# Generalized Low-rank plus Sparse Tensor Estimation by Fast Riemannian Optimization

arXiv: https://arxiv.org/abs/2103.08895 · HTML: https://mlsanity.com/paper/2103.08895/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Mar 2021.

## Built on by (most cited first, 24 of 24)

- [Learned Robust PCA: A Scalable Deep Unfolding Approach for High-Dimensional Outlier Detection](https://mlsanity.com/paper/2110.05649/) (2021)
- [Provable Tensor-Train Format Tensor Completion by Riemannian Optimization](https://mlsanity.com/paper/2108.12163/) (2021)
- [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)
- [Latent Space Model for Higher-order Networks and Generalized Tensor Decomposition](https://mlsanity.com/paper/2106.16042/) (2021)
- [Fast and Provable Tensor Robust Principal Component Analysis via Scaled Gradient Descent](https://mlsanity.com/paper/2206.09109/) (2022)
- [Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruption](https://mlsanity.com/paper/2305.04080/) (2023)
- [Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descent](https://mlsanity.com/paper/2401.11940/) (2024)
- [Provably Accelerating Ill-Conditioned Low-rank Estimation via Scaled Gradient Descent, Even with Overparameterization](https://mlsanity.com/paper/2310.06159/) (2023)
- [Optimal Clustering by Lloyd Algorithm for Low-Rank Mixture Model](https://mlsanity.com/paper/2207.04600/) (2022)
- [Computationally Efficient and Statistically Optimal Robust High-Dimensional Linear Regression](https://mlsanity.com/paper/2305.06199/) (2023)
- [Consensus Knowledge Graph Learning via Multi-view Sparse Low Rank Block Model](https://mlsanity.com/paper/2209.13762/) (2022)
- [High-dimensional Linear Bandits with Knapsacks](https://mlsanity.com/paper/2311.01327/) (2023)
- [Tensor Methods in High Dimensional Data Analysis: Opportunities and Challenges](https://mlsanity.com/paper/2405.18412/) (2024)
- [Statistical Inference in Tensor Completion: Optimal Uncertainty Quantification and Statistical-to-Computational Gaps](https://mlsanity.com/paper/2410.11225/) (2024)
- [Efficient Estimation for Longitudinal Networks via Adaptive Merging](https://mlsanity.com/paper/2211.07866/) (2022)
- [Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent](https://mlsanity.com/paper/2501.13385/) (2025)
- [Outlier-aware Tensor Robust Principal Component Analysis with Self-guided Data Augmentation](https://mlsanity.com/paper/2504.18323/) (2025)
- [Accelerating Large-Scale Regularized High-Order Tensor Recovery](https://mlsanity.com/paper/2506.09594/) (2025)
- [Multi-Dimensional Visual Data Recovery: Scale-Aware Tensor Modeling and Accelerated Randomized Computation](https://mlsanity.com/paper/2602.12982/) (2026)
- [Robust Tensor Regression with Nonconvexity: Algorithmic and Statistical Theory](https://mlsanity.com/paper/2605.07448/) (2026)
- [Dual-Channel Tensor Neural Networks: Finite-Sample Theory and Conformal Structure Selection](https://mlsanity.com/paper/2605.19122/) (2026)
- [Structured Adaptive Tensor Prediction for Streaming Data](https://mlsanity.com/paper/2606.10085/) (2026)
- [Tensor Completion using Subspace Information](https://mlsanity.com/paper/2609.24501/) (2026)

## Abstract (arXiv)

We investigate a generalized framework to estimate a latent low-rank plus sparse tensor, where the low-rank tensor often captures the multi-way principal components and the sparse tensor accounts for potential model mis-specifications or heterogeneous signals that are unexplainable by the low-rank part. The framework is flexible covering both linear and non-linear models, and can easily handle continuous or categorical variables. We propose a fast algorithm by integrating the Riemannian gradient descent and a novel gradient pruning procedure. Under suitable conditions, the algorithm converges linearly and can simultaneously estimate both the low-rank and sparse tensors. The statistical error bounds of final estimates are established in terms of the gradient of loss function. The error bounds are generally sharp under specific statistical models, e.g., the robust tensor PCA and the community detection in hypergraph networks with outlier vertices. Moreover, our method achieves non-trivial error bounds for heavy-tailed tensor PCA whenever the noise has a finite 2+varepsilon moment. We apply our method to analyze the international trade flow dataset and the statistician hypergraph co-authorship network, both yielding new and interesting findings.