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 algorithmic design, our methods complement conventional particle-based methods. Our TT-based approximations naturally define conditional Knothe--Rosenblatt (KR) rearrangements that lead to 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：我们考虑在具有难处理状态转移和观测过程的状态空间模型中，进行序贯状态与参数学习。通过利用低秩张量列（TT）分解，我们提出了在贝叶斯框架下用于联合参数与状态估计的新型序贯学习方法。我们的关键创新在于引入可扩展的函数逼近工具（如TT），以递归学习序贯更新的后验分布。我们方法的函数逼近视角提供了可处理的误差分析，并可能缓解许多基于粒子方法所面临的粒子退化问题。除了算法设计的新见解外，我们的方法还补充了传统的基于粒子的方法。基于TT的逼近自然定义了条件Knothe-Rosenblatt（KR）重排，进而生成伴随我们序贯学习算法的滤波、平滑和路径估计，这为消除潜在逼近偏差打开了大门。我们还探索了基于线性或非线性KR重排的几种预处理技术，以增强TT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和效率，其中我们的方法在估计精度和计算性能上均达到了最先进水平。