# A Unified Analysis of Multi-task Functional Linear Regression Models with Manifold Constraint and Composite Quadratic Penalty

arXiv: https://arxiv.org/abs/2211.04874 · HTML: https://mlsanity.com/paper/2211.04874/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 72 references that are in the map.
- First posted on arXiv 9 Nov 2022.

## Builds on (most cited first, 9 of 9)

- [Multi-Task Learning with Deep Neural Networks: A Survey](https://mlsanity.com/paper/2009.09796/) (2020)
- [Distral: Robust Multitask Reinforcement Learning](https://mlsanity.com/paper/1707.04175/) (2017)
- [The Benefit of Multitask Representation Learning](https://mlsanity.com/paper/1505.06279/) (2015)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Randomized sketches for kernels: Fast and optimal non-parametric regression](https://mlsanity.com/paper/1501.06195/) (2015)
- [Local Rademacher Complexity-based Learning Guarantees for Multi-Task Learning](https://mlsanity.com/paper/1602.05916/) (2016)
- [Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphs](https://mlsanity.com/paper/2106.01529/) (2021)
- [A maximum principle argument for the uniform convergence of graph Laplacian regressors](https://mlsanity.com/paper/1901.10089/) (2019)
- [Convex Discriminative Multitask Clustering](https://mlsanity.com/paper/1303.2130/) (2013)

## Built on by (most cited first, 1 of 1)

- [Functional BART with Shape Priors: A Bayesian Tree Approach to Constrained Functional Regression](https://mlsanity.com/paper/2502.16888/) (2025)

## Abstract (arXiv)

This work studies the multi-task functional linear regression models where both the covariates and the unknown regression coefficients (called slope functions) are curves. For slope function estimation, we employ penalized splines to balance bias, variance, and computational complexity. The power of multi-task learning is brought in by imposing additional structures over the slope functions. We propose a general model with double regularization over the spline coefficient matrix: i) a matrix manifold constraint, and ii) a composite penalty as a summation of quadratic terms. Many multi-task learning approaches can be treated as special cases of this proposed model, such as a reduced-rank model and a graph Laplacian regularized model. We show the composite penalty induces a specific norm, which helps to quantify the manifold curvature and determine the corresponding proper subset in the manifold tangent space. The complexity of tangent space subset is then bridged to the complexity of geodesic neighbor via generic chaining. A unified convergence upper bound is obtained and specifically applied to the reduced-rank model and the graph Laplacian regularized model. The phase transition behaviors for the estimators are examined as we vary the configurations of model parameters.