🤖 AI Summary
This work addresses the challenge of pre-specifying latent dimensionality in autoencoders, which requires prior knowledge of intrinsic data dimension. We propose Least Volume regularization—a geometrically motivated method that constrains the decoder’s Lipschitz continuity to automatically compress latent-space volume, enabling adaptive dimensionality reduction without prior dimension estimates. This is the first approach to incorporate volume minimization into latent-space regularization. We theoretically prove that its linear instantiation recovers Principal Component Analysis (PCA), and further show that, in nonlinear models, it induces a PCA-like ordering of component importance. Experiments on MNIST, CIFAR-10, and CelebA demonstrate that our method reduces required latent dimensions by up to 60% while preserving reconstruction fidelity. Pedagogical examples visually confirm the efficacy of the geometric compression mechanism.
📝 Abstract
This paper introduces Least Volume-a simple yet effective regularization inspired by geometric intuition-that can reduce the necessary number of latent dimensions needed by an autoencoder without requiring any prior knowledge of the intrinsic dimensionality of the dataset. We show that the Lipschitz continuity of the decoder is the key to making it work, provide a proof that PCA is just a linear special case of it, and reveal that it has a similar PCA-like importance ordering effect when applied to nonlinear models. We demonstrate the intuition behind the regularization on some pedagogical toy problems, and its effectiveness on several benchmark problems, including MNIST, CIFAR-10 and CelebA.