  A new method for the unsupervised learning of sparse representations using
autoencoders is proposed and implemented by ordering the output of the hidden
units by their activation value and progressively reconstructing the input in
this order. This can be done efficiently in parallel with the use of cumulative
sums and sorting only slightly increasing the computational costs. Minimizing
the difference of this progressive reconstruction with respect to the input can
be seen as minimizing the number of active output units required for the
reconstruction of the input. The model thus learns to reconstruct optimally
using the least number of active output units. This leads to high sparsity
without the need for extra hyperparameters, the amount of sparsity is instead
implicitly learned by minimizing this progressive reconstruction error. Results
of the trained model are given for patches of the CIFAR10 dataset, showing
rapid convergence of features and extremely sparse output activations while
maintaining a minimal reconstruction error and showing extreme robustness to
overfitting. Additionally the reconstruction as function of number of active
units is presented which shows the autoencoder learns a rank order code over
the input where the highest ranked units correspond to the highest decrease in
reconstruction error.
