Variational Rank Reduction Autoencoder

Deterministic Rank Reduction Autoencoders (RRAEs) enforce by construction a regularization on the latent space by applying a truncated SVD. While this regularization makes Autoencoders more powerful, using them for generative purposes is counter-intuitive due to their deterministic nature. On the other hand, Variational Autoencoders (VAEs) are well known for their generative abilities by learning a probabilistic latent space. In this paper, we present Variational Rank Reduction Autoencoders (VRRAEs), a model that leverages the advantages of both RRAEs and VAEs. Our claims and results show that when carefully sampling the latent space of RRAEs and further regularizing with the Kullback-Leibler (KL) divergence (similarly to VAEs), VRRAEs outperform RRAEs and VAEs. Additionally, we show that the regularization induced by the SVD not only makes VRRAEs better generators than VAEs, but also reduces the possibility of posterior collapse. Our results include a synthetic dataset of a small size that showcases the robustness of VRRAEs against collapse, and three real-world datasets; the MNIST, CelebA, and CIFAR-10, over which VRRAEs are shown to outperform both VAEs and RRAEs on many random generation and interpolation tasks based on the FID score.

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.