# Parsimonious Tensor Response Regression

arXiv: https://arxiv.org/abs/1501.07815 · HTML: https://mlsanity.com/paper/1501.07815/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2015.

## Built on by (most cited first, 20 of 20)

- [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)
- [Guaranteed Nonconvex Factorization Approach for Tensor Train Recovery](https://mlsanity.com/paper/2401.02592/) (2024)
- [Factor Augmented Tensor-on-Tensor Neural Networks](https://mlsanity.com/paper/2405.19610/) (2024)
- [Covariate-assisted Sparse Tensor Completion](https://mlsanity.com/paper/2103.06428/) (2021)
- [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)
- [Conditional Uncertainty Quantification for Tensorized Topological Neural Networks](https://mlsanity.com/paper/2410.15241/) (2024)
- [Tensor-Fused Multi-View Graph Contrastive Learning](https://mlsanity.com/paper/2410.15247/) (2024)
- [A Locally Adaptive Algorithm for Multiple Testing with Network Structure](https://mlsanity.com/paper/2203.11461/) (2022)
- [Bayesian longitudinal tensor response regression for modeling neuroplasticity](https://mlsanity.com/paper/2309.10065/) (2023)
- [Robust Low-rank Tensor Train Recovery](https://mlsanity.com/paper/2410.15224/) (2024)
- [Tensor-on-tensor Regression Neural Networks for Process Modeling with High-dimensional Data](https://mlsanity.com/paper/2510.05329/) (2025)
- [Jointly Modeling and Clustering Tensors in High Dimensions](https://mlsanity.com/paper/2104.07773/) (2021)
- [Regression for matrix-valued data via Kronecker products factorization](https://mlsanity.com/paper/2404.19220/) (2024)
- [Efficient Decision Trees for Tensor Regressions](https://mlsanity.com/paper/2408.01926/) (2024)
- [Decomposable Sparse Tensor on Tensor Regression](https://mlsanity.com/paper/2212.05024/) (2022)
- [Alteration Detection of Tensor Dependence Structure via Sparsity-Exploited Reranking Algorithm](https://mlsanity.com/paper/2310.08798/) (2023)
- [MELCOT: A Hybrid Learning Architecture with Marginal Preservation for Matrix-Valued Regression](https://mlsanity.com/paper/2509.23315/) (2025)
- [Dual-Channel Tensor Neural Networks: Finite-Sample Theory and Conformal Structure Selection](https://mlsanity.com/paper/2605.19122/) (2026)

## Abstract (arXiv)

Aiming at abundant scientific and engineering data with not only high dimensionality but also complex structure, we study the regression problem with a multidimensional array (tensor) response and a vector predictor. Applications include, among others, comparing tensor images across groups after adjusting for additional covariates, which is of central interest in neuroimaging analysis. We propose parsimonious tensor response regression adopting a generalized sparsity principle. It models all voxels of the tensor response jointly, while accounting for the inherent structural information among the voxels. It effectively reduces the number of free parameters, leading to feasible computation and improved interpretation. We achieve model estimation through a nascent technique called the envelope method, which identifies the immaterial information and focuses the estimation based upon the material information in the tensor response. We demonstrate that the resulting estimator is asymptotically efficient, and it enjoys a competitive finite sample performance. We also illustrate the new method on two real neuroimaging studies.