SciPy - Generalization of dot product over sparse and dense matrix

Question

Assume normal dot product:

M3[i,k] = sum_j(M1[i,j] * M2[j,k])

Now I would like to replace the sum by sum other operation, say the maximum:

M3[i,k] = max_j(M1[i,j] * M2[j,k])

This question is parallel to Numpy: Dot product with max instead of sum

Only now consider that the solutions

M3 = np.sum(M1[:,:,None]*M2[None,:,:], axis=1)

or

M3 = np.max(M1[:,:,None]*M2[None,:,:], axis=1)

should refer to a dense matrix M1 and a sparse matrix M2. Unfortunately, 3d sparse matrices are not available in SciPy.

Basically, this would mean that in

M3[i,k] = max_j(M1[i,j] * M2[j,k])

we iterate only over j such that M2[j,k]!=0.

What is the most efficient way to solve this problem?

Answer 1

Here's an approach using one loop that iterated through the common axis of reduction -

from scipy.sparse import csr_matrix
import scipy as sp

def reduce_after_multiply(M1, M2):
    # M1 : Nump array
    # M2 : Sparse matrix
    # Output : NumPy array

    # Get nonzero indices. Get start and stop indices representing 
    # intervaled indices along the axis of reduction containing 
    # the nonzero indices.
    r,c = sp.sparse.find(M2.T)[:2]
    IDs, start = np.unique(r,return_index=1)
    stop = np.append(start[1:], c.size)

    # Initialize output array and start loop for assigning values
    m, n = M1.shape[0], M2.shape[1]
    out = np.zeros((m,n))
    for iterID,i in enumerate(IDs):

        # Non zero indices for each col from M2. Use these to select 
        # M1's cols and M2's rows. Perform elementwise multiplication.
        idx = c[start[iterID]:stop[iterID]]
        mult = M1[:,idx]*M2.getcol(i).data

        # Use the inteneded ufunc along the second axis.
        out[:,i] = np.max(mult, axis=1) # Use any axis supported ufunc here
    return out

Sample run for verification -

In [248]: # Input data
     ...: M1 = np.random.rand(5,3)
     ...: M2 = csr_matrix(np.random.randint(0,3,(3,1000)))
     ...: 
     ...: # For variety, let's make one column as all zero. 
     ...: # This should result in corresponding col as all zeros as well.
     ...: M2[:,1] = 0
     ...: 

In [249]: # Verify
     ...: out1 = np.max(M1[:,:,None]*M2.toarray()[None,:,:], axis=1)

In [250]: np.allclose(out1, reduce_after_multiply(M1, M2))
Out[250]: True

---

Specifically for dot-product, we have a built-in dot method and as such is straight-forward with it. Thus, we can convert the first input which is dense array to a sparse matrix and then use sparse matrix's .dot method, like so -

csr_matrix(M1).dot(M2)

Let's verify this too -

In [252]: # Verify
     ...: out1 = np.sum(M1[:,:,None]*M2.toarray()[None,:,:], axis=1)

In [253]: out2 = csr_matrix(M1).dot(M2)

In [254]: np.allclose(out1, out2.toarray())
Out[254]: True