Sparse Coding: Autoencoder Interpretation
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Observe that with our modified objective function, the objective function <math>J(A, s)</math> given <math>s</math>, that is <math>J(A; s) = \lVert As - x \rVert_2^2 + \gamma \lVert A \rVert_2^2</math> (the L1 term in <math>s</math> can be omitted since it is not a function of <math>A</math>) is simply a quadratic term in <math>A</math>, and hence has an easily derivable analytic solution in <math>A</math>. A quick way to derive this solution would be to use matrix calculus - some pages about matrix calculus can be found in the [[Useful Links | useful links]] section. Unfortunately, the objective function given <math>A</math> does not have a similarly nice analytic solution, so that minimization step will have to be carried out using gradient descent or similar optimization methods. | Observe that with our modified objective function, the objective function <math>J(A, s)</math> given <math>s</math>, that is <math>J(A; s) = \lVert As - x \rVert_2^2 + \gamma \lVert A \rVert_2^2</math> (the L1 term in <math>s</math> can be omitted since it is not a function of <math>A</math>) is simply a quadratic term in <math>A</math>, and hence has an easily derivable analytic solution in <math>A</math>. A quick way to derive this solution would be to use matrix calculus - some pages about matrix calculus can be found in the [[Useful Links | useful links]] section. Unfortunately, the objective function given <math>A</math> does not have a similarly nice analytic solution, so that minimization step will have to be carried out using gradient descent or similar optimization methods. | ||
- | 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. | + | 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 the backpropagation idea | using the backpropagation intuition]] can be helpful. |
== Topographic sparse coding == | == Topographic sparse coding == |