Mathematics for Machine Learning Roadmap & Tutorial

Understanding artificial intelligence (AI) can be a Nightmare for beginners, but the key to unlocking its potential lies in understanding the mathematical foundations of machine learning. This tutorial is designed to demystify the complex equations and algorithms that power AI, making them accessible to enthusiasts and professionals.

If you’re eager to start a career in Machine Learning, whether you’re a newcomer or an experienced professional seeking a new path, it’s essential to first familiarize yourself with several key mathematical concepts.

These include Statistics, Probability Distribution, Linear Algebra and Matrices, Regression, Geometry, Dimensionality Reduction, and Vector Calculus. These concepts are frequently applied in Machine Learning. For instance, in ML, our primary task is to develop a predictive model (Algorithms/Classifiers) based on training data, which we then apply to make predictions for new data.

To assess our model’s performance, we employ a confusion matrix, rooted in the concept of conditional probability—a fundamental mathematical principle. Grasping these mathematical ideas early on simplifies the learning process for Machine Learning concepts.

Thus, mathematics plays an integral role in Machine Learning, establishing its importance in the field.

NOTE.

Machine Learning empowers computers to acquire knowledge and improve from experience without being directly programmed.

Mathematics articulates the ideas embedded within machine learning models, serving as the language that translates complex concepts into a form that computers can understand and act upon. It’s the mathematical algorithms that guide the learning process, enabling machines to identify patterns and make decisions with minimal human intervention.

This  collaborations of math and machine learning is what propels the field forward, driving innovations across various industries.

The Importance of Mathematics in Machine Learning:

At the heart of machine learning lies a suite of mathematical concepts. From linear algebra to probability theory, these are the building blocks that enable machines to learn from data.

Grasping these concepts is not just about solving equations; it’s about developing an intuition for how machines perceive and process information.

Machine Learning Mathematics Tutorial Overview

I》Linear Algebra:

Linear algebra provides the vocabulary for describing data structures and the operations that can be performed on them. Understanding vectors and matrices is crucial, as they represent data points and transformations in machine learning models.

¡¡》Calculus: The Study of Change:

Machine learning is all about optimization. Calculus helps us understand how to adjust model parameters to minimize errors. The gradient descent algorithm, a cornerstone of machine learning, is a direct application of calculus.

¡¡¡》Probability and Statistics:

Probability theory and statistics are the tools we use to make predictions and decisions under uncertainty. They help us quantify the likelihood of events and the reliability of our models.

Complete tutorial & Roadmap

In this complete tutorial, we’ll explore a variety of mathematical concepts, progressing from the basics to more advanced topics, all through the lens of a specific algorithm.

This approach will help us understand how these mathematical principles are applied within machine learning models, providing a practical framework for learning.

1. Linear Algebra and Matrix

Linear Algebra is a mathematical expression with unknown terms. Its an extension to number of dimensions not defined.

Linear Algebra concerns the focus on linear equation systems.

  • Vectors and Matrices
    • Matrix Introduction
    • Matrix Addition
      • Matrix Addition using NumPy Arrays
    • Matrix Multiplication
      • Matrix Multiplication using Python
    • Matrix Manipulation using NumPy Arrays
    • Inverse of a Matrix
      • Evaluating Inverse using NumPy Arrays
    • Transpose of a Matrix
      • Evaluating Transpose using NumPy Arrays
    • Properties of Matrix
    • Determinant
    • Trace
  • System of Linear Equations
    • System of Linear Equation
    • Solving Linear Equations using Gaussian Elimination 
    • LU Decomposition of Linear Equation
    • Matrix Inversion
  • Matrix Factorization
    • Gram-Schmidt Process
    • QR Decomposition
    • Cholesky Decomposition
    • Singular Value Decomposition
    • Matrix Factorization
    • Diagonalization
    • Eigenvalues and Eigenvectors
    • Eigenspace
  • Vector Spaces
    • Vector Operations
    • Vector Spaces and SubSpaces
    • Basis and Dimension
  • Row Echelon Form
  • Linear Mappings
  • Least Square and Curve Fitting
  • Affine Spaces

2. Statistics

Statistics refers to the act of collecting data, presenting in tabular form and interpreting into numerical form. Statistics is aplied mathematics problems that require data collection, analysis, interpretation and presentation.

  • Mean, Standard Deviation, and Variance
    • Calculating Mean, Standard Deviation, and Variance using Numpy Arrays
  • Sample Error and True Error
  • Bias Vs Variance and Its Trade-Off
  • Hypothesis Testing
    • T-test
    • Paired T-test
    • p-value
    • F-Test
    • z-test
  • Confidence Intervals
  • Correlation and Covariance
  • Correlation Coefficient
  • Covariance Matrix
  • Normal Probability Plot
  • Q-Q Plot
  • Residuals Leverage Plot
  • Robust Correlations
  • Hypothesis Testing
    • Null and Alternative Hypothesis
    • Type 1 and Type 2 Errors
    • p-value interaction
    • Parametric Hypothesis Testing
      • T-test
      • Paired Samples t-test
      • ANOVA Test
    • Non-Parametric Hypothesis Testing
      • Mann-Whitney U test
      • Wilcoxon signed-rank test
      • Kruskal-Wallis test
      • Friedman test
  • Theory of Estimation
    • Difference between Estimators and Estimation
    • Methods of Estimation
      • Method of Moments
      • Bayesian Estimation
      • Least Square Estimation
      • Maximum Likelihood Estimation
    • Likelihood Function and Log-Likelihood Function
    • Properties of Estimation
      • Unbiasedness
      • Consistency
      • Sufficiency
      • Completeness
      • Robustness
  • Confidence Intervals

3. Geometry

Geometry is the branch of mathematics that deals with the forms, angles, measurements, and proportions of ordinary objects.

