Math 0-1: Matrix Calculus In Data Science & Machine Learning

A Casual Guide for Artificial Intelligence, Deep Learning, and Python Programmers

What you'll learn

Derive matrix and vector derivatives for linear and quadratic forms

Solve common optimization problems (least squares, Gaussian, financial portfolio)

Understand and implement Gradient Descent and Newton's method

Learn to use the Matrix Cookbook

Requirements

Competence with Calculus and Linear Algebra

Optional: Familiarity with Python, Numpy, and Matplotlib to implement optimization techniques

Description

Welcome to the exciting world of Matrix Calculus, a fundamental tool for understanding and solving problems in machine learning and data science. In this course, we will dive into the powerful mathematics that underpin many of the algorithms and techniques used in these fields. By the end of this course, you'll have the knowledge and skills to navigate the complex landscape of derivatives, gradients, and optimizations involving matrices.Course Objectives:Understand the basics of matrix calculus, linear and quadratic forms, and their derivatives.Learn how to utilize the famous Matrix Cookbook for a wide range of matrix calculus operations.Gain proficiency in optimization techniques like gradient descent and Newton's method in one and multiple dimensions.Apply the concepts learned to real-world problems in machine learning and data science, with hands-on exercises and Python code examples.Why Matrix Calculus? Matrix calculus is the language of machine learning and data science. In these fields, we often work with high-dimensional data, making matrices and their derivatives a natural representation for our problems. Understanding matrix calculus is crucial for developing and analyzing algorithms, building predictive models, and making sense of the vast amounts of data at our disposal.Section 1: Linear and Quadratic Forms In the first part of the course, we'll explore the basics of linear and quadratic forms, and their derivatives. The linear form appears in all of the most fundamental and popular machine learning models, including linear regression, logistic regression, support vector machine (SVM), and deep neural networks. We will also dive into quadratic forms, which are fundamental to understanding optimization problems, which appear in regression, portfolio optimization in finance, signal processing, and control theory.The Matrix Cookbook is a valuable resource that compiles a wide range of matrix derivative formulas in one place. You'll learn how to use this reference effectively, saving you time and ensuring the accuracy of your derivations.Section 2: Optimization Techniques Optimization lies at the heart of many machine learning and data science tasks. In this section, we will explore two crucial optimization methods: gradient descent and Newton's method. You'll learn how to optimize not only in one dimension but also in high-dimensional spaces, which is essential for training complex models. We'll provide Python code examples to help you grasp the practical implementation of these techniques.Course Structure:Each lecture will include a theoretical introduction to the topic.We will work through relevant mathematical derivations and provide intuitive explanations.Hands-on exercises will allow you to apply what you've learned to real-world problems.Python code examples will help you implement and experiment with the concepts.There will be opportunities for questions and discussions to deepen your understanding.Prerequisites:Basic knowledge of linear algebra, calculus, and Python programming is recommended.A strong desire to learn and explore the fascinating world of matrix calculus.Conclusion: Matrix calculus is an indispensable tool in the fields of machine learning and data science. It empowers you to understand, create, and optimize algorithms that drive innovation and decision-making in today's data-driven world. This course will equip you with the knowledge and skills to navigate the intricate world of matrix calculus, setting you on a path to become a proficient data scientist or machine learning engineer. So, let's dive in, embrace the world of matrices, and unlock the secrets of data science and machine learning together!

Overview

Section 1: Introduction

Lecture 1 Introduction and Outline

Lecture 2 How to succeed in this course

Lecture 3 Where to get the code

Section 2: Matrix and Vector Derivatives

Lecture 4 Derivatives - Section Introduction

Lecture 5 Linear Form

Lecture 6 Quadratic Form (pt 1)

Lecture 7 Quadratic Form (pt 2)

Lecture 8 Exercise: Quadratic

Lecture 9 Exercise: Least Squares

Lecture 10 Exercise: Gaussian

Lecture 11 Chain Rule

Lecture 12 Chain Rule in Matrix Form

Lecture 13 Chain Rule Generalized

Lecture 14 Exercise: Quadratic with Constraints

Lecture 15 Left and Right Inverse as Optimization Problems

Lecture 16 Derivative of Determinant

Lecture 17 Derivatives - Section Summary

Lecture 18 Suggestion Box

Section 3: Optimization Techniques

Lecture 19 Optimization - Section Introduction

Lecture 20 Second Derivative Test in Multiple Dimensions

Lecture 21 Gradient Descent (One Dimension)

Lecture 22 Gradient Descent (Multiple Dimensions)

Lecture 23 Newton's Method (One Dimension)

Lecture 24 Newton's Method (Multiple Dimensions)

Lecture 25 Exercise: Newton's Method for Least Squares

Lecture 26 Exercise: Code Preparation

Lecture 27 Gradient Descent and Newton's Method in Python

Lecture 28 Optimization - Section Summary

Section 4: Appendix / FAQ Finale

Lecture 29 BONUS

Students and professionals interested in the math behind AI, data science and machine learning