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      The last equation can be utilized from both directions. Now we can write down

(3.89)

(3.90)

where I is an identity matrix of appropriate size. In Eq. (3.89), the size of I is determined by the number of columns in F, and hence I ∈ ℝ^{D × D}. In Eq. (3.90), For the derivation of the gradient computation rules in a convolution layer, the notation summarized in Table 3.3 will be used.

      Update the parameters – backward propagation: First, we need to compute ∂z/∂vec(x^l) and z/∂vec(F), where the first term will be used for backward propagation to the previous (l − 1)th layer, and the second term will determine how the parameters of the current (l−th) layer will be updated. Keep in mind that f, F, and wⁱ refer to the same thing (modulo reshaping of the vector or matrix or tensor). Similarly, we can reshape y into a matrix ; then y, Y, and x^{l + 1} refer to the same object (again, modulo reshaping).

Table 3.3 Variables, for the derivation of gradient with ϕ ↔ φ_.

	Alias	Size and Meaning
X	x l	Hl Wl × Dl, the input tensor
F	f , w l	HW Dl × D, D kernels, each H × W and Dl channels
Y	y , x l+1	Hl + 1 Wl + 1 × Dl + 1, the output, Dl + 1 = D
ϕ( x l)		Hl + 1 Wl + 1 × HW Dl, the im2row expansion of x l
M		Hl + 1 Wl + 1 HW Dl × Hl Wl Dl, the indictor matrix for ϕ( x l)
		Hl + 1 Wl + 1 × Dl + 1, gradient for y
		HW Dl × D, gradient to update the convolution kernels
		Hl Wl × Dl, gradient for x l, useful for back propagation

From the chain rule, it is easy to compute ∂z/∂vec(F) as

(3.91)

The first term on the right in Eq. (3.91) is already computed in the (l + 1)‐th layer as ∂z/∂(vec(x^{l + 1}))^T. Based on Eq. (3.89), we have

      (3.92)Скачать книгу