Difference between np.dot and np.multiply with np.sum in binary cross-entropy loss calculation

Question

I have tried the following code but didn't find the difference between np.dot and np.multiply with np.sum

Here is np.dot code

logprobs = np.dot(Y, (np.log(A2)).T) + np.dot((1.0-Y),(np.log(1 - A2)).T)
print(logprobs.shape)
print(logprobs)
cost = (-1/m) * logprobs
print(cost.shape)
print(type(cost))
print(cost)

Its output is

(1, 1)
[[-2.07917628]]
(1, 1)
<class 'numpy.ndarray'>
[[ 0.693058761039 ]]

Here is the code for np.multiply with np.sum

logprobs = np.sum(np.multiply(np.log(A2), Y) + np.multiply((1 - Y), np.log(1 - A2)))
print(logprobs.shape)         
print(logprobs)
cost = - logprobs / m
print(cost.shape)
print(type(cost))
print(cost)

Its output is

()
-2.07917628312
()
<class 'numpy.float64'>
0.693058761039

I'm unable to understand the type and shape difference whereas the result value is same in both cases

Even in the case of squeezing former code cost value become same as later but type remains same

cost = np.squeeze(cost)
print(type(cost))
print(cost)

output is

<class 'numpy.ndarray'>
0.6930587610394646

Solution 1

What you're doing is calculating the binary cross-entropy loss which measures how bad the predictions (here: A2) of the model are when compared to the true outputs (here: Y).

Here is a reproducible example for your case, which should explain why you get a scalar in the second case using np.sum

In [88]: Y = np.array([[1, 0, 1, 1, 0, 1, 0, 0]])

In [89]: A2 = np.array([[0.8, 0.2, 0.95, 0.92, 0.01, 0.93, 0.1, 0.02]])

In [90]: logprobs = np.dot(Y, (np.log(A2)).T) + np.dot((1.0-Y),(np.log(1 - A2)).T)

# `np.dot` returns 2D array since its arguments are 2D arrays
In [91]: logprobs
Out[91]: array([[-0.78914626]])

In [92]: cost = (-1/m) * logprobs

In [93]: cost
Out[93]: array([[ 0.09864328]])

In [94]: logprobs = np.sum(np.multiply(np.log(A2), Y) + np.multiply((1 - Y), np.log(1 - A2)))

# np.sum returns scalar since it sums everything in the 2D array
In [95]: logprobs
Out[95]: -0.78914625761870361

Note that the np.dot sums along only the inner dimensions which match here (1x8) and (8x1). So, the 8s will be gone during the dot product or matrix multiplication yielding the result as (1x1) which is just a scalar but returned as 2D array of shape (1,1).

---

Also, most importantly note that here np.dot is exactly same as doing np.matmul since the inputs are 2D arrays (i.e. matrices)

In [107]: logprobs = np.matmul(Y, (np.log(A2)).T) + np.matmul((1.0-Y),(np.log(1 - A2)).T)

In [108]: logprobs
Out[108]: array([[-0.78914626]])

In [109]: logprobs.shape
Out[109]: (1, 1)

---

Return result as a scalar value

np.dot or np.matmul returns whatever the resulting array shape would be, based on input arrays. Even with out= argument it's not possible to return a scalar, if the inputs are 2D arrays. However, we can use np.asscalar() on the result to convert it to a scalar if the result array is of shape (1,1) (or more generally a scalar value wrapped in an nD array)

In [123]: np.asscalar(logprobs)
Out[123]: -0.7891462576187036

In [124]: type(np.asscalar(logprobs))
Out[124]: float

---

ndarray of size 1 to scalar value

In [127]: np.asscalar(np.array([[[23.2]]]))
Out[127]: 23.2

In [128]: np.asscalar(np.array([[[[23.2]]]]))
Out[128]: 23.2

Solution 2

np.dot is the dot product of two matrices.

|A B| . |E F| = |A*E+B*G A*F+B*H|
|C D|   |G H|   |C*E+D*G C*F+D*H|

Whereas np.multiply does an element-wise multiplication of two matrices.

|A B| ⊙ |E F| = |A*E B*F|
|C D|   |G H|   |C*G D*H|

When used with np.sum, the result being equal is merely a coincidence.

>>> np.dot([[1,2], [3,4]], [[1,2], [2,3]])
array([[ 5,  8],
       [11, 18]])
>>> np.multiply([[1,2], [3,4]], [[1,2], [2,3]])
array([[ 1,  4],
       [ 6, 12]])

>>> np.sum(np.dot([[1,2], [3,4]], [[1,2], [2,3]]))
42
>>> np.sum(np.multiply([[1,2], [3,4]], [[1,2], [2,3]]))
23

Solution 3

If Y and A2 are (1,N) arrays, then np.dot(Y,A.T) will produce a (1,1) result. It is doing a matrix multiplication of a (1,N) with a (N,1). The N's are summed, leaving the (1,1).

With multiply the result is (1,N). Sum all values, and the result is a scalar.

If Y and A2 were (N,) shaped (same number of elements, but 1d), the np.dot(Y,A2) (no .T) would also produce a scalar. From np.dot documentation:

For 2-D arrays it is equivalent to matrix multiplication, and for 1-D arrays to inner product of vectors

Returns the dot product of a and b. If a and b are both scalars or both 1-D arrays then a scalar is returned; otherwise an array is returned.

squeeze reduces all size 1 dimensions, but still returns an array. In numpy an array can have any number of dimensions (from 0 to 32). So a 0d array is possible. Compare the shape of np.array(3), np.array([3]) and np.array([[3]]).