似估计选定TT核上的后验分布，并用变分推断（VI）处理精度超参数。实验表明，核心选择与TT秩及特征结构基本无关，且VI可替代交叉验证，在训练速度上最高提升65倍。方法在逆动力学问题上验证了有效性。
原文摘要 · Abstract (English)
To address the scalability limitations of Gaussian process (GP) regression, several approximation techniques have been proposed. One such method is based on tensor networks, which utilizes an exponential number of basis functions without incurring exponential computational cost. However, extending this model to a fully probabilistic formulation introduces several design challenges. In particular, for tensor train (TT) models, it is unclear which TT-core should be treated in a Bayesian manner. We introduce a Bayesian tensor train kernel machine that applies Laplace approximation to estimate the posterior distribution over a selected TT-core and employs variational inference (VI) for precision hyperparameters. Experiments show that core selection is largely independent of TT-ranks and feature structure, and that VI replaces cross-validation while offering up to 65x faster training. The method's effectiveness is demonstrated on an inverse dynamics problem.
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