# Tensor Regression Using Low-rank and Sparse Tucker Decompositions

arXiv: https://arxiv.org/abs/1911.03725 · HTML: https://mlsanity.com/paper/1911.03725/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Nov 2019.

## Built on by (most cited first, 10 of 10)

- [Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements](https://mlsanity.com/paper/2104.14526/) (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)
- [Fast and Provable Tensor Robust Principal Component Analysis via Scaled Gradient Descent](https://mlsanity.com/paper/2206.09109/) (2022)
- [Factor Augmented Tensor-on-Tensor Neural Networks](https://mlsanity.com/paper/2405.19610/) (2024)
- [On High-dimensional and Low-rank Tensor Bandits](https://mlsanity.com/paper/2305.03884/) (2023)
- [Structured Low-Rank Tensors for Generalized Linear Models](https://mlsanity.com/paper/2308.02922/) (2023)
- [Federated Low-Rank Tensor Estimation for Multimodal Image Reconstruction](https://mlsanity.com/paper/2502.02761/) (2025)
- [Information-Theoretic Guarantees for Recovering Low-Rank Tensors from Symmetric Rank-One Measurements](https://mlsanity.com/paper/2502.05134/) (2025)
- [Dual-Channel Tensor Neural Networks: Finite-Sample Theory and Conformal Structure Selection](https://mlsanity.com/paper/2605.19122/) (2026)

## Abstract (arXiv)

This paper studies a tensor-structured linear regression model with a scalar response variable and tensor-structured predictors, such that the regression parameters form a tensor of order d (i.e., a d-fold multiway array) in ℝn1 × n2 × cdots × nd. It focuses on the task of estimating the regression tensor from m realizations of the response variable and the predictors where mll n = prod nolimitsi ni. Despite the seeming ill-posedness of this problem, it can still be solved if the parameter tensor belongs to the space of sparse, low Tucker-rank tensors. Accordingly, the estimation procedure is posed as a non-convex optimization program over the space of sparse, low Tucker-rank tensors, and a tensor variant of projected gradient descent is proposed to solve the resulting non-convex problem. In addition, mathematical guarantees are provided that establish the proposed method linearly converges to an appropriate solution under a certain set of conditions. Further, an upper bound on sample complexity of tensor parameter estimation for the model under consideration is characterized for the special case when the individual (scalar) predictors independently draw values from a sub-Gaussian distribution. The sample complexity bound is shown to have a polylogarithmic dependence on barn = max bigni: i∈ 1,2,ldots,d  big and, orderwise, it matches the bound one can obtain from a heuristic parameter counting argument. Finally, numerical experiments demonstrate the efficacy of the proposed tensor model and estimation method on a synthetic dataset and a collection of neuroimaging datasets pertaining to attention deficit hyperactivity disorder. Specifically, the proposed method exhibits better sample complexities on both synthetic and real datasets, demonstrating the usefulness of the model and the method in settings where n gg m.