IDS.160 / 18.S998 / 9.521 Spring 20

Mathematical Statistics: A non-asymptotic approach.

## Topics covered

- Introduction to high dimensional phenomena.
- Sums of independent random variables.
- Sub-Gaussian random variables. Chernoff bounds. Hoeffding's inequality.
- Sub-Exponential random variables. Bernstein's inequality.

- Linear Regression.
- Gaussian Sequence model. Sparsity. Thresholding.
- High dimensional linear regression.

- Matrix estimation.
- Perturbation analysis. Davis-Kahan sin-theta theorem. Community detection.
- Covariance matrix estimation. Principal component analysis.

- Minimax lower bounds.
- Distance between probability distributions.
- Lower bounds for estimation and detection problems.

- Empirical processes
- Statistical Learning. Prediction. Uniform Convergence.
- Maximal Inequalities. Symmetrization. Covering, packing, chaining.
- VC theory. Scale-sensitive dimensions.

- High-dimensional classification
- Properties of Rademacher averages. Margin analysis for classification.
- Linear classifiers. Neural networks.

- Nonparametric regression
- Well-specified and misspecified models. Reproducing Kernel Hilbert Spaces.
- Bias-variance. Local methods. The interpolation phenomenon.

## Evaluation

Grading will be based on 4 problem sets (80%) and scribe notes (20%).