Rank Reduction Autoencoders -- Enhancing interpolation on nonlinear manifolds

The efficiency of classical Autoencoders (AEs) is limited in many practical situations. When the latent space is reduced through autoencoders, feature extraction becomes possible. However, overfitting is a common issue, leading to ``holes'' in AEs' interpolation capabilities. On the other hand, increasing the latent dimension results in a better approximation with fewer non-linearly coupled features (e.g., Koopman theory or kPCA), but it doesn't necessarily lead to dimensionality reduction, which makes feature extraction problematic. As a result, interpolating using Autoencoders gets harder. In this work, we introduce the Rank Reduction Autoencoder (RRAE), an autoencoder with an enlarged latent space, which is constrained to have a small pre-specified number of dominant singular values (i.e., low-rank). The latent space of RRAEs is large enough to enable accurate predictions while enabling feature extraction. As a result, the proposed autoencoder features a minimal rank linear latent space. To achieve what's proposed, two formulations are presented, a strong and a weak one, that build a reduced basis accurately representing the latent space. The first formulation consists of a truncated SVD in the latent space, while the second one adds a penalty term to the loss function. We show the efficiency of our formulations by using them for interpolation tasks and comparing the results to other autoencoders on both synthetic data and MNIST.