# Regression in projective space?

Is there a method for (nonlinear? kernelized?) regression of functions with output in projective space? That is, given a series of examples $x_i\in\mathbb{R}^n$ (or $x_i\in\mathbb{P}^n$) and $y_i\in\mathbb{P}^m$ (so, $\forall c\in\mathbb{R}, y_i=c\cdot y_i$) such that $f(x_i)=y_i$, can I write down some function that reasonably approximates $f$?

• @Dmitrij By definition, projective space in dimension $m$ consists of equivalence classes of nonzero $m+1$-vectors $\mathbf{y}$ modulo the relation $\mathbf{y} \sim c \cdot \mathbf{y}$ for any scalar $c$. The question therefore says that any estimator $\hat{f}(\mathbf{x}) = c \cdot \mathbf{y}$ should be considered perfectly accurate. The challenge is to find some measure of discrepancy (which comes down to a tiny generalization of circular regression, I believe). – whuber Sep 27 '11 at 4:32
• I found this link with question title as a search term: www-sop.inria.fr/asclepios/events/MFCA11/Proceedings/MFCA11_3_1, maybe it will help. – mpiktas Sep 27 '11 at 8:26
• @whuber, never met one in my practice (neither circular regression), thanks for clarification. Geodesic regression discussion from the above link seems to be also interesting. – Dmitrij Celov Sep 27 '11 at 11:27
• I'll check these out -- thanks for your help! – Justin Solomon Sep 30 '11 at 23:26