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 univariate function g : R → R {\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } , that is called a profile function, with an affine transformation, given by a direction vector a ∈ R d {\displaystyle a\in \mathbb {R} ^{d}} with shift b ∈ R {\displaystyle b\in \mathbb {R} } .
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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 univariate function
g
:
R
→
R
{\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }
, that is called a profile function, with an affine transformation, 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.
A ridge function is not susceptible to the curse of dimensionality, making it an instrumental tool in various estimation problems. This is a direct result of the fact that ridge functions are constant in
d
−
1
{\displaystyle d-1}
directions: Let
a
1
,
…
,
a
d
−
1
{\displaystyle a_{1},\dots ,a_{d-1}}
be
d
−
1
{\displaystyle d-1}
independent vectors that are orthogonal to
a
{\displaystyle a}
, such that these vectors span
d
−
1
{\displaystyle d-1}
dimensions. Then
f
(
x
+
∑
k
=
1
d
−
1
c
k
a
k
)
=
g
(
x
⋅
a
+
∑
k
=
1
d
−
1
c
k
a
k
⋅
a
)
=
g
(
x
⋅
a
+
∑
k
=
1
d
−
1
c
k
0
)
=
g
(
x
⋅
a
)
=
f
(
x
)
{\displaystyle f\left({\boldsymbol {x}}+\sum _{k=1}^{d-1}c_{k}{\boldsymbol {a}}_{k}\right)=g\left({\boldsymbol {x}}\cdot {\boldsymbol {a}}+\sum _{k=1}^{d-1}c_{k}{\boldsymbol {a}}_{k}\cdot {\boldsymbol {a}}\right)=g\left({\boldsymbol {x}}\cdot {\boldsymbol {a}}+\sum _{k=1}^{d-1}c_{k}0\right)=g({\boldsymbol {x}}\cdot {\boldsymbol {a}})=f({\boldsymbol {x}})}
for all
c
i
∈
R
,
1
≤
i
<
d
{\displaystyle c_{i}\in \mathbb {R} ,1\leq i
. In other words, any shift of
x
{\displaystyle {\boldsymbol {x}}}
in a direction perpendicular to
a
{\displaystyle {\boldsymbol {a}}}
does not change the value of
f
{\displaystyle f}
.
Ridge functions play an essential role in amongst others projection pursuit, generalized linear models, and as activation functions in neural networks. For a survey on ridge functions, see. For books on ridge functions, see.
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