Analytical Solution of a Three-layer Network with a Matrix Exponential Activation Function
This provides a theoretical insight into deep learning for researchers, though it is incremental as it focuses on a specific case.
The paper tackles the theoretical understanding of why deeper networks are more powerful by finding an analytical solution for a three-layer network with a matrix exponential activation function, showing it can solve two equations compared to one for a single-layer network.
In practice, deeper networks tend to be more powerful than shallow ones, but this has not been understood theoretically. In this paper, we find the analytical solution of a three-layer network with a matrix exponential activation function, i.e., $$ f(X)=W_3\exp(W_2\exp(W_1X)), X\in \mathbb{C}^{d\times d} $$ have analytical solutions for the equations $$ Y_1=f(X_1),Y_2=f(X_2) $$ for $X_1,X_2,Y_1,Y_2$ with only invertible assumptions. Our proof shows the power of depth and the use of a non-linear activation function, since one layer network can only solve one equation,i.e.,$Y=WX$.