K-mappings and Regression trees

We describe a method for learning a piecewise affine approximation to a mapping f : ℝ
^d
→ R
^p
given a labeled training set of examples {x
₁
,..., x
_n
} = X ⊂ ℝ
^d
and targets {y
₁
= f(x
₁
), ..., y
_n
= f(x
_n
)} = Y ⊂ ℝ
^p
. The method first trains a binary subdivision tree that splits across hyperplanes in X corresponding to high variance directions in Y. A fixed number K of affine regressors of rank q are then trained via a K-means like iterative algorithm, where each leaf must vote on its best fit mapping, and each mapping is updated as the best fit for the collection of leaves that chose it.