Mathematics / research
High-dimensional statistical inference
Study inference when dimension grows with data through sparsity, random design, regularization, and detection limits.
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Module 01 research
High-dimensional statistical inference
Study inference when dimension grows with data through sparsity, random design, regularization, and detection limits.
- 01
High-dimensional scaling and geometric obstructions
Separate nonidentifiability, prediction, and simultaneous noise control when dimension grows.
26 min - 02
Sparse recovery and restricted eigenvalues
Separate sparse identification, null-space recovery, and stable curvature on an error cone.
29 min - 03
Lasso oracle inequalities and tuning
Derive finite-sample sparse error bounds from a noise event and an explicit restricted-eigenvalue convention.
31 min - 04
Support recovery and irrepresentability
Use active and inactive optimality conditions to distinguish accurate estimation from exact variable selection.
31 min - 05
Debiased Lasso and confidence intervals
Expose an approximately Gaussian coordinate by controlling an inverse-Gram remainder rather than ignoring shrinkage.
33 min - 06
Covariance estimation and shrinkage
Choose a covariance loss, quantify spectral error, and separate numerical regularization from statistical accuracy.
30 min - 07
Spiked models and principal components
Distinguish a population eigendirection from its noisy high-dimensional sample counterpart.
32 min - 08
Matrix completion and nuclear norms
Recover missing entries only when low-rank geometry and sampling information support a convex certificate.
33 min - 09
Multiple testing and false discovery rate
Apply a step-up rule while distinguishing valid p-values, dependence assumptions, and expected false-discovery proportions.
30 min - 10
Information-theoretic minimax lower bounds
Reduce estimation to testing separated hypotheses and distinguish statistical impossibility from computational limits.
33 min