Calibrated Multivariate Regression with Application to Neural Semantic Basis Discovery
This work addresses the challenge of high-dimensional multivariate regression for researchers and practitioners in fields like neuroscience, offering an incremental improvement through task-specific calibration.
The authors tackled the problem of fitting high-dimensional multivariate regression models by proposing a calibrated multivariate regression (CMR) method that adjusts regularization per task based on noise levels, achieving improved finite-sample performance and tuning insensitiveness, with numerical simulations showing consistent outperformance over other methods and competitive results in a brain activity prediction application.
We propose a calibrated multivariate regression method named CMR for fitting high dimensional multivariate regression models. Compared with existing methods, CMR calibrates regularization for each regression task with respect to its noise level so that it simultaneously attains improved finite-sample performance and tuning insensitiveness. Theoretically, we provide sufficient conditions under which CMR achieves the optimal rate of convergence in parameter estimation. Computationally, we propose an efficient smoothed proximal gradient algorithm with a worst-case numerical rate of convergence $\cO(1/ε)$, where $ε$ is a pre-specified accuracy of the objective function value. We conduct thorough numerical simulations to illustrate that CMR consistently outperforms other high dimensional multivariate regression methods. We also apply CMR to solve a brain activity prediction problem and find that it is as competitive as a handcrafted model created by human experts. The R package \texttt{camel} implementing the proposed method is available on the Comprehensive R Archive Network \url{http://cran.r-project.org/web/packages/camel/}.