MATH 647 Spring 2026: Random Matrix Theory in Machine Learning

Instructor: Yizhe Zhu, yizhezhu@usc.edu

Class schedule: MWF 1:00-1:50 pm at GFS 112

Office Hours: Friday 12:00-12:50 pm KAP 464B

Course Description: This course provides an introduction to the fundamental methods, concepts, and tools of random matrix theory, with an emphasis on its modern applications in machine learning. We will explore both the asymptotic and non-asymptotic perspectives of random matrices and discuss how these theoretical foundations inform applications in network analysis, signal processing, and deep learning theory.

Homework: Homework will be posted on Brightspace 

                         Course Schedule:  Below is a tentative schedule, to be updated as the semester progresses.

Week 1

• Jan 12: Wigner matrices, semicircle law, moment method proof

• Jan 14: moment method proof continued

• Jan 16: Marchenko-Pastur law 

Week 2

• Jan 19: No class

• Jan 21:  Spectral norm bound via high moment method

• Jan 23:  Stieltjes transform

Week 3

• Jan 26:  Global law with Stieltjes transform

• Jan 28:  Marchenko-Pastur law with relaxed assumptions

• Jan 30: Local law and  eigenvector delocalization

Week 4

• February 2:  Matrix concentration inequalities

• February 4:  Matrix concentration inequalities

• February 6:  Convex Gaussian  min-max theorem and applications

Week 5

• February 9:  Convex Gaussian  min-max theorem and applications

• February 11:  Davis-Kahan inequality 

• February 13:  Low-rank estimation under random perturbation  

Week 6

• February 16: No class

• February 18:  Refinement of Davis-Kahan

• February 20:  Refinement of Davis-Kahan

Week 7

• February 23:  BBP phase transition

• February 25: BBP phase transition

• February 27:  Stochastic block model: Exact recovery and weak recovery

Week 8

• March 2:  Detection threshold

• March 4:  non-backtracking operator

• March 6:   non-backtracking operator

Week 9

• March 9:  non-backtracking operator

• March 11:  Matrix completion

• March 13:  Matrix completion

Week 10

• March 16: No class

• March 18: No class

• March 20: No class

Week 11

• March 23:   Tensor completion

• March 25:   Tensor PCA

• March 27:  Tensor PCA

Week 12

• March 30:  the double descent

• April 1:  the double descent

• April 3:  benign overfitting

Week 13

• April 6:  benign overfitting

• April 8: random feature models

• April 10:  random feature models

Week 14

• April 13:  neural tangent kernel

• April 15:  neural tangent kernel

• April 17: precise asymptotics of generalization error

Week 15

• April 20: precise asymptotics of generalization error

• April 22: high-dimensional learning dynamics

• April 24:  high-dimensional learning dynamics

Week 16

• April 27:  Final presentation

• April 29:  Final presentation

• May 1: Final presentation

A list of papers for the final project:

Concentration inequalities

Afonso Bandeira, Kevin Lucca, Petar Nizić-Nikolac, Ramon van Handel. Matrix Chaos Inequalities and Chaos of Combinatorial Type

March T. Boedihardjo. Injective norm of random tensors with independent entries

Danil Akhtiamov, David Bosch, Reza Ghane, K Nithin Varma, Babak Hassibi. A novel Gaussian min-max theorem and its applications

Aicke Hinrichs, David Krieg, Erich Novak, Joscha Prochno, Mario Ullrich. Random sections of ellipsoids and the power of random information

Afonso S. Bandeira, Giorgio Cipolloni, Dominik Schröder, Ramon van Handel. Matrix Concentration Inequalities and Free Probability II. Two-sided Bounds and Applications

Tatiana Brailovskaya, Ramon van Handel. Extremal random matrices with independent entries and matrix superconcentration inequalities

Spectral clustering

Anderson Ye Zhang. Fundamental Limits of Spectral Clustering in Stochastic Block Models

Alexandra Carpentier, Christophe Giraud, Nicolas Verzelen. Phase Transition for Stochastic Block Model with more than  $\sqrt{n} $ Communities

Spiked model

Alice Guionnet, Justin Ko, Florent Krzakala, Pierre Mergny, Lenka Zdeborová. Spectral Phase Transitions in Non-Linear Wigner Spiked Models

Michael J. Feldman. Spectral Properties of Elementwise-Transformed Spiked Matrices

Zhangsong Li. The Algorithmic Phase Transition in Correlated Spiked Models

Pravesh K. Kothari, Jeff Xu. Smooth Trade-off for Tensor PCA via Sharp Bounds for Kikuchi Matrices

Jingqiu Ding, Samuel B.Hopkins, David Steurer. Estimating Rank-One Spikes from Heavy-Tailed Noise via Self-Avoiding Walks

Nonlinear random matrix model

Lucas Benigni, Elliot Paquette. Eigenvalue distribution of the Neural Tangent Kernel in the quadratic scaling

Chiraag Kaushik, Justin Romberg, Vidya Muthukumar. A general technique for approximating high-dimensional empirical kernel matrices

Deep learning theory

Licong Lin, Jingfeng Wu, Sham Kakade, Peter Bartlett, Jason Lee. Scaling Laws in Linear Regression: Compute, Parameters, and Data

Jimmy Ba, Murat A Erdogdu, Taiji Suzuki, Zhichao Wang, Denny Wu. Learning in the presence of low-dimensional structure: a spiked random matrix perspective

Yatin Dandi, Florent Krzakala, Bruno Loureiro, Luca Pesce, Ludovic Stephan, How two-layer neural networks learn, one (giant) step at a time

High-dimensional statistics

Parham Rezaei, Filip Kovacevic, Francesco Locatello, Marco Mondelli. High-dimensional Analysis of Synthetic Data Selection