Topological Data Analysis of Large Language Models

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  Topological Data Analysis of Large Language Models: A Persistent-Homology Framework for Neural Representation Geometry Abstract Large Language Models (LLMs) have demonstrated remarkable capabilities across a broad spectrum of natural language processing tasks, yet the internal mechanisms governing their representations remain fundamentally opaque. We propose a rigorous methodological framework utilizing Topological Data Analysis (TDA), specifically persistent homology, to characterize the hidden geometric and topological structures of these neural activations. Rather than presenting experimental findings, this paper serves as a comprehensive methodology proposal designed to transition the analysis of LLM embeddings from heuristic geometric approximations to formalized topological invariants. By treating the outputs of self-attention heads and feed-forward networks as dynamic metric spaces, we construct Vietoris-Rips filtrations to trace the birth, persistence, and death of to...
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🎨 Mastering Graph Plotting in Python with Matplotlib

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Welcome to your creative journey into plotting graphs in Python! πŸ“ˆ

Today, we'll dive deep into Matplotlib, a powerful, flexible library that makes data visualization in Python super fun. We'll cover everything from scatter plots to Monte Carlo simulations — with interactive challenges and fun facts along the way!

πŸ“š 1. Plotting Graphs in Python: An Overview

Matplotlib is a comprehensive library for creating static, interactive, and animated graphics in Python.
You can use modules like pyplot and pylab to easily create a variety of graphs.

  • Pylab mixes numpy and pyplot features into a single workspace.
  • But pyplot is generally preferred for non-interactive plotting.

πŸ–Œ️ 1.1 Scatter Plots: Let's Paint Some Dots!

First, let's create a simple scatter plot:

Changing color, size, titles, and labels makes your graph pop:

πŸ“ˆ 1.2 Plotting Explicit Functions

Let's plot the graph of

in the interval

🎨 1.3 Plot Styles: Make it Fancy!

Explore different line styles:

πŸ“Œ Did You Know?

Matplotlib was originally created by John D. Hunter in 2003 to replicate MATLAB’s plotting capabilities — but for free and open-source! πŸš€

🧩 1.4 Titles, Labels, Grids: Polishing Your Plot

Add important plot elements:

A little polish makes a big difference!

🎭 1.5 Plotting Multiple Graphs Together

Sometimes you need to compare different functions on the same graph:

Or with legends:

πŸ–Ό️ 1.6 Subplots: Tiny Worlds in One Figure

Multiple graphs in a grid layout:

πŸ“Œ Did You Know?

You can use  for even easier subplot management. Try it out!

πŸ”— 1.7 Parametric Plots: Beautiful Curves

Example: Figure-Eight Shape!

πŸŒ€ 1.8 Random Walk: Art Made from Randomness

Create a random walk — a path where each step is random:

🎲 1.9 Monte Carlo Simulation: Estimating Pi (Ο€)

Estimate Ο€ by throwing random darts 🎯 inside a square and counting how many land inside the circle:

import numpy as np

import matplotlib.pyplot as plt

from numpy.random import random

 

def Pi_MonteCarlo(maxItr):

    sqrX = [1, -1, -1, 1, 1]

    sqrY = [1, 1, -1, -1, 1]

    cirX, cirY = [], []

    for i in range(361):

        cirX.append(np.cos(np.pi * i/180))

        cirY.append(np.sin(np.pi * i/180))

   

    insideX, insideY, outsideX, outsideY = [], [], [], []

    insideCount = 0

    for i in range(maxItr):

        x = 2*(random() - 0.5)

        y = 2*(random() - 0.5)

        if np.sqrt(x**2 + y**2) <= 1:

            insideCount += 1

            insideX.append(x)

            insideY.append(y)

        else:

            outsideX.append(x)

            outsideY.append(y)

 

    piValue = 4 * insideCount / maxItr

    plt.figure(figsize=(5, 5))

    plt.scatter(insideX, insideY, color='g', marker=".", s=5)

    plt.scatter(outsideX, outsideY, color='b', marker=".", s=5)

    plt.plot(sqrX, sqrY, color='y')

    plt.plot(cirX, cirY, color='r')

    plt.xlabel('x')

    plt.ylabel('y')

    plt.title(f'Approximate Ο€ ≈ {piValue}')

    plt.show()

    print(f'Approximate value of pi is {piValue}')

Pi_MonteCarlo(100000)

πŸ› ️ 1.10 Other Cool Plots to Explore

Matplotlib has even more plot types:

Plot Type

Function

Polar plot

Bar plot

Histogram plot

Pie chart

Box plot

πŸ“Œ Did You Know?

Matplotlib also has a 3D plotting toolkit called mpl_toolkits.mplot3d for stunning 3D graphs! πŸ”₯

🌟 Coming Up Next: Fun Challenges and Creative Plotting Projects!

Get ready to elevate your plotting skills with hands-on practice challenges:

  1. We'll create a stunning parametric curve, diving into the magic of trigonometric and periodic functions.
  2. Discover the beauty of calculus as we plot a function and its derivative side by side.
  3. Explore the power of visualizing real-world data with bar plots and pie charts, focusing on meaningful insights from COVID data.
  4. Dive into statistics by generating normally distributed random numbers, complete with a histogram to visualize their spread.

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