Sparse Coding: Autoencoder Interpretation

(note that the third term, <math>\lVert A \rVert_2^2</math> is the sum of squares of the entries of A, or <math>\sum_r{\sum_c{A_{rc}^2}}</math>)

This objective function presents one last problem - the L1 norm is not differentiable at 0, and hence poses a problem for gradient-based methods. While the problem can be solved using other non-gradient descent-based methods, we will "smooth out" the L1 norm using an approximation which will allow us to use gradient descent. To "smooth out" the L1 norm, we use <math>\sqrt{x + \epsilon}</math> in place of <math>\left| x \right|</math>, where <math>\epsilon</math> is a "smoothing parameter" which can also be interpreted as a sort of "sparsity parameter" (to see this, observe that when <math>\epsilon</math> is large compared to <math>x</math>, the <math>x + \epsilon</math> is dominated by <math>\epsilon</math>, and taking the square root yields approximately <math>\sqrt{\epsilon}</math>). This "smoothing" will come in handy later when considering topographic sparse coding below.  

In theory, optimizing for this objective function using the iterative method as above should (eventually) yield features (the basis vectors of <math>A</math>) similar to those learned using the sparse autoencoder. However, in practice, there are quite a few tricks required for better convergence of the algorithm, and these tricks are described in greater detail in the later section on [[ Sparse Coding: Autoencoder Interpretation#Sparse coding in practice | sparse coding in practice]]. Deriving the gradients for the objective function may be slightly tricky as well, and using matrix calculus or [[Deriving gradients using backpropagation | using the backpropagation intuition]] can be helpful.

Concretely, suppose we (arbitrarily) organized our features into a square matrix. We would then like adjacent features in the matrix to similar. The way this is accomplished is to group these adjacent features together in the smoothed L1 penalty, so that instead of say <math>\sqrt{s_{1,1}^2 + \epsilon}</math>, we use say <math>\sqrt{s_{1,1}^2 + s_{1,2}^2 + s_{1,3}^2 + s_{2,1}^2 + s_{2,2}^2 + s_{3,2}^2 + s_{3,1}^2 + s_{3,2}^2 + s_{3,3}^2 + \epsilon}</math> instead, if we group in 3x3 regions. The grouping is usually overlapping, so that the 3x3 region starting at the 1st row and 1st column is one group, the 3x3 region starting at the 1st row and 2nd column is another group, and so on. Further, the grouping is also usually done wrapping around, as if the matrix were a torus, so that every feature is counted an equal number of times.

<li>For each feature in <math>s</math> (i.e. each column of <math>s</math>), divide the feature by the norm of the corresponding basis in <math>A</math>. That is, if <math>s_{r, c}</math> is the <math>r</math>th feature for the <math>c</math>th example, and <math>A_c</math> is the <math>c</math>th basis vector in <math>A</math>, then set <math>s_{r, c} \leftarrow \frac{ s_{r, c} } { \lVert A_c \rVert }.</math>