What is the reason for the failure of a RBF kernel to generate a covariance matrix that is positive-definite?

edit

The failure of a Radial Basis Function (RBF) kernel to generate a positive-definite covariance matrix can be due to several reasons:

1. Choosing a very small value of the length-scale parameter: The length-scale parameter determines the length scale over which the RBF kernel functions decay. If the length-scale parameter is set to a very small value, then the RBF kernel will have very high-frequency oscillations, causing the covariance matrix to become ill-conditioned (i.e., not positive-definite).

2. Choosing a very large value of the length-scale parameter: If the length-scale parameter is set to a very large value, then the RBF kernel will have very low-frequency oscillations, causing the covariance matrix to become close to singular.

3. Choosing a high noise level: If the noise level is high, then the RBF kernel becomes less sensitive to the data and can result in a poorly conditioned covariance matrix.

4. Poorly conditioned training data: If the training data is poorly conditioned (i.e., contains a high degree of collinearity or redundancy), then the covariance matrix can become ill-conditioned.

Overall, it is crucial to carefully choose the hyperparameters of the RBF kernel and thoroughly preprocess the training data to ensure stable and positive-definite covariance matrices.