Functional-Analytic Foundations for Implicit Kernels in Deep Generalization

Deep learning models exhibit remarkable generalization capabilities, often despite extreme overparameterization. A growing body of theoretical work attempts to explain this phenomenon by connecting deep networks to kernel methods, particularly through the concept of "implicit kernels." These kernels are not explicitly designed but emerge from the architecture and optimization dynamics of deep neural networks. While the Neural Tangent Kernel (NTK) provides a prominent example, a more general and rigorous functional-analytic framework is needed to fully characterize the properties of such implicit kernels and their role in deep generalization. This paper establishes foundational concepts from functional analysis, including Hilbert spaces, Reproducing Kernel Hilbert Spaces (RKHS), and operator theory, to provide a comprehensive mathematical lens for understanding implicit kernels. We investigate how the architectural choices and training processes of deep neural networks implicitly define these kernel functions, analyzing their spectral properties and regularity. Furthermore, we explore how these functional-analytic properties directly relate to generalization bounds, implicit regularization, and the expressive power of deep models. The proposed framework aims to bridge the gap between empirical success and theoretical understanding, offering new avenues for designing more robust and interpretable deep learning systems.