  • Vector Norms
  • Inner, Outer, Cross Products
  • Distance Between Two Points
  • Distance Measures
    • Euclidean Distance
    • Manhattan Distance
    • Minkowski Distance
    • Chebysev Distance
  • Similarity Measures
    • Cosine Similarity
    • Jaccard Similarity
    • Pearson Correlation Coefficient
    • Kendall Rank Correlation Measure
    • Pearson Product-Moment Correlations
    • Spearman’s Rank Correlation Measure
  • Orthogonality and Orthogonal Projections
    • Orthogonality and Orthonormal Vectors
    • Orthogonal Projections
    • Rotations
  • Geometric Algorithms
    • Nearest Neighbor Search
    • Voronoi diagrams
    • Delaunay Triangulation
    • Geometric intersection and Proximity queries
  • Constraints and Splines
  • Box-Cox Transformations
    • Box-Cox Transformation using Python
  • Fourier transformation
    • Properties of Fourier Transform
  • Inverse Fast Fourier Transformation

4. Calculus

Calculus is a branch mathematics that deals with the study of continuous transition. Calculus is also referred to as infinitesimal calculus or “infinite calculus.” The analysis of continuous change /transition of functions is known as classical calculus.

  • Differentiation
    • Implicit Differentiation
    • Inverse Trigonometric Functions Differentiation
    • Logarithmic Differentiation
    • Partial Differentiation
    • Advanced Differentiation
  • Mathematical Intuition Behind Gradients and their usage
    • Implementation of Gradients using Python
    • Optimization Techniques using Gradient Descent
  • Higher-Order Derivatives
  • Multivariate Taylor Series
  • Application of Derivation
    • Application of Derivative – Maxima and Minima
    • Absolute Minima and Maxima
    • Constrained Optimization
    • Unconstrained Optimization
    • Constrained Optimization – Lagrange Multipliers
    • Newton’s Method
  • Uni-variate Optimization
  • Multivariate Optimization
  • Convex Optimization
  • Lagrange’s Interpolation
  • Area Under Curve

5. Probability and Distributions

Probability and distributions are statistical functions that describe all the possible values.

  • Probability
  • Chance and Probability
  • Addition Rule for Probability
  • Law of total probability
  • Bayes’ Theorem
  • Discrete Probability Distributions
    • Discrete Uniform Distribution
    • Bernoulli Distribution
    • Binomial Distribution
    • Poisson Distribution
  • Continuous Probability Distributions
    • Continuous Uniform Distribution
    • Exponential Distribution
    • Normal Distribution
    • Beta Distribution
      • Beta Distribution of First Kind
      • Beta Distribution of Second Kind
    • Gamma Distribution
  • Sampling Distributions
    • Chi-Square Distribution
    • F – Distribution
    • t – Distribution
  • Central Limit Theorem
    • Implementation of Central Limit Theorem
  • Law of Large Numbers
  • Change of Variables/Inverse Transformation

6. Regression

Regression is a statistical process for estimating the relationships between the dependent variables or criterion variables.

  • Parameter Estimation
  • Bayesian Linear Regression
  • Quantile Linear Regression
  • Normal Equation in Linear Regression
  • Maximum Likelihood as Orthogonal Projection

5. Dimensionality Reduction

Dimensionality reduction is a technique to reduce the number of input variables in training data.

  • Introduction to Dimensionality Reduction
  • Projection Perspective in Machine Learning
  • Eigenvector Computation and Low-Rank Approximations
  • Mathematical Intuition Behind PCA
    • PCA implementation in Python
  • Latent Variable Perspective
  • Mathematical Intuition Behind LDA
    • Implementation of Linear Discriminant Analysis (LDA)
  • Mathematical Intuition Behind GDA
    • Implementation of Generalized Discriminant Analysis (GDA)
  • Mathematical Intuition Behind t-SNE Algorithm
    • Implementation of the t-SNE Algorithm

Some Related Articles-

  • Introduction to Machine Learning
  • Machine learning Tutorial
  • Top 50 Machine Learning Interview Questions (2023)

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Conclusion:

The journey through the mathematics of machine learning is challenging but rewarding. By breaking down complex concepts into understandable parts, we can begin to see the elegance and power of the algorithms that drive AI.

As you continue to explore and learn, remember that the beauty of mathematics lies in its ability to simplify the complex and illuminate the path to innovation.

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