Upper bounds for the L^q empirical process via generic chaining

2025/11/09 by Shang, Zong
Mathematics · #Point processes and geometric inequalities #Random Matrices and Applications #Geometry and complex manifolds

paper · doi:10.48550/arxiv.2511.06338

Abstract

Using the generic chaining method, we derive upper bounds for the \(L^q\) process of sub-Gaussian classes when \(1 ≤ q ≤ 2\), thereby resolving an open problem posed by Al-Ghattas, Chen, and Sanz-Alonso in arXiv:2502.16916. Combined with the results of arXiv:2502.16916, this yields upper bounds for the \(L^q\) process for all \(1 ≤ q < ∞\). We also present corollaries of this result in the geometry of Banach spaces, including high-probability bounds on the \(ℓ_q\) norm diameter of random hyperplane sections of convex bodies where the subspaces are not necessarily uniformly distributed on the Grassmannian manifold and the restricted isomorphic property for \(ℓ_q\) norm.

Citations

• On the Dimension-Free Concentration of Simple Tensors via Matrix Deviation
• Sharp Concentration of Simple Random Tensors
• A geometrical viewpoint on the benign overfitting property of the minimum l₂-norm interpolant estimator and its universality
• Dimension-free Bounds for Sums of Independent Matrices and Simple Tensors via the Variational Principle
• Some notes on concentration for α-subexponential random variables
• Upper bounds on product and multiplier empirical processes
• Concentration via chaining method and its applications
• Tail bounds via generic chaining
• Dimension reduction by random hyperplane tessellations
• Empirical processes with bounded ψ₁ diameter
• Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling
• Decoding by Linear Programming

Related

• Empirical processes with bounded ψ₁ diameter
• Tail bounds via generic chaining
• Exact bounds for some quadratic empirical processes with applications
• Uniform mean estimation via generic chaining
• Squared-Norm Empirical Process in Banach Space