Computing / manipulating a numpy array

Question

Want to make this calculation as quickly as possible. I have X as an nxm numpy array. I want to define Y as follows:

Y_11 = 1 / (exp(X_11-X_11) + exp(X_11-X_12) + ... exp(X_11 - X_1N) ).

or for Y_00

1/np.sum(np.exp(X[0,0]-X[0,:]))

So basically Y is also nxm, where i, j is 1 / sum_j 'exp (X_ij - X_ij')

Any tips would be great! Thank.

Sample code as requested:

np.random.seed(111)
J,K = 110,120
X = np.random.rand(J,K)
Y = np.zeros((J,K))
for j in range(J):
    for k in range(K):
        Y[j,k] = 1/np.sum(np.exp(X[j,k]-X[j,:]))

# note each row will sum to 1 under this operation
np.sum(Y,axis=1)

Answer 1

Here are some fully vectorized approaches -

def vectorized_app1(X):
    return 1/np.exp(X[:,None] - X[...,None]).sum(1)

def vectorized_app2(X):
    exp_vals = np.exp(X)
    return 1/(exp_vals[:,None]/exp_vals[...,None]).sum(1)

def vectorized_app3(X):
    exp_vals = np.exp(X)
    return 1/np.einsum('ij,ik->ij',exp_vals,1/exp_vals)

Used Tricks and Lessons Learned

• Expand sizes with None/np.newaxisto enable broadcasting and do everything in vector form.

• np.exp - . , . , : exp(A-B) = exp(A)/exp(B). , np.exp(X) , .

• - np.einsum. , , .

-

In [111]: # Setup input, X
     ...: np.random.seed(111)
     ...: J,K = 110,120
     ...: X = np.random.rand(J,K)
     ...: 

In [112]: %timeit original_approach(X)
1 loop, best of 3: 322 ms per loop

In [113]: %timeit vectorized_app1(X)
10 loops, best of 3: 124 ms per loop

In [114]: %timeit vectorized_app2(X)
100 loops, best of 3: 14.6 ms per loop

In [115]: %timeit vectorized_app3(X)
100 loops, best of 3: 3.01 ms per loop

, einsum 100x+ speedup!

Answer 2

:

def foo2(X):
    Y = np.zeros_like(X)
    for k in range(X.shape[1]):
        Y[:,k]=1/np.exp(X[:,[k]]-X[:,:]).sum(axis=1)
    return Y

, k, , . X[:,[k]]-X[:,:] ( ).

:

Z = np.stack([X[:,[k]]-X for k in range(X.shape[1])],2)
Y = 1/np.exp(Z).sum(axis=1)

( )

Z = X[:,None,:]-X[:,:,None]