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Finding an orthonormal basis for a reproducing kernel Hilbert space.

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Aimless
#1
Jul17-11, 04:59 AM
P: 128
Hello all,

I'm currently working on a problem in which I'm attempting to characterize a centered Gaussian random process \xi(x) on a manifold M given a known covariance function C(x,x') for that process. My current approach is to find a series expansion $\xi(x) = \sum_{n=1}^{\infty} X_n \phi_n(x)$ where the X_n's are Gaussian random variables with the standard distribution and the \phi_n(x)'s are the orthonormal basis functions of the reproducing kernel Hilbert space H generated by C(x,x'). I'm fairly certain that this procedure works so long as C^{-1}(x,x') exists, since that allows me to define the inner product for the RKHS.

However, I'm not sure how to proceed regarding finding an orthonormal basis for the RKHS. My RKHS should be some subset of L^2(M), so I assume that if I have an orthonormal basis on L^2(M) I should be able to find one for H. Is there a general procedure for this?

Thanks,
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Aimless
#2
Jul17-11, 08:32 AM
P: 128
Upon further consideration, I realized that I said something incorrect in the above post, but I think I might have also solved my problem. Comments and criticisms appreciated.

First, the mistake. While the RKHS will be composed of some subset of continuous functions on M, the norm will not in general be L^2.

To give a full definition of the RKHS, I start from the family of functions

[itex]S=\{u : M \to \mathbb{R} : u(\cdot) = \sum_{i=1}^n a_{i} C(\cdot,x_i)\}[/itex]

and define an inner product

[itex] \langle u(x),v(y) \rangle = \int_M \, dx \, dy \sum_{i=1}^n \sum_{j=1}^m a_i b_j C(x,x_i) C^{-1}(x,y) C(y,y_j) = \sum_{i=1}^n \sum_{j=1}^m a_i b_j C(x_i,y_j). [/itex]

The Hilbert space H is thus composed of all elements of S such that [itex] \langle u, u \rangle = \sum_{i=1}^n a_{i}^2 C(x_i,x_i) < \infty . [/itex]

So, as a possible answer to my original question, let the points [itex]\{ z_i \}[/itex] be a countable dense subset of M, and I should have a (non-orthonormal) basis for H in the form of the family of functions [itex] \{ C(x,z_i) \} [/itex]. From there, if I apply the Gram-Schmidt process, that should give me orthonormality, correct?

From there, for the purposes of numerically analyzing the Gaussian process [itex]\xi(x)[/itex], it should just be a matter of calculating the [itex] \{ C(x,z_i) \} [/itex] functions on a lattice of points to obtain an approximate solution.

I know that the above will work if M is simply Euclidean space so long as [itex]C^{-1}[/itex] exists. What about metric spaces?


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