[Question] How reverse engineering RBF-Kernel #1890

[makinzm]

📚 Question

I would like to create odd RBF-kernel which of "x-y" is converted into "x+y".
However, I don't have confidence my code works.
Therefore, if you teach me, I would appreciate it.

# ReverseRBF kernel (start)
from gpytorch.functions import RBFCovariance
from gpytorch.kernels import Kernel
from gpytorch.settings import trace_mode


def postprocess_rbf(dist_mat):
    return dist_mat.div_(-2).exp_()

class ReverseRBFKernel(Kernel):
    has_lengthscale = True

    def forward(self, x1, x2, diag=False, **params):
        if (
            x1.requires_grad
            or x2.requires_grad
            or (self.ard_num_dims is not None and self.ard_num_dims > 1)
            or diag
            or trace_mode.on()
        ):
            x1_ = x1.div(self.lengthscale)
            x2_ = x2.div(self.lengthscale)
            return self.covar_dist(
                -1*x1_, x2_, square_dist=True, diag=diag, dist_postprocess_func=postprocess_rbf, postprocess=True, **params
            )
        return RBFCovariance().apply(
            -1*x1,
            x2,
            self.lengthscale,
            lambda x1, x2: self.covar_dist(
                x1, x2, square_dist=True, diag=False, dist_postprocess_func=postprocess_rbf, postprocess=False, **params
            ),
        )
        
# ReverseRBF kernel (end)

my-example is here : https://github.com/makinzm/UNDONE_local/blob/main/08_oputuna/gp-02.ipynb

Issue : #1890

[jacobrgardner]

Let x1 = [sqrt(2)/4, sqrt(2)/4] and x2 = [-sqrt(2)/4, -sqrt(2)/4], and assume a lengthscale of 1.

Then, ||x1 + x1||^2 = ||x2 + x2||^2 = 1, but ||x1 + x2||^2 = ||x2 + x1||^2 = 0.

So the 2x2 kernel matrix would be exp(-I) = [[1/e, 1], [1, 1/e]].

This matrix has eigenvalues 1/e + 1 and 1/e - 1. 1/e - 1 is negative, so the matrix isn't positive definite.

[makinzm]

Thank you for show this example!