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The Dual Majorizing Measure Theorem for Canonical Processes

Xuanang Hu, Vladimir V. Ulyanov, Hanchao Wang

TL;DR

This work characterizes the boundedness of log-concave-tailed canonical processes $X_t=\sum_i t_i Y_i$ by establishing a dual majorizing measure theorem rooted in a novel parameterized separation-tree framework and a growth condition. The authors develop iterative organized trees to connect geometric tree-structure with probabilistic bounds, proving Size$(T)$ is regularly comparable to $\mathbb{E}\sup_{t\in T} X_t$ and to the Fernique-functional bound $\sup_\mu \int_T I_\mu(t)\,d\mu(t)$; this yields a deterministic polynomial-time algorithm for finite $T$ to approximate the supremum. They instantiate the theory in two key regimes, $p=2$ and $p=1$, showing how the geometry reduces to Gaussian-like and $\ell_1$-type structures, respectively. The results advance both the geometric understanding of canonical processes and the computational tractability of supremum bounds in high dimensions, with implications for high-dimensional statistics, learning theory, and related stochastic-analysis applications.

Abstract

We completely characterize the boundedness of the log-concave-tailed canonical processes. The corresponding new majorizing measure theorem for log-concave-tailed canonical processes is proved using the new tree structure. Moreover, we introduce the new growth condition. Combining this condition with the dual majorizing measure theorem proven in the paper, we have developed a polynomial-time algorithm for computing expected supremum of the log-concave canonical processes.

The Dual Majorizing Measure Theorem for Canonical Processes

TL;DR

This work characterizes the boundedness of log-concave-tailed canonical processes by establishing a dual majorizing measure theorem rooted in a novel parameterized separation-tree framework and a growth condition. The authors develop iterative organized trees to connect geometric tree-structure with probabilistic bounds, proving Size is regularly comparable to and to the Fernique-functional bound ; this yields a deterministic polynomial-time algorithm for finite to approximate the supremum. They instantiate the theory in two key regimes, and , showing how the geometry reduces to Gaussian-like and -type structures, respectively. The results advance both the geometric understanding of canonical processes and the computational tractability of supremum bounds in high dimensions, with implications for high-dimensional statistics, learning theory, and related stochastic-analysis applications.

Abstract

We completely characterize the boundedness of the log-concave-tailed canonical processes. The corresponding new majorizing measure theorem for log-concave-tailed canonical processes is proved using the new tree structure. Moreover, we introduce the new growth condition. Combining this condition with the dual majorizing measure theorem proven in the paper, we have developed a polynomial-time algorithm for computing expected supremum of the log-concave canonical processes.
Paper Structure (11 sections, 30 theorems, 232 equations, 1 figure)

This paper contains 11 sections, 30 theorems, 232 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Growth condition
  4. The proof of the main result
  5. Step (1)
  6. Step (2)
  7. Step (3)
  8. Important example
  9. Case 1: $p=2$.
  10. Case 2: $p=1$.
  11. Application

Key Result

Theorem 1.1

Under conditions 1 and 2, where the supremum is taken over all parameterized separation trees $\mathcal{T}$ on $T$, the notation $A \sim_{r} B$ means that there exist positive constants $c(r)$ and $C(r)$, depending only on $r$, such that $c(r)A \le B \le C(r)A$.

Figures (1)

  • Figure 1: The iterative organized tree.

Theorems & Definitions (65)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.1
  • Definition 2.2: Parameterized separation tree
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.1: Contraction principle
  • proof
  • Remark 3.1
  • ...and 55 more