Latent Space Characterization of Autoencoder Variants (2412.04755v2)

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.

• The paper characterizes latent spaces of CAEs, DAEs, and VAEs by mapping them to symmetric positive semidefinite matrix manifolds to analyze smoothness and stratification.
• The research reveals that CAEs and DAEs have stratified latent manifolds with variable ranks under noise, unlike VAEs which maintain smooth, singular manifolds with stable ranks.
• Understanding these structural differences provides insight for designing autoencoder systems, suggesting VAEs are better suited for applications requiring smooth latent space transitions.

Analyzing Latent Space Characterization of Autoencoder Variants

The paper "Latent Space Characterization of Autoencoder Variants," by Anika Shrivastava, Renu Rameshan, and Samar Agnihotri, explores the latent space structures of various autoencoder models, including Convolutional Autoencoders (CAEs), Denoising Autoencoders (DAEs), and Variational Autoencoders (VAEs). This research provides an in-depth exploration of how these latent spaces are affected by input perturbations and investigates the smoothness of the manifold structures underlying these models.

The authors focus on characterizing the matrix manifolds that arise within the latent spaces of these autoencoder models. They discern a dichotomy between the non-smooth manifold structures found in CAEs and DAEs and the smooth manifold structures of VAEs. This is achieved by aligning the encoded latent tensors with product manifolds in the space of symmetric positive semidefinite (SPSD) matrices, leveraging their ranks to describe the multifaceted nature of these latent manifolds.

Methodology and Contributions

The paper employs a novel approach by mapping the encoded latent spaces onto product manifolds constructed from SPSD matrices. This mapping allows the authors to calculate and analyze manifold ranks, which provides insight into the stratification and smoothness of the latent spaces. A central contribution of this work is the demonstration that CAE and DAE structures result in stratified manifold spaces—each stratum comprising a smooth manifold—whereas the VAE occupies a singular smooth product manifold. Empirical evaluations utilized additive Gaussian noise to examine how these manifold structures react to varying input perturbations. The research reveals that CAEs and DAEs exhibit rank variability leading to multiple strata, while VAEs maintain a stable rank profile across noise levels, ensuring smooth and coherent transitions within their latent spaces.

The assessment of PSNR (Peak Signal-to-Noise Ratio) across models provides additional perspective on how reconstruction fidelity correlates with latent space configurations. The authors also utilize transformations into Hilbert space to preserve distance relationships and analyze changes in dimensionality under noise perturbations, giving a clear visualization of the differences in response among the latent spaces in terms of smoothness and transition continuity.

Numerical Results and Implications

Quantitative results from this paper indicate that VAEs sustain consistent ranks across all noise scenarios, facilitating a smooth traverse through latent spaces. Conversely, CAEs and DAEs display variability in matrix ranks, a characteristic of their stratified manifolds, contributing to dimensionality fluctuations in their corresponding latent spaces as confirmed by Hilbert space dimension changes and principal angle analyses.

These insights bear significant implications for autoencoder-based learning systems, particularly in applications that require robust and smooth data representation such as generative modeling or anomaly detection. Understanding the structural underpinnings of latent spaces can lead to more effective model design and improved performance in domains that demand latent space manipulation.

Conclusion and Future Directions

This research makes significant strides in elucidating the geometric and topological properties of latent spaces in different autoencoder architectures. It successfully characterizes and contrasts the nature of latent manifolds in CAEs, DAEs, and VAEs, highlighting the advantages of smooth latent space transitions in VAEs over the stratified structures in the others.

Future developments in AI and machine learning could build on these findings by incorporating manifold-based insights into the design of new models, leveraging the smoothness of VAEs for more robust feature and representation learning tasks. Moreover, this work opens avenues for further research in expanding the application of manifold theory in neural network interpretability and data representation strategies.