Talk:Positive-definite function

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I'm confused

There seems to be confusion about positive semi-definite versus positive definite. Sam Coskey 03:02, 6 December 2006 (UTC)[reply]

yeah, it's rather unfortunate that there is no standard terminology. two common ones are:

positive definite means ≥ 0 and strictly positive means >0. this seems to be pretty standard in the general context of positive kernels. see positive definite kernel.
positive semidefinite means ≥ 0 and positive means >0. this is commonplace in matrix theory.

so, assuming the above, a positive semidefinite matrix is a particular instance of positive definite kernel. Mct mht 03:31, 1 April 2007 (UTC)[reply]

Too narrow

The explanation of positive definite function seems to be too narrow here.

In general a positive definite function is any function f(x,y) with the property that all finite index sets lead to positive definite matrices. Not, as the article says, just ones of the form f(x-y). Those of the form f(x-y) are called "stationary" or "homogeneous".

Also this is in the category "complex analysis" which I think may be too limiting. The value of the function doesn't necessarily have to be complex to have the property of positive definiteness.

For a reference see, e.g., Matthias Seeger, "Gaussian Processes for Machine Learning".

Baxissimo 06:58, 18 January 2007 (UTC)Baxissimo[reply]

i would agree that the definition is unnecessarily restrictive. what you call "stationary" is sometimes called "Toeplitz", by analogy with Toeplitz matrices. the contributor(s) who wrote that section probably had a reason for this restriction. Toeplitz kernels is intimately related to moment problems and Fourier transforms of positive measures, as they pointed out. Mct mht 05:30, 1 April 2007 (UTC)[reply]

Bochner's theorem?

I don't quite understand -- were you looking for a reference to Bochner's theorem? Why slap on a "fact" tag? Is this a particularly controversial statement or are you just seeking aid in finding a reference?

If you simply want aid finding a source, how about [1], [2], [3], or [4]. Also, if you simply want aid in finding a source, just ask on the talk page; there's no need to mark up the article. Lunch 15:39, 27 March 2007 (UTC)[reply]