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Generic chaining bounds the supremum of a stochastic process by choosing an optimized hierarchy of approximations to its index metric space. A standard complexity iswhere $|T_0|=1$, $|T_n|\le 2^{2^n}$ and $d(t,T_n)={\inf }_{s\in T_n}d(t,s)$. Sparse early nets are cheap; progressively larger nets reduce the approximation error. The infimum allows the hierarchy to adapt to the geometry instead of using a single uniform covering scale.

For a centered Gaussian process, its canonical distance is $d(s,t)=(\mathbb{E}|X_s-X_t{|}^2{)}^{1/2}$. Under standard separability and boundedness assumptions, the expected supremum is comparable, up to universal constants, to ${\gamma }_2(T,d)$. For $N$ independent unit-variance Gaussian variables, this scale is $\sqrt{\log N}$ for $N\ge 2$, explaining why a maximum grows much more slowly than the number of variables. Generic chaining refines chaining bounds that depend only on global covering numbers.^[ref]

Roman Vershynin studies high-dimensional probability theory and its applications to data. His High-Dimensional Probability by Roman Vershynin develops tools including concentration inequalities, random-matrix estimates and generic chaining, which turn geometry and independence into bounds that hold simultaneously over many directions.^[ref]

A representative question is how many random observations are needed to preserve information about a high-dimensional object. Controlling one fixed direction is easier than controlling every direction at once, so the geometry of a whole family of directions becomes part of the probability calculation. His research page describes this connection between probability and mathematical data science; his author-hosted expository work also treats geometric methods for constrained estimation.^[ref][ref]

Sara van de Geer develops the mathematical theory of high-dimensional statistics. Her Statistical Theory for High-Dimensional Models by Sara van de Geer studies prediction bounds for the Lasso and Square-root Lasso, and inference after correcting shrinkage bias.^[ref]

The distinction is useful: an estimator that predicts well may still bias an individual coefficient toward zero. A de-sparsified estimator adds a correction whose leading error can have an approximately normal distribution. Bounds on the remaining error, together with sparsity and design assumptions, then justify confidence intervals. The same notes connect generic chaining and concentration inequalities with control of random-process suprema.^[ref]