Suggestions for improving performance of sparse ishermitian, issym

[KristofferC]

The current implementations for ishermitian and issym for sparse matrices are the full fledged A - A' == 0, see here. However, in many cases it is unnecessary to look at all elements before deciding that the matrix is not hermitian.

Due to the way sparse matrices are stored, checking one element at a time might be longer than the current method. On my computer, checking element by element is about 20% slower than the current method for hermitian matrices.

However, instead of looking at all the elements in an element by element fashion, we can look at only a fraction of the elements and if we find that the matrix is not hermitian we return false. If we do not yet know, we fall back to the full fledged method. This will have an insignificant time in the case that the matrix is hermitian but we can discard obviously non hermitian matrices instantly.

I wrote an example implementation of how this could look here:
https://gist.github.com/KristofferC/27dd9e1577f779c0d177

In the gist is also an example of a is_prob_hermitian which checks that |<Ax, y> - <x, Ay>| < eps for random x and y. I thought this could be called with a flag but maybe these type of functions are unwanted in Base.

I could polish what I have in the gist and make a PR if there is a positive response.

[stevengj]

The A - A' method seems very unsatisfying to me because it involves a lot of allocation and A might be extremely large. It seems better to try to speed up the element-by-element check: 20% is not an overwhelming margin, so I'm guessing that careful code optimization might be able to eliminate that penalty.

[KristofferC]

I agree. I guess one improvement you can do is that for a given column you start the search at the last found element instead of starting from the beginning every time. I will see what difference that makes.

[dhoegh]

There has been a discussion around testing approximate ishermitian in #10369 and #10298

[andreasnoack]

CHOLMOD has a function that checks the structure of a sparse matrix

julia/base/sparse/cholmod.jl

Line 1160 in 5a85e60

function ishermitian(A::Sparse{Complex{Float64}})

and we could use it more extensively. It requires a conversion to CHOLMOD.Sparse but in many cases it is probably faster than the present method you linked to. A better solution would be to have something like that in pure Julia.

As @dhoegh's links show, there might also be reasons beyond speed to prefer an approximative version.

[KristofferC]

This is the best I can do now which seems to be around 10% slower in the hermitian case for the test I did.

function ishermitian_new(A::SparseMatrixCSC)
    m, n = size(A)
    if m != n; return false; end

    colptr = A.colptr
    rowval = A.rowval
    nzval = A.nzval
    start_indices = copy(A.colptr)
    nrnzval = length(nzval)
    @inbounds for col = 1 : A.n
        for j = colptr[col] : colptr[col+1]-1
            row = rowval[j]
            nz = nzval[j]

            r1 = start_indices[row]
            r2 = Int(A.colptr[row+1]-1)
            r1 = searchsortedfirst(A.rowval, col, r1, r2, Forward)
            start_indices[row] = r1
            if nz != ctranspose(A.nzval[r1])
                return false
            end
        end
    end
    true
end

test:

new_herm = Float64[]
old_herm = Float64[]

for i = 1:10
    A = sprand(10^5,10^5, 0.0005);
    Aherm = A + A';
    push!(new_herm, @elapsed ishermitian_new(Aherm))
    push!(old_herm, @elapsed ishermitian(Aherm))
end

julia> mean(new_herm)
3.112574440315789

julia> mean(old_herm)
2.809209260263158

[stevengj]

I wonder if it would be faster to just do a linear search (rather than the binary search in searchsortedfirst) if r2-r1 is sufficently small?

[stevengj]

Also, shouldn't the test be if r1 <= r2 && nz != ctranspose(...), in case the sparsity pattern is not symmetrical?

[KristofferC]

I realized I did not exploit that we only need to check the lower diagonal against the upper.

With this: https://gist.github.com/KristofferC/36c760057b6e911e6258

I get around 2.5 times faster than Base version for Hermitian matrices.

Cholmod is likely a lot faster though, so maybe it is worth converting and ccalling.

@stevengj To be honest, I am not sure. Could you give a small example matrix where you think that you would need that.

Regarding linear / binary search, it would be interesting to benchmark.

[KristofferC]

@stevengj I realized what you meant now. I can't do the optimization I did above.

julia> A = speye(5)
julia> A[1,3] = 1
julia> ishermitian_new(A)
true

:(

[toivoh]

If you can get n ints of workspace to check whether an nxn sparse matrix is symmetric, you don't need to do any searching at all (if each column is stored on order, but it is, right?). If you go over the matrix column by column then, for the transpose access, you can keep an index for the next entry to be accessed in each column. Next time you access that column for the transpose, you check that the entry matches, and increment the index. This works as long as each column is sorted and you go through the columns in order for the non-transpose.

[toivoh]

But maybe those are too many assumptions? I would guess that the algorithm in CHOLMOD does something similar though.

[KristofferC]

Cholmod source code is here but I guess we shouldn't peak too much due to licencing:
https://www.cise.ufl.edu/research/sparse/cholmod/CHOLMOD/MatrixOps/cholmod_symmetry.c

You are right though that no searching is needed, I will try to implement what you described and test it more carefully this time :)

[KristofferC]

Closing this and referring to: #11371