Subhomogeneous deep equilibrium models guarantee unique fixed points by utilizing specific activation functions and normalization layers. Theoretical findings support stable architectures for image classification and nonlinear graph propagation.
Implicit-depth neural networks have emerged as powerful tools in deep learning, defining feature embeddings through nonlinear equations. These models match or exceed traditional neural networks' performance on various tasks, including time series modeling. Despite their advantages, the existence and uniqueness of fixed points in deep equilibrium architectures remain open questions. Monotone operator theory has been a prominent line of analysis for uniqueness in deep equilibrium fixed points. In contrast, the author's work introduces a new analysis based on positive subhomogeneous operators, providing a theorem that guarantees unique fixed points for a broad class of operators. This result allows for the design of stable DEQ models with well-posed fixed-point equations.
The fundamental brick of the analysis is the notion of subhomogeneous operators, extending from homogeneous mappings to provide a more flexible framework for implicit networks. The proposed notion generalizes both homogeneity and strong subhomogeneity, offering stability and uniqueness in deep equilibrium architectures. By introducing activation functions that are subhomogeneous, the author ensures the existence and uniqueness of fixed points in neural networks.
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by Pietro Sitto... at arxiv.org 03-04-2024
https://arxiv.org/pdf/2403.00720.pdfDeeper Inquiries