Empirical approximation of L^p norms for p∈ (1, 2)

Abstract

For p∈[1,2], we prove uniform moment estimates for weighted sums ∑_j=1^mζ_jw_j|⟨bf v_j,y⟩|^p of independent subgaussian random variables ζ_j, where the unit vectors {bf v_j}_j=1^m and nonnegative weights {w_j}_j=1^m form an isotropic frame. This allows us to obtain optimal sampling discretization of L_p norms for each fixed p∈[1,2]. Namely, for an N-dimensional space X_N of bounded functions satisfying |f|_∞≤√(KN) |f|_L₂ for every f∈ X_N, we show that m≥ Cε⁻²KNlog(2N) independent samples ξ¹,…,ξ^m provide the discretization (1−ε)|f|_{L_p}^p ≤1/m∑_j=1^m|f(ξ^j)|^p ≤(1+ε)|f|_{L_p}^p ∀ f∈ X_N with probability at least 1−exp(−cmε²/KN). This implies a corresponding weighted discretization result that, for 1<p<2, improves Talagrand's bound on the dimension m needed to embed N-dimensional subspaces of L_p into ℓ_p^m with distortion at most 1+ε, giving m≤ Cε⁻²Nlog(2N).

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BibTeX RIS

BibTeX

@misc{dai2026empirical,
  title = {{Empirical approximation of $L^p$ norms for $p\in (1, 2)$}},
  author = {Feng Dai and Egor Kosov},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05930},
  eprint = {2611.05930},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05930},
}

RIS

TY  - UNPB
TI  - Empirical approximation of L^p norms for p∈ (1, 2)
AU  - Feng Dai
AU  - Egor Kosov
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.05930
UR  - https://arxiv.org/abs/2611.05930
AB  - For p∈[1,2], we prove uniform moment estimates for weighted sums ∑_j=1^mζ_jw_j|⟨bf v_j,y⟩|^p of independent subgaussian random variables ζ_j, where the unit vectors {bf v_j}_j=1^m and nonnegative weights {w_j}_j=1^m form an isotropic frame. This allows us to obtain optimal sampling discretization of L_p norms for each fixed p∈[1,2]. Namely, for an N-dimensional space X_N of bounded functions satisfying |f|_∞≤√(KN) |f|_L_2 for every f∈ X_N, we show that m≥ Cε^−2KNlog(2N) independent samples ξ^1,…,ξ^m provide the discretization (1−ε)|f|_L_p^p ≤1/m∑_j=1^m|f(ξ^j)|^p ≤(1+ε)|f|_L_p^p ∀ f∈ X_N with probability at least 1−exp(−cmε^2/KN). This implies a corresponding weighted discretization result that, for 1<p<2, improves Talagrand's bound on the dimension m needed to embed N-dimensional subspaces of L_p into ℓ_p^m with distortion at most 1+ε, giving m≤ Cε^−2Nlog(2N).
ER  -

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