# $\mathbb{R}^m\to\mathbb{R}^n$ regression

The naive approach to a $$\mathbb{R}^m\to\mathbb{R}^n$$ regression problem is simply to solve $$n$$ distinct $$\mathbb{R}^m\to\mathbb{R}$$ regression problems. However, suppose that the output of the regressor $$f:\mathbb{R}^m\to\mathbb{R}^n$$ must satisfy some constraint, such as lying on some manifold or belonging to some convex set. Have such regression problems been studied? Pointers to theoretical results much welcome.

[Previously posted at cstheory, now deleted from there.]

• You may find the discussion in R's systemfit package vignette useful. Dec 17, 2018 at 13:42
• This is called multivariate multiple regression Dec 17, 2018 at 13:50
• I am interested in theoretical results—any pointers? Dec 17, 2018 at 14:43
• The restriction to lie on a (non-affine) manifold is extremely broad, because perforce it makes this a nonlinear regression problem. These things are studied one at a time because their behaviors can be so incredibly varied.
– whuber
Dec 17, 2018 at 16:16