[Paper Review] Provable Finite Data Generalization with Group Autoencoder.

Romain Cosentino, Randall Balestriero+2 more|arXiv (Cornell University)|Sep 20, 2020

Generative Adversarial Networks and Image Synthesis49 references4 citations

TL;DR

This paper proposes a novel regularization for deep autoencoders (AEs) based on Lie group structure to ensure provable generalization in the finite training data regime. By analyzing AEs through a spline framework and enforcing group constraints, the method achieves improved generalization and reconstruction guarantees, validated empirically across multiple datasets.

ABSTRACT

Deep Autoencoders (AEs) provide a versatile framework to learn a compressed, interpretable, or structured representation of data. As such, AEs have been used extensively for denoising, compression, data completion as well as pre-training of Deep Networks (DNs) for various tasks such as classification. By providing a careful analysis of current AEs from a spline perspective, we can interpret the input-output mapping, in turn allowing us to derive conditions for generalization and reconstruction guarantee. By assuming a Lie group structure on the data at hand, we are able to derive a novel regularization of AEs, allowing for the first time to ensure the generalization of AEs in the finite training set case. We validate our theoretical analysis by demonstrating how this regularization significantly increases the generalization of the AE on various datasets.

Motivation & Objective

• To address the lack of generalization guarantees in deep autoencoders when trained on finite datasets.
• To establish theoretical conditions for generalization and reconstruction in autoencoders using a spline-based analysis.
• To develop a novel regularization technique grounded in Lie group structure to enforce structural inductive bias on data manifolds.
• To validate the theoretical framework through empirical evaluation on diverse datasets, demonstrating improved generalization performance.

Proposed method

• Analyzes the input-output mapping of autoencoders through a spline interpolation perspective to derive generalization conditions.
• Assumes the data manifold has a Lie group structure, enabling the formulation of group-consistent constraints on the latent space.
• Introduces a novel regularization term that enforces group invariance and consistency in the autoencoder's encoder and decoder mappings.
• Derives theoretical bounds on reconstruction error and generalization error under the group structure assumption.
• Applies the regularized autoencoder to learn structured, compressed representations that generalize well to unseen finite data.
• Validates the method on multiple datasets, comparing generalization performance against standard AE baselines.

Experimental results

Research questions

• RQ1 Can we derive theoretical conditions for generalization and reconstruction in autoencoders trained on finite datasets?
• RQ2 How can Lie group structure on data be leveraged to improve generalization in autoencoders?
• RQ3 What regularization mechanism ensures that autoencoder representations generalize beyond the training set?
• RQ4 To what extent does group-aware regularization improve generalization compared to standard autoencoders?
• RQ5 Can the spline-based analysis framework be used to derive provable generalization guarantees for autoencoders?

Key findings

• The proposed group-aware regularization enables the first provable generalization guarantee for autoencoders in the finite training set scenario.
• Theoretical analysis using a spline framework provides conditions under which autoencoders generalize and reconstruct accurately.
• Empirical results show significant improvements in generalization performance across multiple datasets compared to standard autoencoders.
• The method maintains strong reconstruction quality while enhancing generalization, demonstrating the effectiveness of group structure regularization.
• The regularization is effective even with limited training data, highlighting its utility in low-data regimes.
• Theoretical and empirical results confirm that enforcing Lie group consistency in the latent space leads to better generalization.