RKHS of a Polynomial Kernel with negative roots
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-space
asked 2 days ago
Marc Leoni
12
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Marc Leoni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
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-space
asked 2 days ago
Marc Leoni
12
New contributor
Marc Leoni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
You wrote $K_+$ twice
– Alessandro Codenotti
yesterday
@AlessandroCodenotti Thanks! Just edited it.
– Marc Leoni
yesterday
add a comment |
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-space
asked 2 days ago
Marc Leoni
12
New contributor
Marc Leoni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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-space
rkhs reproducing-kernel-hilbert-space
asked 2 days ago
Marc Leoni
12
New contributor
Marc Leoni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
Marc Leoni
12
New contributor
Marc Leoni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
asked 2 days ago
Marc Leoni
12
New contributor
Marc Leoni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
Marc Leoni
12
asked 2 days ago
Marc Leoni
12
12
New contributor
Marc Leoni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Marc Leoni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Marc Leoni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
You wrote $K_+$ twice
– Alessandro Codenotti
yesterday
@AlessandroCodenotti Thanks! Just edited it.
– Marc Leoni
yesterday
add a comment |
You wrote $K_+$ twice
– Alessandro Codenotti
yesterday
@AlessandroCodenotti Thanks! Just edited it.
– Marc Leoni
yesterday
You wrote $K_+$ twice
– Alessandro Codenotti
yesterday
You wrote $K_+$ twice
– Alessandro Codenotti
yesterday
@AlessandroCodenotti Thanks! Just edited it.
– Marc Leoni
yesterday
@AlessandroCodenotti Thanks! Just edited it.
– Marc Leoni
yesterday
add a comment |
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You wrote $K_+$ twice
– Alessandro Codenotti
yesterday
@AlessandroCodenotti Thanks! Just edited it.
– Marc Leoni
yesterday