(c) (12 pts) In this section, we are going to work with 1-dimensional convolu- tions. Discrete convolution of 1-dimensional input x[n] and kernel k[n] is defined as follows: s[n] = (x * k)[n] = _x[n - m]k[m] However, in machine learning convolution is usually implemented as cross- correlation, which is defined as follows: s[n] = (x * k)[n] = _x[n + mlk[m] Note the difference in signs, which will get the network to learn an "flipped" kernel. In general it doesn't change much, but it's important to keep it in mind. In convolutional neural networks, the kernel k[n] is usually 0 everywhere, except a few values near 0: Vini>Mk[n] = 0. Then, the formula becomes: s[n] = (x* k)[n] = _ x[n+m]k[m] m=-M Let's consider an input x[n] E R', with 1 s n s 11, e.g. it is a length 11 sequence with 7 channels. We consider the convolutional layer fw with one filter, with kernel size 3, stride of 2, no dilation, and no padding. The only parameters of the convolutional layer is the weight W, WE Rlx7x3, there's no bias and no non-linearity. (i) (1 pts) What is the dimension of the output fw(x)? Provide an expres- sion for the value of elements of the convolutional layer output fw(x). Example answer format here and in the following sub-problems: fw(x) e R42x42x42, fw(x)[i, j,k] = 42. (ii) (4 pts) What is the dimension of 0/W12? Provide an expression for the values of the derivative ofw(x) ow (iii) (4 pts) What is the dimension of 0/W62? Provide an expression for the values of the derivative ofw(x) ox 2 (iv) (5 pts) Now, suppose you are given the gradient of the loss ( w.r.t. the output of the convolutional layer fw(x), i.e. afw(x). What is the dimension of aw? Provide an expression for aw. Explain similarities and differences of this expression and expression in (i)