# Optimization on the Extended Tensor-Train Manifold with Shared Factors

## TL;DR

This paper develops Riemannian optimization algorithms for tensors in the Extended Tensor Train format with shared factors, addressing the challenges posed by factor constraints and demonstrating practical effectiveness in approximation and eigenvalue problems.

## Contribution

It introduces a smooth manifold framework and efficient Riemannian algorithms for ETT tensors with shared factors, a novel approach in tensor optimization.

## Key findings

- Algorithms effectively handle shared factor constraints.
- Successful tensor approximation and eigenvalue computations.
- Manifold smoothness is theoretically established.

## Abstract

This paper studies tensors that admit decomposition in the Extended Tensor Train (ETT) format, with a key focus on the case where some decomposition factors are constrained to be equal. This factor sharing introduces additional challenges, as it breaks the multilinear structure of the decomposition. Nevertheless, we show that Riemannian optimization methods can naturally handle such constraints and prove that the underlying manifold is indeed smooth. We develop efficient algorithms for key Riemannian optimization components, including a retraction operation based on quasi-optimal approximation in the new format, as well as tangent space projection using automatic differentiation. Finally, we demonstrate the practical effectiveness of our approach through tensor approximation tasks and multidimensional eigenvalue problem.