Re: questions about ichol

[siko1056]

Hi Juan,


Juan Pablo Carbajal-2 wrote
> The squared exponential function is one of the archetypes positive
> definite kernels
> https://en.wikipedia.org/wiki/Covariance_function#Parametric_families_of_covariance_functions
> So maybe your are working with a different definition of positive
> definite 

No, we are working with the same definition of positive definite, and
numerically in double precision it is not. Also Matlab fails on your data,
so I assumed with more background information to find a better solution.


Juan Pablo Carbajal-2 wrote
> or the method you are using just can handle small
> eigenvalues, which was the point I was trying to rise.
> 
> The incomplete Cholesky factorization has been used for positive
> kernels and that is how I ended in your algorithm[1], but it seems the
> implementation is not general enough.
> 
> 
> [1] Fine, S. and Scheinberg, K. (2002). Efficient SVM Training Using
> Low-Rank Kernel Representations. Journal of Machine Learning Research,
> 2(2):243–264.

After reading some of [1] I got stuck with the sentence of Figure 6 and 7
using a variant of incomplete Cholesky factorization "input dim 10". As I am
not a domain expert, does it mean an input

d = 10; t=linspace(0,1,d) ?

In this case, I assume there are already better approaches available for
larger problem dimensions. Using your data with a full Cholesky
factorization, up to dimension d=40, I had no problems. Using a higher
dimension d quickly results in numerical difficulties, including the claim
of non-positive definitness.

Additionally in [1] they used an incomplete Cholesky factoriztation with
symmetric pivoting, which is not implemented in ichol. Try the dense
Cholesky factorization with chol(k, "vector") to get this permutation.

Kai



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