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

TL;DR

This paper introduces a unified framework for multi-task functional linear regression with manifold constraints and composite penalties, providing theoretical analysis and convergence bounds applicable to various models.

Contribution

It proposes a general multi-task regression model with double regularization, unifying several existing approaches and analyzing their theoretical properties.

Findings

Unified convergence upper bound derived

Phase transition behaviors analyzed

Applicable to reduced-rank and graph Laplacian models

Abstract

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.