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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 ...

Taming the Infinite: Singularities, Regularization, and Analytic Continuation Explained

Taming the Infinite: Singularities, Regularization, and Analytic Continuation Explained Matrix Space Toolkit in SageMath Taming the Infinite – How We Make Sense of "Impossible" Functions! Introduction: The Mystery of the Infinite (and Why We Care!) What happens when a function tries to break mathematics? Can we ever truly understand something that goes to infinity? These aren’t just philosophical musings. In both pure math and applied science, functions that misbehave—spiking to infinity or becoming undefined—are everywhere. And yet, they’re essential. But how do we work with something that shouldn’t be computable? Take 1/x. It's fine—until you hit x = 0, where it suddenly becomes undefined. This is what mathematicians c...

Unlocking Distribution Theory: Understanding Generalized Functions & derivatives

Unlocking Distribution Theory: Understanding Generalized Functions & Derivatives Matrix Space Toolkit in SageMath From Smooth Functions to Distributions: What Happens When You Differentiate a Functional? Introduction: More Than Just Derivatives If you've ever taken a calculus class, you know how to differentiate a function. But what if you're not differentiating a function—but a functional? Even more mind-bending: what if the object you're working with isn't even a function in the traditional sense, but a generalized function or distribution? Welcome to the magical world of distribution theory, where even the Dirac delta "function" makes perfect sense, and derivatives can be defined in a way that bypasses a...

Generalized Functions & Differential Equations: Exploring the Infinite & the Unexpected

Understanding Delta Function Approximations: Lorentzian, Gaussian, and Sinc Compared Matrix Space Toolkit in SageMath Differential Equations for Generalized Functions: When Calculus Meets the Infinite and the Weird In the previous blog, we understood the Understanding Delta Function Approximations: Lorentzian, Gaussian, and Sinc Compared . Let's take another one step and explore the Differential Equations for Generalized Functions: What happen When Calculus Meets the Infinite and the Weird What happens when you mix the familiar world of differential equations with the strange universe of generalized functions—those magical creatures that extend what we usually call a function? It turns out, you get a whole new playground where class...

Understanding Delta Function Approximations: Lorentzian, Gaussian, and Sinc Compared

Understanding Delta Function Approximations: Lorentzian, Gaussian, and Sinc Compared Matrix Space Toolkit in SageMath Delta-Convergent Sequences — Refined Blog with SageMath Symbolics, Physics Insights, and Cleaner Code In the previous blog, we understood the Lorentzian Delta Sequence (Cauchy Kernel) , Gaussian Approximation (Heat Kernel) and Sinc Delta Sequence . Let's take another one step and explore the Understanding Delta Function Approximations: Lorentzian, Gaussian, and Sinc Compared. Why Study These Approximations? Delta functions are central in many fields: Signal Processing: Ideal impulse, filter response Physics: Point charges/masses, Green's functions Spectral Theory: Lorentzian prof...

Understanding Delta Function Approximations: Sinc-Based Approximation (Fourier Kernel)

Understanding Delta Function Approximations: Sinc-Based Approximation (Fourier Kernel) Matrix Space Toolkit in SageMath Delta-Convergent Sequences — Refined Blog with SageMath Symbolics, Physics Insights, and Cleaner Code In the previous blog, we understood the Lorentzian Delta Sequence (Cauchy Kernel) and Gaussian Approximation (Heat Kernel) . Let's take another one step and explore the Sinc-Based Approximation (Fourier Kernel). Why Study These Approximations? Delta functions are central in many fields: Signal Processing: Ideal impulse, filter response Physics: Point charges/masses, Green's functions Spectral Theory: Lorentzian profiles in resonance Diffusion Models: Gaussians arise from ...

Understanding Delta Function Approximations: Gaussian Delta Sequence (Heat Kernel)

Understanding Delta Function Approximations: Gaussian Delta Sequence (Heat Kernel) Matrix Space Toolkit in SageMath Delta-Convergent Sequences — Refined Blog with SageMath Symbolics, Physics Insights, and Cleaner Code In the previous blog, we understood the Lorentzian Delta Sequence (Cauchy Kernel) . Let's take another step and explore the Gaussian Delta Sequence (Heat Kernel). Why Study These Approximations? Delta functions are central in many fields: Signal Processing: Ideal impulse, filter response Physics: Point charges/masses, Green's functions Spectral Theory: Lorentzian profiles in resonance Diffusion Models: Gaussians arise from the heat equation Numerics: Regularizing singular...

Understanding Delta Function Approximations: Lorentzian Delta Sequence (Cauchy Kernel)

Understanding Delta Function Approximations: Lorentzian Delta Sequence Matrix Space Toolkit in SageMath Delta-Convergent Sequences — Refined Blog with SageMath Symbolics, Physics Insights, and Cleaner Code The Dirac delta function isn’t a “normal” function — it’s an idealization used to represent a point source. It's infinitely narrow, infinitely tall, and yet integrates to 1. We approximate it using delta-convergent sequences: real functions depending on a parameter that becomes increasingly peaked at zero as the parameter vanishes. This post explores the three most common delta-approximating sequences using SageMath, including plots, integration checks, and real-world meaning. Why Study These Approximations? Delta function...

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