The Math Behind AI

 

Introduction

Artificial Intelligence and Machine Learning may sound like complicated topics filled with difficult math and smart computers, but at the core, they are all about mathematics.

Whenever we train an AI model, try to improve its performance, or use methods like gradient descent, we are actually using math. It is the language that helps machines learn and make decisions.

Before moving to advanced AI topics, it is important to start with the basics, and one of the most important foundations is Algebra.

Algebra helps us find unknown values, understand relationships, and discover patterns, just like how AI learns patterns from data.

In this post, we will see how simple algebra equations can help solve real life problems and how we can use Python to visualize them.

What is Algebra

Solving for unknows within system of linear equations is Algebra

Problem Statement

The Cop has started to catch bank robber in a van of speed 180km/h. Bank robber has started for rubbery 5 minutes ahead of police van in a car of speed 150 km/h. How long the cop takes to catch the rubber and what distance they have travelled? Ignore the traffic and acceleration etc.

Solution

There are unknows in this problem statement which can be solved by algebra.
Consider d for distance and t for time.
Police van has a speed of 150 km/h
=>d/t = 150 mk/h
=>d/t = 2.5 km/minute ---------------------------- Equation1

Bank robber van has a speed 180km/h and started 5 minutes later than police van.
=>d/(t-5) = 180 km/h
=>d/(t-5) = 3 km/h
=>d = 3t – 15 -----------------------------------------Equation2

From the above 2 equations
=>3t-15 = 2.5t
=>0.5t = 15
=>t = 30 minutes

Using the value of t in equation 2
d = 3x30 – 15
=>d = 75 kilometers

From the above result we are clear that after a distance of 75 kilometers police can catch the rubber in 30 minutes of drive.

Using pen and paper we can plot the graph for the above to equations to get the unknowns. Same can be done using python code as well.
Let’s try using python programming.

Find the unknowns in Python program using NumPy and matplotlib

Download the following packages if not downloaded already.
pip install NumPy
pip install matplotlib

import numpy as np
import matplotlib.pyplot as plt
t = np.linspace(0, 40, 1000) # start, finish, n points
dr = 2.5 * t
dc = 3*t - 15
#Matplotlib command that creates a Figure and Axes — the two main objects used for plotting.
fig, ax = plt.subplots()
plt.title('Bank Robber Caught Sample Plot')
plt.xlabel('time in minutes')
plt.ylabel('distance in km')
ax.set_xlim([0, 40])
ax.set_ylim([0, 100])
ax.plot(t, dr, c='green')
ax.plot(t, dc, c='blue')
plt.axvline(x=30, color='yellow', linestyle='--')
plt.axhline(y=75, color='yellow', linestyle='--')
# To show the graph 
plt.show()

What is Tensor?

A tensor is a mathematical object that generalizes scalars, vectors and matrices to higher dimensions. Tensors are widely used in fields like physics, engineering and machine learning to represent and manipulate data.

Dimension	Mathematical Name	Description
0	Scalar	Only magnitude of x
1	Vector	An array. [x, y]
2	Matrix	Square or a table
3	3-Tensor	Cube or 3D table
n	n-Tensor	Nth dimension

 

Scalars in Python

Scalar is just a value like integer 25
In python declare a variable and assign 25 which is a scalar value.

Widely used two automatic differentiation libraries in Python are Pytorch and TensorFlow.Let’s see the scalar value in these libraries

Scalar in Pytorch

Install libraries if not install already.
pip install TensorFlow
pip install torch torchaudio

Scalar in TensorFlow

>>> import tensorflow as tf
>>> x = tf.Variable(25, dtype=tf.int32)
>>> print(x) #returns <tf.Variable 'Variable:0' shape=() dtype=int32, numpy=25>
>>> print(x.shape) #returns ()

Vectors

The vector is 1-dimensional array of numbers. Example [x1, x2] = [10,20]
The vector of length 2 represents a location in 2-dimensional matrix
The vector of length 3 represents a location in 3-dimensional matrix
The vector of length n represents a location in n-dimensional tensor

Vector Transposition  

The concept of converting row vector to column vector and vice versa is called vector transposition.
Example: [x1, x2, x3] ^T = [x1
                                              x2
                                              x3]

Let’s see the example in Python code

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