Periodic kernel for multivariate input

🐛 Bug

Implementation of PeriodicKernel is not positive definite in case of multivariate input Here is how I checked it:

** Code snippet to reproduce **

import torch
from gpytorch import kernels

X = torch.randn(1000, 2, dtype=torch.float64)
kernel = kernels.PeriodicKernel().double()
with torch.no_grad():
    K = kernel(X, X).evaluate()
    eig = torch.symeig(K)[0]

    negative_eig = eig[eig < 0]
    assert torch.allclose(negative_eig, torch.zeros_like(negative_eig))

** Stack trace/error message **

---------------------------------------------------------------------------
AssertionError                            Traceback (most recent call last)
<ipython-input-2-1d738d7c73d9> in <module>
      9 
     10     negative_eig = eig[eig < 0]
---> 11     assert torch.allclose(negative_eig, torch.zeros_like(negative_eig))

AssertionError: 

Expected Behavior

Positive definite kernel must have non-negative eigenvalues.

The implementation is correct for one-dimensional input. For multidimensional input it should be product of one-dimensional kernels, however, it calculates the norm first and applies sine and exponent to the norm which is not correct: see eq. (47) on page 15 of

MacKay, D. J. (1998). Introduction to Gaussian processes. NATO ASI Series F Computer and Systems Sciences, 168, 133-166.

System information

Please complete the following information:

• gpytorch=0.3.4
• torch=1.1.0
• Ubuntu 19.04

[jacobrgardner]

The periodic kernel is by definition a kernel defined in 1D. To extend to multiple dimensions use e.g. ProductKernel

[gpleiss]

@tawe141 I don’t think we ever got around to this. If you’re interested and willing, we’d love a PR from you!