# An Optimal Statistical and Computational Framework for Generalized Tensor Estimation

arXiv: https://arxiv.org/abs/2002.11255 · HTML: https://mlsanity.com/paper/2002.11255/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2020.

## Built on by (most cited first, 30 of 39)

- [Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements](https://mlsanity.com/paper/2104.14526/) (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)
- [On the Complexity of First-Order Methods in Stochastic Bilevel Optimization](https://mlsanity.com/paper/2402.07101/) (2024)
- [Fast and Provable Tensor Robust Principal Component Analysis via Scaled Gradient Descent](https://mlsanity.com/paper/2206.09109/) (2022)
- [Long Random Matrices and Tensor Unfolding](https://mlsanity.com/paper/2110.10210/) (2021)
- [Guaranteed Nonconvex Factorization Approach for Tensor Train Recovery](https://mlsanity.com/paper/2401.02592/) (2024)
- [Learning Good State and Action Representations via Tensor Decomposition](https://mlsanity.com/paper/2105.01136/) (2021)
- [Tensor Principal Component Analysis in High Dimensional CP Models](https://mlsanity.com/paper/2108.04428/) (2021)
- [Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCA](https://mlsanity.com/paper/2303.06198/) (2023)
- [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)
- [On High-dimensional and Low-rank Tensor Bandits](https://mlsanity.com/paper/2305.03884/) (2023)
- [Beyond the Signs: Nonparametric Tensor Completion via Sign Series](https://mlsanity.com/paper/2102.00384/) (2021)
- [Multiway Spherical Clustering via Degree-Corrected Tensor Block Models](https://mlsanity.com/paper/2201.07401/) (2022)
- [Computational and Statistical Guarantees for Tensor-on-Tensor Regression with Tensor Train Decomposition](https://mlsanity.com/paper/2406.06002/) (2024)
- [A Scalable Factorization Approach for High-Order Structured Tensor Recovery](https://mlsanity.com/paper/2506.16032/) (2025)
- [Sharp Analysis of Power Iteration for Tensor PCA](https://mlsanity.com/paper/2401.01047/) (2024)
- [Robust Low-rank Tensor Train Recovery](https://mlsanity.com/paper/2410.15224/) (2024)
- [Multivariate Analysis for Multiple Network Data via Semi-Symmetric Tensor PCA](https://mlsanity.com/paper/2202.04719/) (2022)
- [Mode-wise Principal Subspace Pursuit and Matrix Spiked Covariance Model](https://mlsanity.com/paper/2307.00575/) (2023)
- [Convergence Analysis for Learning Orthonormal Deep Linear Neural Networks](https://mlsanity.com/paper/2311.14658/) (2023)
- [Tensor Methods in High Dimensional Data Analysis: Opportunities and Challenges](https://mlsanity.com/paper/2405.18412/) (2024)
- [Efficient Estimation for Longitudinal Networks via Adaptive Merging](https://mlsanity.com/paper/2211.07866/) (2022)
- [Statistical and computational rates in high rank tensor estimation](https://mlsanity.com/paper/2304.04043/) (2023)
- [Federated Low-Rank Tensor Estimation for Multimodal Image Reconstruction](https://mlsanity.com/paper/2502.02761/) (2025)
- [Accelerating Large-Scale Regularized High-Order Tensor Recovery](https://mlsanity.com/paper/2506.09594/) (2025)
- [Optimal Bias-variance Tradeoff in Matrix and Tensor Estimation](https://mlsanity.com/paper/2509.17382/) (2025)

## Abstract (arXiv)

This paper describes a flexible framework for generalized low-rank tensor estimation problems that includes many important instances arising from applications in computational imaging, genomics, and network analysis. The proposed estimator consists of finding a low-rank tensor fit to the data under generalized parametric models. To overcome the difficulty of non-convexity in these problems, we introduce a unified approach of projected gradient descent that adapts to the underlying low-rank structure. Under mild conditions on the loss function, we establish both an upper bound on statistical error and the linear rate of computational convergence through a general deterministic analysis. Then we further consider a suite of generalized tensor estimation problems, including sub-Gaussian tensor PCA, tensor regression, and Poisson and binomial tensor PCA. We prove that the proposed algorithm achieves the minimax optimal rate of convergence in estimation error. Finally, we demonstrate the superiority of the proposed framework via extensive experiments on both simulated and real data.