Nonparametric Reduced Rank Regression
This work addresses the challenge of high-dimensional multivariate regression with nonlinear dependencies, particularly relevant for applications like genomics, though it appears incremental as it extends existing reduced rank regression concepts to nonparametric settings.
The authors tackled the problem of multivariate nonparametric regression by generalizing reduced rank regression to handle nonlinear relationships, proposing a method that controls model complexity using a functional nuclear norm to achieve low-rank function estimates. They derived backfitting algorithms, provided theoretical guarantees with oracle inequalities on excess risk, and demonstrated the method on gene expression data.
We propose an approach to multivariate nonparametric regression that generalizes reduced rank regression for linear models. An additive model is estimated for each dimension of a $q$-dimensional response, with a shared $p$-dimensional predictor variable. To control the complexity of the model, we employ a functional form of the Ky-Fan or nuclear norm, resulting in a set of function estimates that have low rank. Backfitting algorithms are derived and justified using a nonparametric form of the nuclear norm subdifferential. Oracle inequalities on excess risk are derived that exhibit the scaling behavior of the procedure in the high dimensional setting. The methods are illustrated on gene expression data.