How to create original kernel : k(flip(x),y)

[makinzm]

First : what I would like to

I would like to create the following kernel : like k(flip(x),y).
For example, FlipRBF([1,2],[3,4])=RBF([2,1],[3,4]).
(This kernel may not be symmetry (k(flip(x),y)≠ k(flip(y),x)).However, Flip-RBF satisfy kernel function requirement.)

And Flip-kernel is the same when the dimension of input is 1 because flip(x)=x.

Second : what I do

I create this Gaussian Regression Sample :
Reference : https://github.com/makinzm/UNDONE_local/blob/main/08_oputuna/gp-03.ipynb

WEIGHT=0
minus_Mode=False

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

class SymmetryRBFKernel2D(Kernel):
    has_lengthscale = True

    def forward(self, x1, x2, diag=False,  weight=WEIGHT,**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)
            if not minus_Mode:
                return self.covar_dist(
                    x1_[torch.arange(len(x1_)-1,-1,-1) ], x2_, square_dist=True, diag=diag, dist_postprocess_func=postprocess_rbf, postprocess=True, **params
                )+weight*self.covar_dist(
                    x1_, x2_, square_dist=True, diag=diag, dist_postprocess_func=postprocess_rbf, postprocess=True, **params
                )
            else:
                return self.covar_dist(
                    -1*x1, x2_, square_dist=True, diag=diag, dist_postprocess_func=postprocess_rbf, postprocess=True, **params
                )+weight*self.covar_dist(
                    x1_, x2_, square_dist=True, diag=diag, dist_postprocess_func=postprocess_rbf, postprocess=True, **params
                )
        if not minus_Mode:
            return RBFCovariance().apply(
                x1,
                x2,
                self.lengthscale,
                lambda x1, x2: self.covar_dist(
                    x1[torch.arange(len(x1)-1,-1,-1) ], x2, square_dist=True, diag=False, dist_postprocess_func=postprocess_rbf, postprocess=False, **params
                )+weight*self.covar_dist(
                    x1, x2, square_dist=True, diag=False, dist_postprocess_func=postprocess_rbf, postprocess=False, **params
                ),
            )
        else:
            return RBFCovariance().apply(
                x1,
                x2,
                self.lengthscale,
                lambda x1, x2: self.covar_dist(
                    -1*x1, x2, square_dist=True, diag=False, dist_postprocess_func=postprocess_rbf, postprocess=False, **params
                )+weight*self.covar_dist(
                    x1, x2, square_dist=True, diag=False, dist_postprocess_func=postprocess_rbf, postprocess=False, **params
                ),
            )

However, the result is different even though the dimension is 1 :

I would appreciate it if you could tell me what the problem is.

[gpleiss]

This kernel may not be symmetry (k(flip(x),y)≠ k(flip(y),x)

Kernels must be symmetric and positive definite.