Method of matrix multiplication between two numpy matrices of log2 elements

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I have two numpy matrices a_log2 and b_log2 of log2 elements and I want to perform matrix multiplication between them.

a = np.array([[0.4, 0.4, 0.2],
          [0.1, 0.5, 0.4]])

b = np.array([[0.3, 0.7],
          [0.5, 0.5],
          [0.2, 0.8]])

a_log2 = np.log2(a)
b_log2 = np.log2(b)

I used to perform matrix multiplication of e-based logarithms using scipy.special.logsumexp. Here is the code I use (thanks to Erik Parkinson for his answer in this thread Handling matrix multiplication in log space in Python):

def log_space_product(A,B):
   Astack = np.stack([A]*A.shape[0]).transpose((2,1,0))
   Bstack = np.stack([B]*B.shape[1]).transpose((1,0,2))
   log_sum_exp = logsumexp(Astack+Bstack, axis=0)
   return log_sum_exp

Now, I need your help in performing matrix multiplication between a_log2 and b_log2 because scipy.special.logsumexp was not defined for base-2 logarithms.

Note: I was firstly planning to convert the matrix elements into natural logarithms using a_loge[i, j] = np.log(2**a_log2[i, j]) and b_loge[i, j] = np.log(2**b_log2[i, j]), and then using the aforementioned log_space_product() method to perform the matrix multiplication.

But I resisted myself doing that because the matrices I will eventually work with have > 1000 rows and > 20 columns. (Don't get confused with the numbers of rows and the number of columns here. I do ensure that the matrix multiplication properties are maintained.)

logarithm matrix numpy python
1 Answers

Remember that log2(x) = ln(x)/ln(2). We can rewrite that as ln(x) = c log2(x) with c = ln(2). So conversion to/from log spaces with a different base is literally one multiplication.

from scipy.special import logsumexp

def log_space_product(A, B, base=np.e):
    c = np.log(base)
    Astack = np.stack([A]*A.shape[0]).transpose(2,1,0)
    Bstack = np.stack([B]*B.shape[1]).transpose(1,0,2)
    return (1/c) * logsumexp(c*(Astack+Bstack), axis=0)
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