# Grassmannian Optimization for Online Tensor Completion and Tracking with the t-SVD

arXiv: https://arxiv.org/abs/2001.11419 · HTML: https://mlsanity.com/paper/2001.11419/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2020.

## Built on by (most cited first, 6 of 6)

- [Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descent](https://mlsanity.com/paper/2401.11940/) (2024)
- [Low-Rank Tensor Completion via Novel Sparsity-Inducing Regularizers](https://mlsanity.com/paper/2310.06233/) (2023)
- [Low-Rank Tensor Decompositions for the Theory of Neural Networks](https://mlsanity.com/paper/2508.18408/) (2025)
- [Effective Streaming Low-tubal-rank Tensor Approximation via Frequent Directions](https://mlsanity.com/paper/2108.10129/) (2021)
- [GOCPT: Generalized Online Canonical Polyadic Tensor Factorization and Completion](https://mlsanity.com/paper/2205.03749/) (2022)
- [Guaranteed Low-Rank Tensor Recovery from Modewise Measurements via Normalized Block-Weighted Riemannian Gradient Descent](https://mlsanity.com/paper/2609.24679/) (2026)

## Abstract (arXiv)

We propose a new fast streaming algorithm for the tensor completion problem of imputing missing entries of a low-tubal-rank tensor using the tensor singular value decomposition (t-SVD) algebraic framework. We show the t-SVD is a specialization of the well-studied block-term decomposition for third-order tensors, and we present an algorithm under this model that can track changing free submodules from incomplete streaming 2-D data. The proposed algorithm uses principles from incremental gradient descent on the Grassmann manifold of subspaces to solve the tensor completion problem with linear complexity and constant memory in the number of time samples. We provide a local expected linear convergence result for our algorithm. Our empirical results are competitive in accuracy but much faster in compute time than state-of-the-art tensor completion algorithms on real applications to recover temporal chemo-sensing and MRI data under limited sampling.