
An Introduction to the Course. This course is best used in Google Colab, however, jupyter notebook can be used as well. Packages used in the course:
1. Numpy
2. Scipy
3. Texttable
There will be others as well. If you don't have a particular package installed, you can do so by using
!pip install --specific package. You will need to do so on occasion. Thanks.
Topics Discussed.
-Mathematical Operations in Python
-Min, Max, Argmin, Argmax, Argsort
- Lists, Tuples, Iterable Objects
-For Loop
Topics Discussed.
-For Loop Continued
-For Loop: Inside the Loop and Outside the Loop
-For-If Loop
-Double For Loop
-List Comprehension
Topics Discussed.
-Recursion
-Control Flow:If Statement
-Control Flow:If Else and If-Elif-Else Statement
-Control Flow:While Loop
-Advanced:Control Flow:For If Loop
Topics Discussed.
- Advanced:Control Flow:While If Else Loop
- Advanced:For-While Loop
- Advanced:While For Loop
- Cartesian Product
Topics Discussed
-Expressions
-Manipulating Expressions
Topics Discussed
-Numerical Evaluation
-Differentiation
-Integration
Topics Discussed.
- Introduction to Series
-Taylor Series Approximation
Topics Discussed
-Convergent and Divergent Series
Topics Discussed.
-Limits
Topics Discussed.
- Relationships between Limits, Differentiation, and Convergence
Topics Discussed.
-Relationship between limits, derivatives, convergence, and python for loop.
Topics Discussed.
-Polynomial Power Series Expansion
-Obtain Polynomial Coefficients
-Evaluate Function
-Find Roots
-Find Stationary Points of Polynomial
Topics Discussed.
-Simple Convex and Concave Function
-Log Functions
-Example: Log Function
-Exponential Functions
-Example: Exponential Functions
-More Mathematical Functions
-Non-Linear or Non-Convex Functions
Topics Discussed.
-Interactive univariate functions
-Multivariate Functions
Topics Covered.
-Array Object
-Arrays with Constant Values
Topics Covered.
-Matrix Operations Inner Product vs Dot Product
-Matrix Operations:Outer Product
-Orthogonality
-Square Matrix
-Identity Matrix
-Transpose of a Matrix
-Inverse of a Matrix
-Symmetric Matrix
Topics Covered.
-Determinants
-Vector and Matrix Operations
-Elementwise Functions
-Aggregate Functions
Topics Covered.
-Reshape
-Vector Norms
-Linear Combinations
-Linear Independence
-Orthogonal Matrix
-Orthogonal and Independence
Topics Covered.
-Matrix Multiplication
-Rank of Matrix
-Determinant of a Matrix
-Matrix Inverse
-Solving for A System of Linear Equations
-Finding Eigenvalues and Eigenvectors
Topics Covered.
-Symmetric Matrix
-Positive Definite Matrix
-Covariance Matrix
-Matrix Decomposition SVD
Topics Covered.
-Partial Derivatives
-Gradient
-Jacobian Matrix
-Hessian Matrix
Topics Covered.
-Linear Algebra Concepts used in linear regression
Topics Discussed.
-Combinations and Permutations
-Random Variables,Conditional Probability, Independence
-Measures of Location expected value, mean, median, mode etc.
Topics Discussed.
-Calculating Quantiles
-Measures of Variability variance,covariance,correlation
-Skewness, Kurtosis
Topics Discussed.
-Probability Mass Function, Probability Density Function, Cumulative Distribution Function
Topics Discussed.
-Normal Distribution
-LogNormal Distribution
-Exponential
-Beta Distribution
-Gamma Distribution
Topics Discussed.
-Chi-Square Distribution
-Student-T Distribution
-Logistic Distribution
-F Distribution
Topics Discussed.
-Bernoulli Distribution
-Binomial Distribution
-Poisson Distribution
-Uniform Distribution
-Geometric Distribution
Topics Discussed
-Multivariate Normal Distribution
Topics Discussed.
-Consistency
- Weak Law of Large Numbers
- Strong Law of Large Numbers
- Consistency
-Central Limit Theorem
-CLT Connection to Statistical Methods
Topics Discussed.
-Practical Importance of Sampling
-Sampling with and without Replacement
-Differences between Sample and Population
-Representative Sample
-Sample Size
-Population Mean and Sample Mean
-Population Variance and Sample Variance
-Bootstrap Sampling
Topics Discussed.
-Stratified Sampling
-Under and Over Sampling
Topics Discussed.
-Point Estimate
-Degrees of Freedom In Estimation
-Standard Error
-Confidence Interval
-QQ-Plot
Topics Discussed.
-Hypothesis Tests
-One Tail vs Two Tail
Topics Discussed
-Inference About Difference Between Two Means
-Inference About Population Variance
-ANOVA
Topics Discussed.
-MLE Definition
-Drawbacks
-Normal Distribution MLE
-Binomial Distribution MLE
-Exponential Distribution MLE
Topics Discussed
-Gradient Descent
-Gradient Descent with Non Linear Function
Topics Discussed.
-Univariate Case: Taylor Series and Newton's Method
-Multivariate Case: Newton's Method and Hessian
Topics Discussed
-Advanced:BFGS
-Advanced:L-BFGS
This course provides a comprehensive foundation in the mathematical concepts essential for understanding and implementing machine learning algorithms from first principles. Students will explore Linear Algebra, covering vectors, matrices, eigenvalues, and singular value decomposition—critical for data representation and transformations. Multivariable Calculus will focus on gradients, Jacobians, and Hessians, which are fundamental to optimization techniques used in training models.
The course also introduces Probability and Statistics, covering key topics such as random variables, probability distributions, expectation, variance, and fundamental statistical inference techniques. Optimization methods, including gradient descent and related algorithms, will be explored to understand how machine learning models learn from data. Additionally, students will develop problem-solving skills by working through mathematical proofs and derivations that underpin these techniques.
Throughout the course, students will gain hands-on experience with NumPy and SciPy, leveraging these powerful Python libraries to implement mathematical concepts programmatically. Rather than applying models to real-world datasets, the focus will be on understanding and building the mathematical foundations necessary for machine learning. By the end of the course, students will have the necessary mathematical and computational tools to derive and implement machine learning techniques from scratch, preparing them for deeper study in artificial intelligence and data science, as well as advanced mathematical modeling.