Improve efficiency of kernel calculation

[joannajzou]

Small change to reduce the cost of computing symmetric kernel matrices. Only one half of the matrix terms [Kij] are computed, then copied over to [Kji].

The code is verified to return the same matrix as the original code.

[vchuravy]

In theory we wouldn't even need to store the Lower triangle since it is being wrapped in Symmetric. Symmetric disregards the lower triangle already.

https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#LinearAlgebra.Symmetric

Note that Supper will not be equal to Slower unless A is itself symmetric (e.g. if A == transpose(A)).

[emmanuellujan]

@joannajzou To save memory allocations, I would try using SparseArrays. The current implementation still allocates the lower triangular part of the matrix.

[vchuravy]

For symmetric arrays the 2x memory overhead is better than the memory cost of a 50% filled sparse arrays. Remember that sparse arrays need to allocate two more int64 arrays of the same size for the indices.

@joannajzou I think your pull-request is still an improvement? Since it reduces the runtime of the initialization by half?

[emmanuellujan]

For symmetric arrays the 2x memory overhead is better than the memory cost of a 50% filled sparse arrays. Remember that sparse arrays need to allocate two more int64 arrays of the same size for the indices.

True, but there should be an abstraction for sparse symmetric matrices somewhere, right? In problems related to clustering, where we need to compute a matrix of distances where N is of the order of millions, we need to save as much memory as possible. Do you know which is the proper abstraction to tackle this?

@joannajzou I think your pull-request is still an improvement? Since it reduces the runtime of the initialization by half?

It is