Tensor-train methods for sequential state and parameter learning in state-space models

TL;DR

This work introduces tensor-train (TT) based recursive Bayesian learning for sequential state-space models with intractable transitions and observations. By representing the evolving joint posterior over states and unknown parameters with TT decompositions and employing Knothe–Rosenblatt (KR) transport maps, the authors design online filtering, parameter estimation, path estimation, and smoothing without relying on particle ensembles. A squared-TT formulation preserves nonnegativity, enabling robust debiasing and the construction of KR rearrangements to implement particle filtering and smoothing within the TT framework. The paper also develops error bounds for TT approximations, and preconditioning strategies (Gaussian bridging, tempering) to enhance approximation power, with numerical demonstrations on linear, stochastic volatility, high-dimensional SIR, and predator–prey models. Overall, the approach delivers competitive estimation accuracy and improved computational efficiency, offering a scalable alternative to traditional particle-based methods for uncertainty quantification in sequential inference.

Abstract

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.

Figures (18)

• Figure 1: Conditional dependency structures of the joint map $\mathcal{F}$ used in particle smoothing. The blue entries indicate the conditional dependency on the states, and the orange entries indicate the conditional dependency on the parameters. The $2\times 2$ sub-matrix in the top left corner represents the structure of the first two blocks of the map $\mathcal{F}^l_T$.
• Figure 2: Left: approximation of $q_t$ (gray contours) using a single layer of TT-based approximation (red contours), in which Fourier basis of order 30 and a TT rank 24 is used. Middle: preconditioned density using the tempering technique (cf. Section \ref{['sec:nonlinear_precond']}), in which the nonlinear preconditioning transform is defined by a TT with Fourier basis of order 30 and a rank 12. Right: approximation of $q_t$ (gray contours) using the preconditioned approximation (blue contours), in which Fourier basis of order 30 and a TT rank 12 is used in the preconditioned approximation.
• Figure 3: Linear Kalman filter with unknown parameters. The relative $L^1$ error of approximations of posterior parameter densities built by Alg. \ref{['alg:basic']} and \ref{['alg:stt']}.
• Figure 4: Linear Kalman filter with unknown parameters. The Hellinger distances (blue lines) between the theoretical posterior parameter density and its TT approximation, and the ESS (red lines) for the joint posterior of the states and the parameters at time $t \in \{30, 50\}$. Left: changing the number of degrees of freedom of the basis functions $\ell$ with a fixed maximum TT rank $r = 15$. Right: changing the maximum TT rank $r$ with a fixed number of degrees of freedom of the basis functions $\ell = 33$.
• Figure 5: Linear Kalman filter with unknown parameters. Left: the Hellinger distance between the theoretical posterior parameter density $p(\boldsymbol \theta|\boldsymbol y_{1:t})$ and its TT approximations versus time. Middle: the ESS of the path estimation (Alg. \ref{['alg:stt_smooth']}) for sampling the joint posterior $p(\boldsymbol \theta, \boldsymbol x_{0:t}|\boldsymbol y_{1:t})$ versus time. Right: the ESS for sampling the posterior parameter density $p(\boldsymbol \theta|\boldsymbol y_{1:t})$ versus time.

Theorems & Definitions (16)

• Example 1
• Lemma 1
• Proposition 2
• Remark 3
• Proposition 4
• Proposition 5
• Proposition 6
• Theorem 7
• Theorem 8
• Proposition 9