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Bernal Jimenez edited section_Descending_the_Alternate_Sparse__.tex
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\section{Descending the Alternate Sparse Coding Objective}\label{descend}
The original SAILnet paper \cite{Zylberberg_2011} maintains the reconstruction part of the sparse coding objective but
instead of L1 regularization, replaces the sparse prior
was turned into with a set of two constraints: homeostasis for the individual neuron's firing rates and decorrelation for the pairwise statistics:
\begin{equation}\label{eq:1}
E(X, a; \Phi, W, \theta) = \frac{1}{2}\sum_i(X_i-\sum_j\Phi_{ij}a_j)^2 + \sum_i\theta_i(a_i-p) + \sum_{ij}W_{ij}(a_ia_j-p^2)
\end{equation}