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There has been major progress on the representation of kernel matrices over the last five years. Notably, the KeOps library is an extension for PyTorch/NumPy/etc. that allows you to perform the operations you're thinking of very quickly (10-100 faster than a standard GPU implementation with PyTorch), with low memory usage.
As a satisfied customer (thanks!), was about to recommend KeOps as well. It might also be worth looking into falkon which builds on KeOps and leverages Nystrom approximation and conjugate gradient optimisation to further scale kernel operations.