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Syntax ref: Numpy einsum

$a_m = m^{th} \text{ element of } \mathbf{a}$

$\mathbf{a}_m = \sum_m{\mathbf{a}}$

$\mathbf{a}_{(m)} =\text{ all of } \mathbf{a} \text{’s } m \text{ elements} $

One-dimensional array:

>>> a = np.array([1, 2, 3, 4])
>>> np.einsum("m->", a)
10
>>> np.einsum("m->m", a)
array([1, 2, 3, 4])

One-dimensional arrays:

>>> a = np.array([1, 2, 3])
>>> b = np.array([-4, 5, -6])
>>> # dot product
>>> a @ b
-12
>>> np.einsum("m,m->", a, b)
-12
>>> # Element-wise product
>>> a * b
array([-4, 10, -18])
>>> np.einsum("m,m->m", a, b)
array([-4, 10, -18])

Higher-dimension arrays:

  • Einsum form of $ A_{(i),m} B_{m,(j)} = im,mj -> ij $
>>> A = np.array([[1, 2], [-3, 4]])
>>> B = np.array([[0, 1], [1, 0]])
>>> A @ B
array([[ 2,  1],
       [ 4, -3]])
>>> np.einsum("im,mj->ij", A, B)
array([[ 2,  1],
       [ 4, -3]])

Equations ref

Classical ML Equations in LaTeX