[Paper Review] The Marcinkiewicz-type discretization theorems

Vladimir Temlyakov|arXiv (Cornell University)|Mar 10, 2017

Mathematical Approximation and Integration12 references7 citations

TL;DR

This paper establishes new Marcinkiewicz-type discretization theorems for $L_1$ norms of functions in finite-dimensional subspaces using a novel combination of probabilistic chaining techniques and entropy number estimates in the uniform norm. The key result shows that discretization with $m \ll N(\log N)^{7/2}$ points is possible, nearly matching the ideal $m = N$ case, significantly improving prior bounds for $L_1$ and $L_\infty$ settings.

ABSTRACT

The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. This problem is very important in applications but there is no systematic study of it. We present here a new technique, which works well for discretization of the integral norm. It is a combination of probabilistic technique, based on chaining, with results on the entropy numbers in the uniform norm.

Motivation & Objective

• To address the fundamental problem of discretizing integral norms of functions in finite-dimensional subspaces, particularly in $L_1$ and $L_\infty$.
• To develop a systematic framework for Marcinkiewicz-type discretization theorems with explicit bounds on the number of sampling points.
• To bridge the gap between theoretical discretization and practical numerical integration by establishing conditions under which equal-weight cubature rules are effective.
• To extend existing results on submatrices of orthogonal matrices to $L_1$ and $L_\infty$ norms, especially for function classes of mixed smoothness.
• To analyze the role of entropy numbers in $L_\infty$ for $X_N$ in enabling discretization, particularly for trigonometric and hyperbolic cross polynomials.

Proposed method

• Combines probabilistic chaining techniques with entropy number estimates in the uniform norm to control suprema of random processes over function classes.
• Uses the entropy numbers $\varepsilon_k(X_N^1, L_\infty)$ as a central tool to derive bounds on the number of sampling points required for discretization.
• Applies the chaining method to control the deviation of empirical $L_1$ norms from their true $L_1$ norms over finite-dimensional subspaces.
• Leverages known results on submatrices of orthogonal matrices (e.g., Rudelson and Batson-Spielman-Srivastava theorems) as a foundation for $L_2$-based insights.
• Introduces a dictionary-based approximation scheme using normalized Dirichlet kernels to estimate entropy numbers in higher dimensions.
• Derives bounds on the number of points $m$ needed for $\mathcal{M}(m,1,\varepsilon)$ and $\mathcal{M}^{w}(m,1,\varepsilon)$ theorems via entropy and chaining arguments.

Experimental results

Research questions

• RQ1 Can the number of sampling points required for $L_1$ norm discretization be bounded below $N(\log N)^{7/2}$ for finite-dimensional subspaces?
• RQ2 To what extent do entropy numbers in $L_\infty$ control the feasibility of Marcinkiewicz-type discretization in $L_1$?
• RQ3 Is it possible to achieve $\mathcal{M}(m,1)$ with $m \asymp |Q_n|$ for hyperbolic cross polynomials, or is the $n^{7/2}$ factor unavoidable?
• RQ4 How do the results for $L_1$ compare to known negative results in $L_\infty$, where $m \gg |Q_n|^{1+c}$ is necessary?
• RQ5 Can the entropy number estimate $\varepsilon_k(\mathcal{T}(Q_n)_1, L_\infty) \ll n^{3/2}(|Q_n|/k)\log(4|Q_n|/k)$ be extended to $d > 2$?

Key findings

• For $L_1$ discretization, the number of sampling points $m$ satisfies $m \ll N(\log N)^{7/2}$, nearly matching the ideal $m = N$ case.
• The entropy number estimate $\varepsilon_k(\mathcal{T}(Q_n)_1, L_\infty) \ll n^{1/2}(|Q_n|/k)\log(4|Q_n|/k)$ for $d=2$ is used to derive optimal bounds for $L_1$ discretization.
• The bound $\varepsilon_k(\mathcal{T}(Q_n)_1, L_\infty) \ll n^{3/2}(|Q_n|/k)\log(4|Q_n|/k)$ holds for all $d$, though it is weaker than the $d=2$ version.
• The chaining technique contributes a factor of $n^2$ to the final bound, while entropy estimates contribute $n^{3/2}$, leading to the $n^{7/2}$ factor in the total point count.
• For $L_\infty$ discretization, the paper confirms a known negative result: $m \gg |Q_n|^{1+c}$ is necessary for $\mathcal{M}(m,\infty)$, highlighting the fundamental difficulty of $L_\infty$ discretization.
• The result $X_N(\Omega_M) \in \mathcal{M}^w(m,2,\varepsilon)$ for $m \geq CN\varepsilon^{-2}$ demonstrates the robustness of weighted discretization in $L_2$, though the focus remains on $L_1$.