[WIP] Sparse implementation of EMD

[tvayer]

The goal is to return a sparse transport in ot.emd plan when the cost matrix matrix is itself sparse which corresponds to infinite cost between points, or when no value is given i.e. we know that no match is allowed.

Wishlist

• Accelerate the EMD when the cost is sparse
• Stay in O(nnz(C)) in memory

What we have

• Working cython wrapper for coo cost matrix, it returns the same result at numerical precision

Working example:

from ot.lp import emd_wrap

# COO representation of cost
iM2 = sM2.row
jM2 = sM2.col
vM2 = sM2.data

maxitr= 1000000
g,ig,jg, cost, alpha, beta, result_code = emd_wrap.emd_c_sparse(a,b,iM2,jM2,vM2, np.int64(maxitr))

# recover the sparse plan
Gs = sp.sparse.coo_matrix((g, (ig, jg)), shape=(m,n))

What we want

• Change the cpp of NetworkSimplex to remove the O(n^2) memory when associating the cost to each edge of the network flow