In last few layers of a neural network, what is the benefit of connecting several affine layers instead of just one? It is a question regarding some fundamental structure of the neural network. I now take the famous LENET network for MNIST data as an example. In LENET, the last few layers are the affine layers. In the below image, they are C5, F6 and Output respectively, which are 400×120, 120×84 and 84×10 layers.
In this sense, at the output of these layers we are using the vectors of 120 dimensions, 84 dimensions and 10 dimensions to try to represent each MNIST image (or to match the features).
My question is, why can't we just use the two layers of 400×84 and 84×10 to replace the above? Or at an extreme, can we even just use a single layer of 400×10 to replace the above all three layers?
In the view of parameters, the said two/single layers are lighter than three layers (and are easier to train). For image processing sense, the previous CNN/pooling layers have already extracted the 400-dimensional features from each MNIST image.
So, what is the benefit of using the above three layers instead of the said two/single layers?
Or to be more general, what is the benefit of connecting several separated affine layers instead of combining the layers?

Many thanks!
 A: Those aren't affine layers. A fully connected layer like the ones shown in above diagram typically computes something like $y = a(Wx+b)$ where $x$ is the input, $y$ is the output, and $W$ and $b$ are learnable parameters. $a$ is some nonlinear activation function.
The benefit of multiple fully connected layers as opposed to just one is  similar to that of any change which increases the model complexity -- increased expressive power to model more complicated nonlinear functions at the cost of increased computation and variance.
A: With the architecture in your diagram you have 40,320,000 connections for layers 400×120, 120×84 and 84×10. In addition you have two nonlinear transformations between the layers.
If you replace it with 400x10 layer, then you will remove the nonlinearity altogether in this section of NN and have only 4,000 connection.
If you empirically show that this new architecture works just as well or better, then go ahead and change the architecture. It's a massive reduction in complexity. There's no intuitive reasoning to justify this replacement. Why should it be better or the same in terms of performance?
