A continuous analogue of the tensor-train decomposition

We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents functions using a tensor-train ansatz by replacing the three-dimensional TT cores with univariate matrix-valued functions. The main contribution of this paper is a framework to compute the FT that employs adaptive approximations of univariate fibers, and that is not tied to any tensorized discretization. The algorithm can be coupled with any univariate linear or nonlinear approximation procedure. We demonstrate that this approach can generate multivariate function approximations that are several orders of magnitude more accurate, for the same cost, than those based on the conventional approach of compressing the coefficient tensor of a tensor-product basis. Our approach is in the spirit of other continuous computation packages such as Chebfun, and yields an algorithm which requires the computation of "continuous" matrix factorizations such as the LU and QR decompositions of vector-valued functions. To support these developments, we describe continuous versions of an approximate maximum-volume cross approximation algorithm and of a rounding algorithm that re-approximates an FT by one of lower ranks. We demonstrate that our technique improves accuracy and robustness, compared to TT and quantics-TT approaches with fixed parameterizations, of high-dimensional integration, differentiation, and approximation of functions with local features such as discontinuities and other nonlinearities

Tensor Train for Global Optimization Problems in Robotics

The convergence of many numerical optimization techniques is highly dependent on the initial guess given to the solver. To address this issue, we propose a novel approach that utilizes tensor methods to initialize existing optimization solvers near global optima. Our method does not require access to a database of good solutions. We first transform the cost function, which depends on both task parameters and optimization variables, into a probability density function. Unlike existing approaches, the joint probability distribution of the task parameters and optimization variables is approximated using the Tensor Train model, which enables efficient conditioning and sampling. We treat the task parameters as random variables, and for a given task, we generate samples for decision variables from the conditional distribution to initialize the optimization solver. Our method can produce multiple solutions (when they exist) faster than existing methods. We first evaluate the approach on benchmark functions for numerical optimization that are hard to solve using gradient-based optimization solvers with a naive initialization. The results show that the proposed method can generate samples close to global optima and from multiple modes. We then demonstrate the generality and relevance of our framework to robotics by applying it to inverse kinematics with obstacles and motion planning problems with a 7-DoF manipulator.Comment: 25 pages, 21 figure

Efficient High-Dimensional Stochastic Optimal Motion Control using Tensor-Train Decomposition

Abstract-Stochastic optimal control problems frequently arise as motion control problems in the context of robotics. Unfortunately, all existing approaches that guarantee arbitrary precision suffer from the curse of dimensionality: the computational effort invested by the algorithm grows exponentially fast with increasing dimensionality of the state space of the underlying dynamic system governing the robot. In this paper, we propose a novel algorithm that utilizes compressed representations to efficiently solve stochastic optimal control problems with arbitrary precision. The running time of the new algorithms scale linearly with increasing dimensionality of the state space! The running time also depends polynomially on the "rank" of the value function, a measure that quantifies the intrinsic geometric complexity of the value function, due to the geometry and physics embedded in the problem instance at hand. The new algorithms are based on the recent analysis and algorithms for tensor decomposition, generalizing matrix decomposition algorithms, e.g., the singular value decomposition, to three or more dimensions. In computational experiments, we show the computational effort of the new algorithm also scales linearly with the discretization resolution of the state space. We also demonstrate the new algorithm on a problem involving the perching of an aircraft, represented by a nonlinear non-holonomic longitudinal model with a sevendimensional state space, the full numerical solution to which was not obtained before. In this example, we estimate that the proposed algorithm runs more than seven orders of magnitude faster, when compared to the naive value iteration

Perception-driven optimal motion planning under resource constraints

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Ocean Science & Engineering at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2019.Over the past few years there has been a new wave of interest in fully autonomous robots operating in the real world, with applications from autonomous driving to search and rescue. These robots are expected to operate at high speeds in unknown, unstructured environments using only onboard sensing and computation, presenting significant challenges for high performance autonomous navigation. To enable research in these challenging scenarios, the first part of this thesis focuses on the development of a custom high-performance research UAV capable of high speed autonomous flight using only vision and inertial sensors. This research platform was used to develop stateof-the-art onboard visual inertial state estimation at high speeds in challenging scenarios such as flying through window gaps. While this platform is capable of high performance state estimation and control, its capabilities in unknown environments are severely limited by the computational costs of running traditional vision-based mapping and motion planning algorithms on an embedded platform. Motivated by these challenges, the second part of this thesis presents an algorithmic approach to the problem of motion planning in an unknown environment when the computational costs of mapping all available sensor data is prohibitively high. The algorithm is built around a tree of dynamically feasible and free space optimal trajectories to the goal state in configuration space. As the algorithm progresses it iteratively switches between processing new sensor data and locally updating the search tree. We show that the algorithm produces globally optimal motion plans, matching the optimal solution for the case with the full (unprocessed) sensor data, while only processing a subset of the data. The mapping and motion planning algorithm is demonstrated on a number of test systems, with a particular focus on a six-dimensional thrust limited model of a quadrotor