  An autoencoder is a neural network which data projects to and from a lower
dimensional latent space, where this data is easier to understand and model.
The autoencoder consists of two sub-networks, the encoder and the decoder,
which carry out these transformations. The neural network is trained such that
the output is as close to the input as possible, the data having gone through
an information bottleneck : the latent space. This tool bears significant
ressemblance to Principal Component Analysis (PCA), with two main differences.
Firstly, the autoencoder is a non-linear transformation, contrary to PCA, which
makes the autoencoder more flexible and powerful. Secondly, the axes found by a
PCA are orthogonal, and are ordered in terms of the amount of variability which
the data presents along these axes. This makes the interpretability of the PCA
much greater than that of the autoencoder, which does not have these
attributes. Ideally, then, we would like an autoencoder whose latent space
consists of independent components, ordered by decreasing importance to the
data. In this paper, we propose an algorithm to create such a network. We
create an iterative algorithm which progressively increases the size of the
latent space, learning a new dimension at each step. Secondly, we propose a
covariance loss term to add to the standard autoencoder loss function, as well
as a normalisation layer just before the latent space, which encourages the
latent space components to be statistically independent. We demonstrate the
results of this autoencoder on simple geometric shapes, and find that the
algorithm indeed finds a meaningful representation in the latent space. This
means that subsequent interpolation in the latent space has meaning with
respect to the geometric properties of the images.
