We consider sequential state and parameter learning in state-space models with intractable state transition and observation processes. By exploiting low-rank tensor train (TT) decompositions, we propose new sequential learning methods for joint parameter and state estimation under the Bayesian framework. Our key innovation is the introduction of scalable function approximation tools such as TT for recursively learning the sequentially updated posterior distributions. The function approximation perspective of our methods offers tractable error analysis and potentially alleviates the particle degeneracy faced by many particle-based methods. In addition to the new insights into the algorithmic design, our methods complement conventional particle-based methods. Our TT-based approximations naturally define conditional Knothe--Rosenblatt (KR) rearrangements that lead to parameter estimation, filtering, smoothing and path estimation accompanying our sequential learning algorithms, which open the door to removing potential approximation bias. We also explore several preconditioning techniques based on either linear or nonlinear KR rearrangements to enhance the approximation power of TT for practical problems. We demonstrate the efficacy and efficiency of our proposed methods on several state-space models, in which our methods achieve state-of-the-art estimation accuracy and computational performance.

翻译：我们研究了状态转移与观测过程难以处理的状态空间模型中的序列状态与参数学习问题。通过利用低秩张量列车分解，我们在贝叶斯框架下提出了联合参数与状态估计的新序列学习方法。我们的核心创新在于引入可扩展的函数逼近工具（如张量列车）来递归学习序列更新的后验分布。本方法的函数逼近视角提供了可处理的误差分析，并有望缓解许多基于粒子方法所面临的粒子退化问题。除了为算法设计提供新视角外，我们的方法与传统基于粒子的方法形成互补。基于张量列车的逼近自然定义了条件Knothe–Rosenblatt重排，从而可在序列学习算法中同步实现参数估计、滤波、平滑与路径估计，这为消除潜在逼近偏差开辟了新途径。我们还探索了基于线性与非线性Knothe–Rosenblatt重排的多种预处理技术，以增强张量列车在实际问题中的逼近能力。我们在多个状态空间模型上验证了所提方法的有效性与效率，实验表明本方法在估计精度与计算性能方面均达到最先进水平。