Jean-Philippe Bouchaud, Capital Fund Management & Ecole Polytechnique, France

Random Matrix Theory and Big Data Cleaning (4.5h)

1. Cleaning large noisy matrices: motivations

2. A short introduction to RMT Resolvants, HCIZ integrals, Dyson Brownian motion

3. Eigenvalue distribution of noisy matrices

a) The additive case. Replicas and R-transforms

b) The multiplicative case (correlation matrices). Replicas and S-transforms

c) Rectangular matrices and SVD

4. Eigenvector deformation in the presence of noise

a) Overlaps in the additive case. Replicas and Dyson's motion

b) Overlaps in the multiplicative case

c) Eigenspace stability

d) The case of spikes

5. Optimal Rotational Invariant Estimators

a) Rotational Invariance and Oracle estimator

b) The additive case

c) The multiplicative case and the Ledoit-Péché estimator

d) Numerical/Empirical illustrations