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.]

  • $\begingroup$ You may find the discussion in R's systemfit package vignette useful. $\endgroup$ Dec 17, 2018 at 13:42
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    $\begingroup$ This is called multivariate multiple regression $\endgroup$ Dec 17, 2018 at 13:50
  • $\begingroup$ I am interested in theoretical results—any pointers? $\endgroup$
    – Aryeh
    Dec 17, 2018 at 14:43
  • $\begingroup$ 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. $\endgroup$
    – whuber
    Dec 17, 2018 at 16:16


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