High-Dimensional Statistics

Sparsity, regularization, high-dimensional regression, covariance estimation, minimax limits, and inference when the ambient dimension is large relative to sample size.
Modified

April 26, 2026

Keywords

high-dimensional statistics, sparsity, lasso, compressed sensing, minimax

1 Why This Module Matters

Classical statistics usually imagines that sample size is comfortably larger than the number of parameters.

High-dimensional statistics studies the regime where that intuition breaks:

  • the number of features can be comparable to or much larger than sample size
  • naive least squares can become non-unique or unstable
  • recovery becomes possible only with additional structure
  • guarantees depend on geometry, concentration, and regularization all at once

That is why modern papers keep talking about:

  • sparsity
  • shrinkage
  • restricted eigenvalues
  • random design
  • minimax lower bounds
  • inference after selection or regularization

This module is the bridge from ordinary estimation language to the research-facing world of p >> n, sparse recovery, and high-dimensional inference.

Prerequisites Statistics should come first. Optimization matters because many estimators are posed as penalized optimization problems. Matrix Analysis and High-Dimensional Probability become important as soon as spectral control, random design, and concentration enter the story.

Unlocks Sparse regression, compressed sensing, covariance estimation, minimax arguments, post-selection and debiased inference

Research Use Reading papers on lasso, high-dimensional regression, sparse recovery, covariance/PCA, uncertainty in large-feature regimes, and modern statistical limits

2 First Pass Through This Module

The intended first-pass spine for this module is:

  1. Sparsity and Regularization
  2. Lasso and Compressed Sensing Basics
  3. Design Geometry: Restricted Eigenvalues, Coherence, and RIP
  4. High-Dimensional Regression
  5. Covariance, PCA, and Spectral Estimation in High Dimension
  6. Minimax and Lower Bounds
  7. Inference in High Dimension

The module now opens with a complete seven-page first-pass spine, moving from ill-posedness and regularization to sparse estimators, then to the geometry conditions that make sparse theorems work, and then onward to regression rates, covariance/PCA, minimax limits, and inference after selection or shrinkage.

3 How To Use This Module

Read this module in spine order.

For a clean first pass, that means:

  1. start with Sparsity and Regularization
  2. continue to Lasso and Compressed Sensing Basics
  3. continue to Design Geometry: Restricted Eigenvalues, Coherence, and RIP
  4. continue to High-Dimensional Regression
  5. continue to Covariance, PCA, and Spectral Estimation in High Dimension
  6. continue to Minimax and Lower Bounds
  7. continue to Inference in High Dimension
  8. then use nearby live pages in Statistics, Optimization, Learning Theory, Matrix Analysis, and High-Dimensional Probability whenever a page talks about concentration, operator control, or estimation geometry

This module should stay focused on the structural ideas that make estimation possible in large-feature regimes, rather than trying to become a full statistical-learning textbook.

4 Core Concepts

5 Proof Patterns In This Module

  • Structure defeats ill-posedness: identify which assumption makes recovery or estimation possible.
  • Geometry plus concentration: combine random design control with matrix or norm inequalities.
  • Upper bound versus lower bound: separate what an estimator achieves from what the problem fundamentally allows.

6 Applications

6.1 Sparse Regression And Recovery

Many modern regression problems only become statistically meaningful when the model is sparse, approximately sparse, or otherwise low-complexity.

6.2 Spectral Estimation

Covariance estimation, PCA, and related matrix problems depend on the same operator and concentration language already built elsewhere on the site.

6.3 ML Theory And Modern Regimes

High-dimensional statistics gives concrete versions of questions that also appear in ML theory: overparameterization, shrinkage, stability, bias, variance, and recoverability.

7 Go Deeper By Topic

The strongest adjacent live pages right now are:

8 Optional Deeper Reading After First Pass

The strongest current references connected to this module are:

9 Sources and Further Reading

Back to top