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Latent Space Characterization of Autoencoder Variants

Anika Shrivastava, Renu Rameshan, Samar Agnihotri

TL;DR

The paper addresses how latent spaces differ across autoencoder variants by modeling latent tensors as points on a product manifold of SPSD matrices and embedding them into a Hilbert space via a distance-based kernel to compare structure under input perturbations. It finds that CAEs and DAEs exhibit stratified, non-smooth manifolds due to varying ranks, while VAEs form a smooth product manifold with fixed ranks, a conclusion supported by both manifold-space and Hilbert-space analyses and visualizations. These insights illuminate why VAEs favor smooth interpolation and generation, and suggest avenues for improved denoising and restoration by exploiting latent-space geometry. The work provides a principled framework for comparing latent spaces across architectures and offers guidance for designing robust representation learning methods.

Abstract

Understanding the latent spaces learned by deep learning models is crucial in exploring how they represent and generate complex data. Autoencoders (AEs) have played a key role in the area of representation learning, with numerous regularization techniques and training principles developed not only to enhance their ability to learn compact and robust representations, but also to reveal how different architectures influence the structure and smoothness of the lower-dimensional non-linear manifold. We strive to characterize the structure of the latent spaces learned by different autoencoders including convolutional autoencoders (CAEs), denoising autoencoders (DAEs), and variational autoencoders (VAEs) and how they change with the perturbations in the input. By characterizing the matrix manifolds corresponding to the latent spaces, we provide an explanation for the well-known observation that the latent spaces of CAE and DAE form non-smooth manifolds, while that of VAE forms a smooth manifold. We also map the points of the matrix manifold to a Hilbert space using distance preserving transforms and provide an alternate view in terms of the subspaces generated in the Hilbert space as a function of the distortion in the input. The results show that the latent manifolds of CAE and DAE are stratified with each stratum being a smooth product manifold, while the manifold of VAE is a smooth product manifold of two symmetric positive definite matrices and a symmetric positive semi-definite matrix.

Latent Space Characterization of Autoencoder Variants

TL;DR

The paper addresses how latent spaces differ across autoencoder variants by modeling latent tensors as points on a product manifold of SPSD matrices and embedding them into a Hilbert space via a distance-based kernel to compare structure under input perturbations. It finds that CAEs and DAEs exhibit stratified, non-smooth manifolds due to varying ranks, while VAEs form a smooth product manifold with fixed ranks, a conclusion supported by both manifold-space and Hilbert-space analyses and visualizations. These insights illuminate why VAEs favor smooth interpolation and generation, and suggest avenues for improved denoising and restoration by exploiting latent-space geometry. The work provides a principled framework for comparing latent spaces across architectures and offers guidance for designing robust representation learning methods.

Abstract

Understanding the latent spaces learned by deep learning models is crucial in exploring how they represent and generate complex data. Autoencoders (AEs) have played a key role in the area of representation learning, with numerous regularization techniques and training principles developed not only to enhance their ability to learn compact and robust representations, but also to reveal how different architectures influence the structure and smoothness of the lower-dimensional non-linear manifold. We strive to characterize the structure of the latent spaces learned by different autoencoders including convolutional autoencoders (CAEs), denoising autoencoders (DAEs), and variational autoencoders (VAEs) and how they change with the perturbations in the input. By characterizing the matrix manifolds corresponding to the latent spaces, we provide an explanation for the well-known observation that the latent spaces of CAE and DAE form non-smooth manifolds, while that of VAE forms a smooth manifold. We also map the points of the matrix manifold to a Hilbert space using distance preserving transforms and provide an alternate view in terms of the subspaces generated in the Hilbert space as a function of the distortion in the input. The results show that the latent manifolds of CAE and DAE are stratified with each stratum being a smooth product manifold, while the manifold of VAE is a smooth product manifold of two symmetric positive definite matrices and a symmetric positive semi-definite matrix.

Paper Structure

This paper contains 13 sections, 3 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Product manifold structure
  4. Model Architectures
  5. Latent tensors as points in a product manifold
  6. Transformation to Hilbert space
  7. Experimental Setup
  8. Results and analysis
  9. Latent space structure in manifold space
  10. Latent space structure in Hilbert space
  11. Visualizing the latent tensors
  12. Discussion
  13. CONCLUSION AND FUTURE WORK

Figures (6)

  • Figure 1: The SCTC block used in CAE and DAE models.
  • Figure 2: Pipeline of the proposed approach.
  • Figure 3: Histograms of ranks of $S^{(1)},\,S^{(2)},\,S^{(3)}$ for the three models on $300$ test samples. Left side is for clean and right for noisy with standard deviation $0.1$. From top to bottom: CAE, DAE, VAE.
  • Figure 4: The Hilbert space dimensionality and PSNR versus noise level for CAE, DAE, and VAE.
  • Figure 5: Principal angle variations for CAE, DAE, VAE
  • ...and 1 more figures