Mixing
RKHS of a Polynomial Kernel with negative roots
$begingroup$
Wikipedia, and actually all books and ressources I could find, define a polynomlial kerenel as:
$$
K: x,y mapsto (x^Ty + c)^2,
$$
where $cge 0$. What happens if $c < 0$? Take the following kernels for example:
$$
forall x,y in mathbb{R}
begin{cases}
K_+: (x,y)mapsto (xy + 1)^2 \
K_-: (x,y)mapsto (xy - 1)^2
end{cases}
$$
Both $K_+$ and $K_-$ are positive definite kernel and therefore both have an RKHS. Shouldn't $K_-$ be included in the polynomial kernel definition?
Another question I have about this example is: although it is quite easy to find $K_+$'s RKHS, it seems way more difficult to find a closed form expression of $K_-$'s RKHS.
An idea could be to see $K_-$ as a $90º$ rotation of $K_+$, but I couldn't get from this to a proper expression of the associated Hilbert space.
Any help would be greatly appreciated :)
rkhs reproducing-kernel-hilbert-spaces
asked Jan 5 at 9:27
Marc LeoniMarc Leoni
12
$endgroup$
add a comment |
$begingroup$
Wikipedia, and actually all books and ressources I could find, define a polynomlial kerenel as:
$$
K: x,y mapsto (x^Ty + c)^2,
$$
where $cge 0$. What happens if $c < 0$? Take the following kernels for example:
$$
forall x,y in mathbb{R}
begin{cases}
K_+: (x,y)mapsto (xy + 1)^2 \
K_-: (x,y)mapsto (xy - 1)^2
end{cases}
$$
Both $K_+$ and $K_-$ are positive definite kernel and therefore both have an RKHS. Shouldn't $K_-$ be included in the polynomial kernel definition?
Another question I have about this example is: although it is quite easy to find $K_+$'s RKHS, it seems way more difficult to find a closed form expression of $K_-$'s RKHS.
An idea could be to see $K_-$ as a $90º$ rotation of $K_+$, but I couldn't get from this to a proper expression of the associated Hilbert space.
Any help would be greatly appreciated :)
rkhs reproducing-kernel-hilbert-spaces
asked Jan 5 at 9:27
Marc LeoniMarc Leoni
12
$endgroup$
$begingroup$
You wrote $K_+$ twice
$endgroup$
– Alessandro Codenotti
Jan 5 at 11:25
$begingroup$
@AlessandroCodenotti Thanks! Just edited it.
$endgroup$
– Marc Leoni
Jan 5 at 13:56
add a comment |
$begingroup$
Wikipedia, and actually all books and ressources I could find, define a polynomlial kerenel as:
$$
K: x,y mapsto (x^Ty + c)^2,
$$
where $cge 0$. What happens if $c < 0$? Take the following kernels for example:
$$
forall x,y in mathbb{R}
begin{cases}
K_+: (x,y)mapsto (xy + 1)^2 \
K_-: (x,y)mapsto (xy - 1)^2
end{cases}
$$
Both $K_+$ and $K_-$ are positive definite kernel and therefore both have an RKHS. Shouldn't $K_-$ be included in the polynomial kernel definition?
Another question I have about this example is: although it is quite easy to find $K_+$'s RKHS, it seems way more difficult to find a closed form expression of $K_-$'s RKHS.
An idea could be to see $K_-$ as a $90º$ rotation of $K_+$, but I couldn't get from this to a proper expression of the associated Hilbert space.
Any help would be greatly appreciated :)
rkhs reproducing-kernel-hilbert-spaces
asked Jan 5 at 9:27
Marc LeoniMarc Leoni
12
$endgroup$
Wikipedia, and actually all books and ressources I could find, define a polynomlial kerenel as:
$$
K: x,y mapsto (x^Ty + c)^2,
$$
where $cge 0$. What happens if $c < 0$? Take the following kernels for example:
$$
forall x,y in mathbb{R}
begin{cases}
K_+: (x,y)mapsto (xy + 1)^2 \
K_-: (x,y)mapsto (xy - 1)^2
end{cases}
$$
Both $K_+$ and $K_-$ are positive definite kernel and therefore both have an RKHS. Shouldn't $K_-$ be included in the polynomial kernel definition?
Another question I have about this example is: although it is quite easy to find $K_+$'s RKHS, it seems way more difficult to find a closed form expression of $K_-$'s RKHS.
An idea could be to see $K_-$ as a $90º$ rotation of $K_+$, but I couldn't get from this to a proper expression of the associated Hilbert space.
Any help would be greatly appreciated :)
rkhs reproducing-kernel-hilbert-spaces
rkhs reproducing-kernel-hilbert-spaces
asked Jan 5 at 9:27
Marc LeoniMarc Leoni
12
asked Jan 5 at 9:27
Marc LeoniMarc Leoni
12
edited Jan 5 at 13:56
Marc Leoni
asked Jan 5 at 9:27
Marc LeoniMarc Leoni
12
asked Jan 5 at 9:27
Marc LeoniMarc Leoni
12
asked Jan 5 at 9:27
Marc LeoniMarc Leoni
12
12
$begingroup$
You wrote $K_+$ twice
$endgroup$
– Alessandro Codenotti
Jan 5 at 11:25
$begingroup$
@AlessandroCodenotti Thanks! Just edited it.
$endgroup$
– Marc Leoni
Jan 5 at 13:56
add a comment |
$begingroup$
You wrote $K_+$ twice
$endgroup$
– Alessandro Codenotti
Jan 5 at 11:25
$begingroup$
@AlessandroCodenotti Thanks! Just edited it.
$endgroup$
– Marc Leoni
Jan 5 at 13:56
$begingroup$
You wrote $K_+$ twice
$endgroup$
– Alessandro Codenotti
Jan 5 at 11:25
$begingroup$
You wrote $K_+$ twice
$endgroup$
– Alessandro Codenotti
Jan 5 at 11:25
$begingroup$
@AlessandroCodenotti Thanks! Just edited it.
$endgroup$
– Marc Leoni
Jan 5 at 13:56
$begingroup$
@AlessandroCodenotti Thanks! Just edited it.
$endgroup$
– Marc Leoni
Jan 5 at 13:56
add a comment |
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$begingroup$
You wrote $K_+$ twice
$endgroup$
– Alessandro Codenotti
Jan 5 at 11:25
$begingroup$
@AlessandroCodenotti Thanks! Just edited it.
$endgroup$
– Marc Leoni
Jan 5 at 13:56