Consider the following neural...

Consider the following neural network. W11 W12 W21 X W22 Layer 1 W31 W32 Layer 2 W12 W22 y Assume there are no activation functions and so each neuron functions as a linear combination of the inputs. The output of layer 1 is $a_1$ and that of layer 2 is $a_2$. The final output is $y$. Come up with a transformation matrix $W$ for the vector $x$ that produces the variable $y$ such that: $Wx = y$

Consider a three-layer fully-connected network with n1, n2, and n3 neurons in the three layers respectively. Inputs are fed into the first layer. The loss is the mean squared error, and the non-linearity is a sigmoid function. Let the label vector be t of size n3. Let each layer output vector be y and input vector be x, both of size ni. Let the weight matrix between layer i and layer i + 1 be W(i+1). The j-th element in yi is defined by yj = o(c(i-1)W(i-1),j-2,3), the same for the weight connecting the k-th and l-th neuron in layers i and i + 1, which is defined by W(i+1)kl. You do not need to consider bias in this problem. Input #1 Output #1 Input #2 Output #2 Input #3 Output #3 Input #4 Output #4 Figure 2: A three-layer fully-connected MLP network: Here is a summary of the notation: o(c) denotes the activation function for L2 and L3. There is no activation applied to the input layer. o(c(i-1)W(i-1),j-2,3) = o(c(i-1)W(i-1),j-2,3).

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