Deep learning / matrix product error backpropagation

1.First of all

The error back propagation of matrix multiplication was difficult to understand, so I will summarize it.

2. Scalar product error backpropagation

Reviewing the error backpropagation of the scalar product,  Assuming that the object to be gradient is L and $ \ frac {\ partial L} {\ partial y} $ is known in advance, from the chain rule This is no problem, isn't it?

3. Matrix product error back propagation

However, when it comes to matrix multiplication, it changes with intuition.

Somehow, it doesn't come with a pin. So, I will confirm it concretely. The setting is considered to be connected to neuron Y via the inner product of two neurons X and four weights W. ** 1) First, find $ \ frac {\ partial L} {\ partial X} $. ** First, calculate these in advance. While using this calculation on the way

** 2) Next, find $ \ frac {\ partial L} {\ partial y} $. ** First, calculate these in advance. While using this calculation on the way

4. Matrix product forward and back propagation code

If x1 = X, x2 = Y, grad = $ \ frac {\ partial L} {\ partial y} $,

class MatMul(object):
    def __init__(self, x1, x2):
        self.x1 = x1
        self.x2 = x2

    def forward(self):
        y = np.dot(self.x1, self.x2)
        self.y = y
        return y

    def backward(self, grad):
        grad_x1 = np.dot(grad, self.x2.T)
        grad_x2 = np.dot(self.x1.T, grad)
        return (grad_x1, grad_x2)