I need help with part c) of the question please. A 1D convolution layer within a

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I need help with part c) of the question please.

A 1D convolution layer within a neural network is given as: Z_i = sigma _j = 0^k -1 W_j X_i + j for ii sin [1, N], Where K is the filter (kernel) size. Suppose the error function (or the objective function for minimization) is denoted by L and partial differential L/partial differentialZ, is known. What is partial differential L/partial differential W ? Show that is actually the result of a convolution. Consider a convolution network with an error function denoted by L has pooling units that take n values x_i(ii sin [1, n]) and compute a scalar output whose value is invariant to permutations of the inputs. The L_p-pooliug module takes positive inputs and computes y = (sigma_i = 1^n x_i^p)^1/p Assuming we know partial differential L/partial differential y, what is partial differential L/partial differential x_i? (p is simply a fixed parameter.) The log-pooling module computes y = 1/beta log(1/n sigma_i = 1^n e^betax_i). Assuming we know partial differential L/partial differential y, what is partial differential L/partial differential x_i? (beta is simply a fixed parameter.) Consider N neural nets G{X_k, W_k) (k isin [1, N]), where the X_k axe a bunch of input vectors, and the W_k a bunch of parameter vectors. The outputs of the networks are combined and fed to an error function L. As it turns out, we want all the networks to share the same weight vector W_k = W (forallk). Assuming that one can compute the individual partial differential L/partial differential W_k, what is partial differential L/partial differential W?

Explanation / Answer

c) In the given N neural networks G(Xk, Wk), with k from 1 to N, where Xk are a bunch of input vectors, and the Wk a bunch of parameter vectors,

dl/dW is the vector of all the vectors dl/dWk for all k from 1 to N.

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