Internship - Data Science - Gradtwin

📊 Mathematics for Data Science and Machine Learning¶

🧠 Table of Contents¶

1. Linear Algebra
2. Calculus
3. Probability and Statistics
4. Optimization
5. Information Theory
6. Discrete Mathematics
7. Numerical Methods
8. Graph Theory
9. Mathematical Foundations of ML
10. Resources and Roadmaps

1. Linear Algebra¶

Topics:

Vectors, matrices, tensors
Matrix operations (addition, multiplication, inverse, transpose)
Eigenvalues and eigenvectors
Singular Value Decomposition (SVD)
Orthogonality, projection
Vector spaces and norms

Resources:

2. Calculus¶

Topics:

Limits and continuity
Derivatives and gradients
Partial derivatives
Chain rule and multivariable calculus
Integration
Jacobian and Hessian matrices

Resources:

3. Probability and Statistics¶

Topics:

Descriptive statistics: mean, median, variance, standard deviation
Probability distributions (Gaussian, Bernoulli, Binomial, Poisson)
Bayes’ theorem
Conditional probability
Expectation, variance
Law of large numbers, Central Limit Theorem
Hypothesis testing, p-values
Confidence intervals

Resources:

4. Optimization¶

Topics:

Gradient descent, stochastic gradient descent
Convex vs non-convex functions
Lagrange multipliers
Loss functions and cost surfaces
Backpropagation and automatic differentiation
Constrained optimization

Resources:

5. Information Theory¶

Topics:

Entropy, cross-entropy
Kullback-Leibler divergence
Mutual information
Bits and data compression

Resources:

6. Discrete Mathematics¶

Topics:

Sets, relations, functions
Combinatorics
Logic and boolean algebra
Proof techniques (induction, contradiction)

Resources:

7. Numerical Methods¶

Topics:

Root finding (Newton-Raphson, bisection)
Numerical differentiation/integration
Linear system solvers (LU, QR)
Stability, convergence

Resources:

8. Graph Theory¶

Topics:

Graphs, nodes, edges
BFS, DFS
Dijkstra’s algorithm
PageRank
Adjacency matrices and applications in ML

Resources:

9. Mathematical Foundations of ML¶

Topics:

Bias-variance tradeoff
VC dimension
Overfitting, underfitting
Regularization (L1, L2)
Kernel methods
PCA and dimensionality reduction
SVD in ML

Resources:

10. Mathematics with Python¶

Core Libraries:

NumPy: Linear algebra, arrays
SciPy: Integration, optimization, numerical solvers
SymPy: Symbolic algebra (derivatives, integrals)
Matplotlib, Seaborn: Visualization
scikit-learn: ML + metrics
statsmodels: Statistical testing and models
PyMC or TensorFlow Probability: Probabilistic modeling
cvxpy: Convex optimization
NetworkX: Graphs and graph algorithms

Topics + Code Examples:

🔹 Linear Algebra¶

import numpy as np

A = np.array([[1, 2], [3, 4]])
eigvals, eigvecs = np.linalg.eig(A)

🔹 Calculus (Symbolic)¶

from sympy import symbols, diff

x = symbols('x')
f = x**2 + 3*x
df = diff(f, x)

🔹 Probability & Stats¶

import scipy.stats as stats

mean = 0
std = 1
prob = stats.norm.cdf(1.96, loc=mean, scale=std)

🔹 Optimization¶

from scipy.optimize import minimize

f = lambda x: x**2 + 3*x + 2
res = minimize(f, x0=0)

🔹 Plotting¶

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-10, 10, 100)
y = x**2
plt.plot(x, y)
plt.title("y = x^2")
plt.show()

11. Resources and Roadmaps¶

📚 Books¶

Mathematics for Machine Learning by Deisenroth, Faisal, Ong
Deep Learning by Goodfellow, Bengio, Courville
Pattern Recognition and Machine Learning by Bishop

🎓 Courses¶

🧾 Cheat Sheets¶

Connect with Me¶

🐦 Website
💼 GitHub
🧠 Hugging Face
📸 ORCiD
💬 Kaggle

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