ELL 409: Machine Intelliegnce and Learning

Instructors: Prof. Sandeep Kumar (SDK)

3 credits (3-0-0)Pre-requisites: Machine Learning and Deep Learning Semester II: 2024-2025
Evaluation: Assignment and Quizzes (25 %), Minor Exam (35%) and Major Exam (40 %).

Course overview

Modern AI systems increasingly operate over relational and interconnected data arising in domains such as social networks, communication systems, neuroscience, biology, finance, recommendation systems, and knowledge graphs. Graphs provide a powerful mathematical and computational framework for modeling such structured interactions, while modern machine learning enables scalable learning, inference, reasoning, and decision-making over these relational systems.

This course presents an advanced, mathematically rigorous, and research-oriented treatment of Graph Machine Learning (Graph ML) and its emerging role in Relational AI. The course integrates ideas from spectral graph theory, graph signal processing, probabilistic inference, optimization, geometric learning, deep graph learning, graph transformers, reinforcement learning over graphs, and graph foundation models within a unified framework for learning over structured relational data.

Unlike standard application-driven Graph ML courses, this course places strong emphasis on mathematical rigor, derivation-driven understanding, and the development of a unified theoretical perspective connecting spectral methods, probabilistic inference, geometric learning, and deep graph architectures. The course further encourages critical analysis of contemporary research literature, scalable algorithmic thinking, and exploration of frontier research problems that are shaping the next generation of graph machine learning and relational AI systems.

By the end of the course, students will be able to:

Understand the mathematical foundations of Graph ML
Analyze graphs using spectral and probabilistic tools
Develop graph representation learning algorithms
Understand and derive modern Graph Neural Networks
Analyze scalability, robustness, and trustworthiness in Graph ML
Read and critique modern Graph ML research literature
Explore reinforcement learning and agentic AI over graph-structured systems

Course Modules and Lecture Flow

Module I : Mathematical Foundations

Lecture 1: Graphs as Relational Inductive Bias

Why Euclidean ML fails on relational systems
Symmetry and permutation equivariance
Graphs vs manifolds
Applications across domains

Lecture 2: Matrix Analysis for Graph ML

Graph matrices, PSD matrices, Spectral decomposition, Rayleigh quotient, Eigenvalue inequalities

Lecture 3: Probability and Inference on Graphs

Probabilistic graphical models, Markov properties, Energy-based formulations

Lecture 4: Optimization Foundations

Convexity, Variational formulations, Graph regularization

PART II — Spectral and Geometric Foundations

Lecture 5: Spectral Graph theory

Combinatorial and normalized Laplacians, Spectral properties, Algebraic connectivity

Lecture 6: Smoothness and Dirichlet Energy

Graph smoothness, Harmonic functions, Spectral regularization

Lecture 7: Graph Fourier Analysis

Graph signals, Spectral basis, Graph Fourier Transform

Lecture 8: Diffusion and Heat Kernels

Diffusion dynamics, Heat kernels, Spectral filtering

Lecture 9: Laplace–Beltrami Operator

Discrete-continuous bridge, Manifold hypothesis, Harmonic analysis

Lecture 10: Graph Signal Processing

Sampling and reconstruction, Graph filtering, Spectral wavelets

PART III — Representation Learning on Graphs

Lecture 11: Spectral Embeddings

Laplacian Eigenmaps
Diffusion maps
Multidimensional Scaling (MDS)

Lecture 12: Random Walk Representation Learning

DeepWalk
Node2Vec
LINE

Lecture 13: Matrix Factorization Perspectives

PMI factorization
Implicit matrix factorization
Connections to word embeddings

Lecture 14: Self-Supervised and Contrastive Learning

Contrastive objectives
InfoNCE
Graph augmentations

Lecture 15: Knowledge Graph Embeddings

TransE and RotatE
Relational reasoning
Knowledge graph learning

PART IV — Probabilistic and Generative Graph ML

Lecture 16: Probabilistic Graphical Models

MRFs
CRFs
Factor graphs

Lecture 17: Variational Inference on Graphs

ELBO
Graph VAEs
Latent graph learning

Lecture 18: Generative Graph Models

Diffusion graph models
Autoregressive generation
Molecular graph generation

PART V — Deep Graph Learning

Lecture 19: Message Passing Neural Networks

Message passing framework
Aggregation and update operators
Expressive power

Lecture 20: Spectral Foundations of GCNs

Spectral graph convolutions
Chebyshev approximations
Derivation of GCNs

Lecture 21: Advanced GNN Architectures

GraphSAGE
GAT
GIN

Lecture 22: Expressivity and Limitations

Weisfeiler–Lehman hierarchy
Over-smoothing
Over-squashing

Lecture 23: Graph Transformers

Graph attention
Positional encoding
Transformer architectures for graphs

PART VI — Scalable and Trustworthy Graph ML

Lecture 24: Graph Coarsening and Pooling

Multiscale graph representations
Graph pooling
Scalable graph learning

Lecture 25: Robustness, Fairness, and Privacy

Adversarial attacks on graphs
Fairness in GNNs
Federated graph learning

Lecture 26: Dynamic and Temporal Graphs

Temporal graph learning
Event-driven graphs
Dynamic message passing

PART VII — Relational AI, RL, and Foundation Models

Lecture 27: Reinforcement Learning on Graphs

Graph traversal and exploration
RL for combinatorial optimization
Multi-agent graph systems

Lecture 28: Graph Foundation Models and Agentic AI

Graph-language models
Memory graphs
Agentic reasoning systems
Neuro-symbolic relational AI

Research Seminar Component

The seminar component forms a major part of the course.

Students will:

present research papers,
reproduce selected results,
critique assumptions,
identify limitations,
and discuss open research problems.

Indicative topics include:

Spectral clustering
Graph embeddings
Graph Neural Networks
Graph transformers
Dynamic graph learning
Graph reinforcement learning
Graph foundation models

Textbooks:

Murphy, K. P. (2022). Probabilistic Machine Learning: An Introduction. MIT Press.
Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press.
Alpaydin, E. (2020). Introduction to Machine Learning (4th ed.). MIT Press.
Hastie, Trevor. The elements of statistical learning: data mining, inference, and prediction." (2009).
Bishop, Christopher M., and Nasser M. Nasrabadi. Pattern recognition and machine learning. Vol. 4. No. 4. New York: springer, 2006.

Additional Readings:

Gilbert Strang, Linear Algebra and Learning from Data Wellesley-Cambridge Press, (2019).
John Hopcroft and Ravi Kannan, Foundations of Data Science, Hind Book Agency, TRIM Series, (2014).
Ekman, M. (2021). Learning deep learning: Theory and practice of neural networks, computer vision, natural language processing, and transformers using TensorFlow. Addison-Wesley Professional.