# A Riemannian trust-region method for low-rank tensor completion

arXiv: https://arxiv.org/abs/1703.10019 · HTML: https://mlsanity.com/paper/1703.10019/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Mar 2017.

## Built on by (most cited first, 8 of 8)

- [Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay](https://mlsanity.com/paper/2206.08756/) (2022)
- [Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence](https://mlsanity.com/paper/2104.12031/) (2021)
- [Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties](https://mlsanity.com/paper/2211.14645/) (2022)
- [A Riemannian Newton Trust-Region Method for Fitting Gaussian Mixture Models](https://mlsanity.com/paper/2104.14957/) (2021)
- [Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds](https://mlsanity.com/paper/2510.21468/) (2025)
- [Higher Order Reduced Rank Regression](https://mlsanity.com/paper/2503.06528/) (2025)
- [A JoLT for the KV cache: Near-Lossless KV Cache Compression via Joint Rank-bit Allocation](https://mlsanity.com/paper/2607.12550/) (2026)
- [Tensor Decomposition of Transformer Key-Value Caches: Spectral Structure and Format Comparison](https://mlsanity.com/paper/2609.28029/) (2026)

## Abstract (arXiv)

The goal of tensor completion is to fill in missing entries of a partially known tensor (possibly including some noise) under a low-rank constraint. This may be formulated as a least-squares problem. The set of tensors of a given multilinear rank is known to admit a Riemannian manifold structure, thus methods of Riemannian optimization are applicable. In our work, we derive the Riemannian Hessian of an objective function on the low-rank tensor manifolds using the Weingarten map, a concept from differential geometry. We discuss the convergence properties of Riemannian trust-region methods based on the exact Hessian and standard approximations, both theoretically and numerically. We compare our approach to Riemannian tensor completion methods from recent literature, both in terms of convergence behaviour and computational complexity. Our examples include the completion of randomly generated data with and without noise and recovery of multilinear data from survey statistics.