Ridge function

In mathematics, a ridge function is any function 
 
 
 
 f
 :
 
 
 R
 
 
 d
 
 
 →
 
 R
 
 
 
 {\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} }
 
 that can be written as the composition of an <a href="/facts/Univariate/PYwC7u5H">univariate</a> function 
 
 
 
 g
 :
 
 R
 
 →
 
 R
 
 
 
 {\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }
 
, that is called a profile function, with an <a href="/facts/Affine_transformation/BD7yDr10">affine transformation</a>, given by a direction vector 
 
 
 
 a
 ∈
 
 
 R
 
 
 d
 
 
 
 
 {\displaystyle a\in \mathbb {R} ^{d}}
 
 with shift 
 
 
 
 b
 ∈
 
 R
 
 
 
 {\displaystyle b\in \mathbb {R} }
 
.
Then, the ridge function reads

f
 (
 x
 )
 =
 g
 (
 
 x
 
 ⊤
 
 
 a
 +
 b
 )
 
 
 {\displaystyle f(x)=g(x^{\top }a+b)}
 
 for 
 
 
 
 x
 ∈
 
 
 R
 
 
 d
 
 
 
 
 {\displaystyle x\in \mathbb {R} ^{d}}
 
. 
Coinage of the term 'ridge function' is often attributed to B.F. Logan and L.A. Shepp.

Ridge function open-in-new

Ridge function