Randomly initialized autoencoders: fixed points and edge-of-chaos
Autoencoders are analyzed through the lens of fixed points, with performance characterized by existence, stability, and basins of attraction of these fixed points The paper introduces a modified notion of Edge-of-Chaos (EoC) specifically for autoencoders, distinguishing between local EoC (controlling small perturbations) and global EoC (controlling arbitrary perturbations) Local EoC analysis relies on spectral techniques from Random Matrix Theory (RMT), while global EoC employs the Sudakov-Ferni
Analysis
TL;DR
- Autoencoders are analyzed through the lens of fixed points, with performance characterized by existence, stability, and basins of attraction of these fixed points
- The paper introduces a modified notion of Edge-of-Chaos (EoC) specifically for autoencoders, distinguishing between local EoC (controlling small perturbations) and global EoC (controlling arbitrary perturbations)
- Local EoC analysis relies on spectral techniques from Random Matrix Theory (RMT), while global EoC employs the Sudakov-Fernique inequality for Gaussian processes
- Initialization at or near the edge-of-chaos regime provides theoretical and practical advantages, including robustness to input perturbations
- The work extends prior mean-field averaging methods for EoC to the specific architectural context of autoencoders
Why It Matters
This research provides a rigorous theoretical foundation for understanding autoencoder stability, which is critical for practitioners building deep representation learning systems. By formalizing local and global edge-of-chaos for autoencoders, the paper offers actionable guidance on initialization strategies that can improve robustness and generalization. The connection to Random Matrix Theory also opens new analytical tools for studying nonlinear neural network behavior.
Technical Details
- Autoencoders are treated as a special class of deep neural networks where performance is characterized via fixed points; the study addresses existence, stability, and basins of attraction through contractive properties
- Two distinct EoC notions are introduced: local EoC controls sensitivity to small (local) input perturbations using spectral techniques from Random Matrix Theory, while global EoC handles arbitrary (global) perturbations via the Sudakov-Fernique inequality for Gaussian processes
- The analysis situates autoencoder stability within nonlinear problems in Random Matrix Theory, with MSC classifications 60B20 and 15B52 indicating the probabilistic and matrix-theoretic foundations
- The work builds on and modifies prior EoC frameworks that were developed for broad classes of neural networks using mean-field averaging methods, adapting them to the encoder-decoder architecture of autoencoders
Industry Insight
- Practitioners designing autoencoder-based systems should consider edge-of-chaos initialization principles to enhance model robustness against input noise and adversarial perturbations
- The distinction between local and global EoC suggests that different initialization regimes may be optimal depending on whether the application prioritizes fine-grained stability or resilience to large input variations
- The theoretical framework presented here could inform the development of new initialization schemes and stability guarantees for deep generative models, particularly in safety-critical applications where fixed-point behavior is paramount
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