Understanding the Efficacy of Over-Parameterization in Neural Networks

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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 ...
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🎨 2D Plotting in SageMath: A Beginner-Friendly, Interactive Guide

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SageMath offers a powerful set of tools for 2D plotting — whether you're visualizing simple functions, multiple graphs, piecewise definitions, parametric curves, or polar plots.

In this guide, you’ll not only learn the basics but also see the plots, try mini-challenges, and discover real-life applications.

1. Plotting 2D Graphs in SageMath

1.1 Graph of Explicit Functions

Let's start by plotting the function:

πŸ”₯ Mini-Challenge:

  • Change the color to 'green'
  • Increase the thickness to 2
  • Add a

2. Plotting Multiple Graphs Together

Now, let’s plot

together:

3. Piecewise Functions

3.1 Basic Piecewise Example

Real-Life Connection: 

Piecewise functions often model mechanical systems where behavior changes suddenly — like a car’s suspension after a speed bump.

πŸ”₯ Mini-Challenge:

  • Modify it so the right side is  instead of .
  • Add a title to the plot.

4. Implicit Plots

Curves defined implicitly like can create beautiful shapes.

Real-Life Connection: 

Implicit plots are used in physics for phase diagrams and energy surfaces.

5. Parametric Plotting

Example:

Real-Life Connection: 

Parametric curves describe particle trajectories, such as the path of an electron in a magnetic field.

6. Polar Plots

Plot beautiful spirals and flowers!

πŸ”₯ Mini-Challenge:

  • Try to make (Replace cos by sin and see the magic!)
  • Try different values of s like 1.5, 2.5, and 5.

 

 

  • Experiment with  to make smoother or rougher plots.

7. Region Plots

Shade regions defined by inequalities!

Real-Life Connection: 

Region plots are used in optimization and economics for feasible regions.

8. Plotting Polygons

9. Graphics Arrays

Combine multiple plots side-by-side:

πŸ–Ό️ Visual Example:

πŸš€ Wrapping Up

SageMath’s 2D plotting features are rich, colorful, and incredibly flexible — perfect for math students, physicists, engineers, and data enthusiasts.

✨ Your Challenge:

  • Create a custom graphics array using parametric and polar plots combined.
  • Try plotting a real-world model, like a pendulum's path or a signal wave!

πŸš€ What's Next?

Now that you've mastered 2D plotting in SageMath, get ready to take things to the next level — 3D Plotting with SageMath! 🎨🌍
We'll explore how to create beautiful surfaces, curves in space, implicit 3D shapes, and even animated 3D visualizations.
Stay tuned for the next post where your graphs will come to life in three dimensions!

 

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