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Understanding the Efficacy of Over-Parameterization in Neural Networks

Understanding the Efficacy of Over-Parameterization in Neural Networks Understanding the Efficacy of Over-Parameterization in Neural Networks: Mechanisms, Theories, and Practical Implications Introduction Deep neural networks (DNNs) have become the cornerstone of modern artificial intelligence, driving advancements in computer vision, natural language processing, and myriad other domains. A key, albeit counter-intuitive, property of contemporary DNNs is their immense over-parameterization: these models often contain orders of magnitude more parameters than the number of training examples, yet they generalize remarkably well to unseen data. This phenomenon stands in stark contrast to classical statistical learning theory, which posits that models with excessive complexity relative to the available data are prone to overfitting and poor generalization. Intriguingly, empirical evidence shows that increasing the number of parameters in DNNs can lead ...

Advanced Perspectives in Mathematical Physics and Quantum Theory

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Advanced Perspectives in Mathematical Physics and Quantum Theory Abstract Quantum Field Theory (QFT) stands as one of the most profound achievements of modern physics, unifying quantum mechanics and special relativity into a single framework for describing fundamental interactions. This chapter provides a comprehensive examination of QFT’s mathematical underpinnings, from its origins in classical field theory to operator-valued distributions, Hilbert space formalism, and renormalization techniques. It explores both the successes and limitations of canonical quantization, path integrals, and algebraic approaches, emphasizing the role of functional analysis and distribution theory in providing rigor. Conceptual challenges such as ultraviolet divergences, gauge redundancy, observer dependence, and the absence of a complete non-perturbative formulation in four dimensions are critically assessed. By tracing historical developments and contemporary advances—including applications in particle...

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