# High-Dimensional Stochastic Optimal Control using Continuous Tensor Decompositions

arXiv: https://arxiv.org/abs/1611.04706 · HTML: https://mlsanity.com/paper/1611.04706/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Nov 2016.

## Built on by (most cited first, 13 of 13)

- [Stochastic Optimal Control Matching](https://mlsanity.com/paper/2312.02027/) (2023)
- [Solving high-dimensional parabolic PDEs using the tensor train format](https://mlsanity.com/paper/2102.11830/) (2021)
- [TTOpt: A Maximum Volume Quantized Tensor Train-based Optimization and its Application to Reinforcement Learning](https://mlsanity.com/paper/2205.00293/) (2022)
- [Value Gradient Guidance for Flow Matching Alignment](https://mlsanity.com/paper/2512.05116/) (2025)
- [Conditional Imitation Learning for Multi-Agent Games](https://mlsanity.com/paper/2201.01448/) (2022)
- [A Taxonomy of Loss Functions for Stochastic Optimal Control](https://mlsanity.com/paper/2410.00345/) (2024)
- [Gradient-augmented Supervised Learning of Optimal Feedback Laws Using State-dependent Riccati Equations](https://mlsanity.com/paper/2103.04091/) (2021)
- [From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEs](https://mlsanity.com/paper/2307.15496/) (2023)
- [Near-Linear Time and Fixed-Parameter Tractable Algorithms for Tensor Decompositions](https://mlsanity.com/paper/2207.07417/) (2022)
- [Low-Rank Tensors for Multi-Dimensional Markov Models](https://mlsanity.com/paper/2411.02098/) (2024)
- [Addressing Finite-Horizon MDPs via Low-Rank Tensor Value Approximation](https://mlsanity.com/paper/2501.10598/) (2025)
- [Error Propagation in Dynamic Programming: From Stochastic Control to American Option Pricing](https://mlsanity.com/paper/2509.20239/) (2025)
- [A Schr"odinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control](https://mlsanity.com/paper/2603.23173/) (2026)

## Abstract (arXiv)

Motion planning and control problems are embedded and essential in almost all robotics applications. These problems are often formulated as stochastic optimal control problems and solved using dynamic programming algorithms. Unfortunately, most existing algorithms that guarantee convergence to optimal solutions suffer from the curse of dimensionality: the run time of the algorithm grows exponentially with the dimension of the state space of the system. We propose novel dynamic programming algorithms that alleviate the curse of dimensionality in problems that exhibit certain low-rank structure. The proposed algorithms are based on continuous tensor decompositions recently developed by the authors. Essentially, the algorithms represent high-dimensional functions (e.g., the value function) in a compressed format, and directly perform dynamic programming computations (e.g., value iteration, policy iteration) in this format. Under certain technical assumptions, the new algorithms guarantee convergence towards optimal solutions with arbitrary precision. Furthermore, the run times of the new algorithms scale polynomially with the state dimension and polynomially with the ranks of the value function. This approach realizes substantial computational savings in "compressible" problem instances, where value functions admit low-rank approximations. We demonstrate the new algorithms in a wide range of problems, including a simulated six-dimensional agile quadcopter maneuvering example and a seven-dimensional aircraft perching example. In some of these examples, we estimate computational savings of up to ten orders of magnitude over standard value iteration algorithms. We further demonstrate the algorithms running in real time on board a quadcopter during a flight experiment under motion capture.