Value Function Iteration Using Tensor Train Decomposition

This paper presents a novel approach to solving dynamic programming problems using value function iteration based on the tensor train decomposition. The tensor train decomposition approximates high-dimensional functions by expressing them as a series of interconnected cores, producing an approximation that separates by variables. This approach is well-suited for approximating and integrating high-dimensional functions, such as a value function. We apply the method to a range of models and compare its performance against established sparse-grid techniques involving Smolyak and hyperbolic cross polynomials and neural networks. For models with as few as three state variables, the tensor train method is shown to be faster and/or more accurate than the leading sparse-grid alternatives. This paper shows how tensor train methods can be used to solve dynamic optimization problems in Economics, offering a powerful approach to solve high-dimensional macroeconomic models.