DeepAI AI Chat
Log In Sign Up

Randomized algorithms for rounding in the Tensor-Train format

by   Hussam Al Daas, et al.

The Tensor-Train (TT) format is a highly compact low-rank representation for high-dimensional tensors. TT is particularly useful when representing approximations to the solutions of certain types of parametrized partial differential equations. For many of these problems, computing the solution explicitly would require an infeasible amount of memory and computational time. While the TT format makes these problems tractable, iterative techniques for solving the PDEs must be adapted to perform arithmetic while maintaining the implicit structure. The fundamental operation used to maintain feasible memory and computational time is called rounding, which truncates the internal ranks of a tensor already in TT format. We propose several randomized algorithms for this task that are generalizations of randomized low-rank matrix approximation algorithms and provide significant reduction in computation compared to deterministic TT-rounding algorithms. Randomization is particularly effective in the case of rounding a sum of TT-tensors (where we observe 20x speedup), which is the bottleneck computation in the adaptation of GMRES to vectors in TT format. We present the randomized algorithms and compare their empirical accuracy and computational time with deterministic alternatives.


Solving high-dimensional parabolic PDEs using the tensor train format

High-dimensional partial differential equations (PDEs) are ubiquitous in...

Parallel Algorithms for Tensor Train Arithmetic

We present efficient and scalable parallel algorithms for performing mat...

Randomized algorithms for low-rank tensor decompositions in the Tucker format

Many applications in data science and scientific computing involve large...

Multi-resolution Low-rank Tensor Formats

We describe a simple, black-box compression format for tensors with a mu...

Tensor train based isogeometric analysis for PDE approximation on parameter dependent geometries

This work develops a numerical solver based on the combination of isogeo...

Structured Matrix Approximations via Tensor Decompositions

We provide a computational framework for approximating a class of struct...

On some orthogonalization schemes in Tensor Train format

In the framework of tensor spaces, we consider orthogonalization kernels...

Code Repositories