# Bias in matrix form of convolutional neural network

With no bias nodes, the matrix form of the input data and the lowest-level parameters of following convolutional neural net

is the following:

$$\left[\begin{array}{cccc} x_{11} & x_{12} & x_{14} & x_{14} \\ x_{21} & x_{22} & x_{24} & x_{24} \\ \vdots\\ x_{n1} & x_{n2} & x_{n4} & x_{n4} \\ \end{array}\right]\left[\begin{array}{cccc} w_1 & 0 & w_3 & 0\\ w_2 & 0 & w_4 & 0\\ 0 & w_1 & 0 & w_3\\ 0 & w_2 & 0 & w_4\\ \end{array}\right]$$

If I were to add bias terms, should it look like this:

$$\left[\begin{array}{ccccc} 1 & x_{11} & x_{12} & x_{14} & x_{14} \\ 1 & x_{21} & x_{22} & x_{24} & x_{24} \\ &\vdots\\ 1 & x_{n1} & x_{n2} & x_{n4} & x_{n4} \\ \end{array}\right]\left[\begin{array}{cccc} b_1 &b_2 &b_3 &b_4 \\ w_1 & 0 & w_3 & 0\\ w_2 & 0 & w_4 & 0\\ 0 & w_1 & 0 & w_3\\ 0 & w_2 & 0 & w_4\\ \end{array}\right]$$

or this

$$\left[\begin{array}{ccccc} 1 & x_{11} & x_{12} & x_{14} & x_{14} \\ 1 & x_{21} & x_{22} & x_{24} & x_{24} \\ &\vdots\\ 1 & x_{n1} & x_{n2} & x_{n4} & x_{n4} \\ \end{array}\right]\left[\begin{array}{cccc} b_1 &b_1 &b_2 &b_2 \\ w_1 & 0 & w_3 & 0\\ w_2 & 0 & w_4 & 0\\ 0 & w_1 & 0 & w_3\\ 0 & w_2 & 0 & w_4\\ \end{array}\right]$$

And why?

The first version has unique bias parameters for each time a linear function is applied to a region of the input data, while the second has a unique bias for each linear function.