Project 31: Random Matrix Theory in High-Dimensional ML

Build a spectrum-analysis lab that separates noise geometry from true signal in high dimensions.

Quick Reference

Attribute Value
Difficulty Level 5: Master (The First-Principles Wizard)
Time Estimate 2-3 weeks
Main Programming Language Python
Alternative Programming Languages Julia, MATLAB, C++
Coolness Level Level 5: Pure Magic (Super Cool)
Business Potential 1. The “Resume Gold” (Educational/Personal Brand)
Knowledge Area Random Matrix Theory / High-Dimensional Statistics
Main Book “High-Dimensional Probability” by Roman Vershynin

1. Learning Objectives

  1. Simulate high-dimensional covariance spectra under controlled regimes.
  2. Compare empirical spectra against Marchenko-Pastur predictions.
  3. Detect signal spikes in spiked-covariance settings.
  4. Use spectral diagnostics to guide dimensionality choices.

2. All Theory Needed (Per-Concept Breakdown)

Concept A: High-Dimensional Concentration

Fundamentals When p is comparable to n, geometric intuitions from low dimensions fail.

Deep Dive into the concept Distances and eigen-spectra concentrate in ways that can make noise look like structure unless corrected by theory-informed thresholds.

Concept B: Marchenko-Pastur Spectrum

Fundamentals Pure-noise covariance eigenvalues occupy predictable bulk support asymptotically.

Deep Dive into the concept Bulk edges provide practical baselines for identifying outlier signal components.

Concept C: Spiked Models and Shrinkage

Fundamentals Low-rank signal plus noise creates outlier eigenvalues above bulk.

Deep Dive into the concept Detecting spikes supports better component retention and regularization strategies.

3. Build Blueprint

  1. Generate Gaussian and spiked-covariance synthetic datasets.
  2. Compute empirical covariance eigen-spectra across p/n regimes.
  3. Compare to theoretical bulk support and detect outliers.
  4. Measure downstream performance impact of spectrum-informed truncation.

4. Real-World Outcome (Target)

$ python rmt_lab.py --n 4000 --p 2000 --model spiked_cov --spikes 6,4,2

gamma = 0.500
MP bulk support: [0.09, 2.96]
Detected spikes above bulk edge: 3
Recommendation: retain only outlier components

5. Core Design Notes from Main Guide

Core Question

“How do we tell real structure from high-dimensional noise geometry?”

Common Pitfalls

  • Wrong covariance normalization
  • Over-trusting explained-variance heuristics
  • Ignoring finite-sample edge fluctuations

Definition of Done

  • Spectral simulations run across multiple p/n regimes
  • MP bulk comparison and outlier detection implemented
  • False-positive spike rates measured on pure-noise runs
  • At least one ML decision is improved using spectral diagnostics

6. Extensions

  1. Add bootstrap edge calibration.
  2. Add covariance shrinkage estimators.
  3. Add overparameterized linear-model risk comparison.