I am trying some new kernel functions.
For some new kernel functions, I have checked the eigen values of corresponding Gram matrix(UCI bench mark data set). The eigen values are positive and for one kernel function it is mixture of positive and negative. Can I conclude that the kernel functions corresponding all positive Eigen values are PD and with mixture is indefinite. I read somewhere that the kernel functions which are not PD may give rise to the Gram matrix with all positive eigen values , Because of this sentence I have confusion.
I have checked the nature of the eigen values taking the parameter values which give good classification accuracy.
I have also confusion between CPD and PD kernels. What is the eigen values of CPD kernels?
I have read "Bernhard Scholkopf, and Alexander J. Smola. Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT press, 2001" . But, still want some easy explanation and proof regrading this.
These works might be of interest in the construction of kernel matrices for approaching engineering applications.
Conference Paper An approach to water supply clusters by semi-supervised learning
Article Metamodel-assisted optimization based on multiple kernel reg...
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regarding CDP vs PD kernel, the following link may prove useful :
https://stats.stackexchange.com/questions/149889/prove-that-a-kernel-is-conditionally-positive-definite
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i know of no easy eigenvalue-based test of the "CPD-ness" of a kernel (except the detour by its exponential, which is definitely not easy !)
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for an admissible kernel (definite or indefinite positive), the eigenvalues of the Gram matrix should be non negative
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Gram matrices are just functions of data sets ; as such, the positiveness of the entries of a set of Gram matrices do not **prove** anything on the positiveness of the kernel (your data set may have some structure which makes the entries all positive for a specific non positive kernel)
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on the other hand, negative entries in the Gram matrix disprove the positivity of the kernel
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