A line in first paragraph says "Defining P(y=1|x)=sigmoid(z) is equivalent to defining P(y=1|x) = softmax(z)1 with two-dimensional z and z1=0."

Could anybody help me understand this line?

After Equation 2.67, the new paragraph says that since we're using the same matrix D to decode all the points, we can no longer consider the points in isolation. Then the author uses the Frobenius norm of the matrix instead of the L2 norm used before. So, why is that?

Also, what are the limits of summation in the norm and how are we defining the transformation on a scalar, ie r(x⁽ⁱ⁾)_j

I have a question about the sigmoid function as the question.

I wish some help on this problem.