# Multivariate Functional Regression via Nested Reduced-Rank Regularization

arXiv: https://arxiv.org/abs/2003.04786 · HTML: https://mlsanity.com/paper/2003.04786/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Mar 2020.

## Built on by (most cited first, 2 of 2)

- [Parallel integrative learning for large-scale multi-response regression with incomplete outcomes](https://mlsanity.com/paper/2104.05076/) (2021)
- [Simultaneous Heterogeneity and Reduced-rank Learning for Multivariate Response Regression](https://mlsanity.com/paper/2512.06514/) (2025)

## Abstract (arXiv)

We propose a nested reduced-rank regression (NRRR) approach in fitting regression model with multivariate functional responses and predictors, to achieve tailored dimension reduction and facilitate interpretation/visualization of the resulting functional model. Our approach is based on a two-level low-rank structure imposed on the functional regression surfaces. A global low-rank structure identifies a small set of latent principal functional responses and predictors that drives the underlying regression association. A local low-rank structure then controls the complexity and smoothness of the association between the principal functional responses and predictors. Through a basis expansion approach, the functional problem boils down to an interesting integrated matrix approximation task, where the blocks or submatrices of an integrated low-rank matrix share some common row space and/or column space. An iterative algorithm with convergence guarantee is developed. We establish the consistency of NRRR and also show through non-asymptotic analysis that it can achieve at least a comparable error rate to that of the reduced-rank regression. Simulation studies demonstrate the effectiveness of NRRR. We apply NRRR in an electricity demand problem, to relate the trajectories of the daily electricity consumption with those of the daily temperatures.