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I devised a distance function similar to this form. K(x,y)=(-||x-y||+xy+1)/2

And now I want to prove it is a kernel function.I have read about Mercer's condition and positive semi definiteness, but I still do not know how to prove it?

If it is not a kernel function, can it be kernel function when x and y are 1 or 0?

Any help would be greatly appreciated :)

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To answer your first question, the kernel function must be symmetric, i.e.

$K(\mathbf{x}, \mathbf{y}) = K(\mathbf{y}, \mathbf{x})$

and positive semi definite, which we can prove if we can factorise the kernel function in the form of a dot product

$K(\mathbf{x}, \mathbf{y}) = <f(\mathbf{x}),f(\mathbf{y})>$

where $f(x)$ is a vector function. In this case, I suspect your input is scalar, so in this case becomes:

$K(x, y) = <g(x),g(y)>$

where g is a scalar function, meaning we need to determine if there exists a function $g(x)$ which is the square root of

$(-|x-y|+xy+1)/2 = (-\sqrt{(x-y)^2}+xy+1)/2$.

The trickiest part here is showing that $-\sqrt{(x-y)^2}$ can be written as a product of two sets of symmetric terms, one in x and one in y. That might be possible, but it would involve imaginary elements since we have a minus sign at the front of the $-\sqrt{(x-y)^2}$ term.

Instead we can check for positive semi-definiteness by trying to find points that violate this rule, i.e. where the function is decreasing. This is quite easy to do, for example if $y = 0$ and $x = 10$, then $K(x,y) = -9/2$. If $y$ remains equal to 0 then as $x$ increases to 100, we see that $K(x,y)$ decreases to $-99/2$. Therefore the function is not positive semi-definite. You can check this property more rigorously by taking the partial derivatives of the kernel function and show whether or not they are non-negative for all x and y.

To answer your second question, "If it is not a kernel function, can it be kernel function when x and y are 1 or 0?" the answer is no, since you would have to define the function only valid for x = y = 1 = 0, or x = y = 1, (since the above must hold for all values of the input domain due to symmetry requirement) by restricting the domain of the inputs to a single value, and then there is no concept of positive semi-definiteness (no gradient).

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