I'm not sure quite what you mean, but you can do the linear part of one layer all together, then the nonlinear part, etc.:

    for weight,bias in layers:
        a = sigmoid(np.dot(weight, a) + bias)

So my question is how large is the linear part? Granted, I was hoping it was big enough to bring these algorithms into consideration. Looks like I'm right in that they are larger than 2x2. However, they are still not 8kx8k. :(

They can be pretty big! I don't know what's normal these days, though.