A Uniform Framework of Yau-Yau Algorithm Based on Deep Learning With the Capability of Overcoming the Curse of Dimensionality

• Published in: IEEE transactions on automatic control Vol. 70; no. 1; pp. 339 - 354
• Main Authors: Chen, Xiuqiong, Sun, Zeju, Tao, Yangtianze, Yau, Stephen S.-T.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2025; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Covariance matrices; Curse of dimensionality; Deep learning; Filter design (mathematics); Filtering algorithms; Filters; Kalman filters; Machine learning; nonlinear filtering; Nonlinear filters; Probability density function; Recurrent neural networks; recurrent neural networks (RNNs); Stochastic processes; Yau–Yau algorithm
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2024.3424628

In numerous application areas, high-dimensional nonlinear filtering is still a challenging problem. The introduction of deep learning and neural networks has improved the efficiency of classical algorithms and they perform well in many practical tasks. However, a theoretical interpretation of their feasibility is still lacking. In this article, we exploit the representational ability of recurrent neural networks (RNNs) and provide a computationally efficient and optimal framework for nonlinear filter design based on the Yau-Yau algorithm and RNNs. Theoretically, it can be proved that the size of the neural network required in this algorithm increases only polynomially rather than exponentially with dimension, which implies that the Yau-Yau algorithm based on RNNs has the ability to overcome the curse of dimensionality. Numerical results also show that our method is more competitive than classical algorithms for high-dimensional problems.