[Paper Review] Gaussian marginals of convex bodies with symmetries

Mark W. Meckes|arXiv (Cornell University)|Jun 3, 2006

Point processes and geometric inequalities24 references5 citations

TL;DR

This paper establishes Gaussian approximation theorems for k-dimensional marginals of symmetric convex bodies using a multivariate Stein's method. It proves that isotropic, log-concave random vectors on 1-unconditional bodies or regular simplices have k-dimensional projections close to Gaussian, with total variation error bounds of order n^{-1/4} for k ≥ 2, improving on prior results via a refined smoothing argument and concentration inequalities.

ABSTRACT

We prove Gaussian approximation theorems for specific $k$-dimensional marginals of convex bodies which possess certain symmetries. In particular, we treat bodies which possess a 1-unconditional basis, as well as simplices. Our results extend recent results for 1-dimensional marginals due to E. Meckes and the author.

Motivation & Objective

• To extend previous 1-dimensional Gaussian approximation results for symmetric convex bodies to higher-dimensional marginals (k ≥ 2).
• To develop a constructive method for identifying specific high-dimensional subspaces where marginals are approximately Gaussian, overcoming the non-constructive nature of Dvoretzky-type theorems.
• To improve quantitative error bounds in Gaussian approximation by combining multivariate Stein's method with concentration inequalities and smoothing techniques from prior work.
• To demonstrate that for isotropic, log-concave measures on symmetric convex bodies (e.g., 1-unconditional bodies and simplices), k-dimensional projections converge to Gaussian with explicit error rates.

Proposed method

• Adapts a multivariate Stein's method of exchangeable pairs developed by Chatterjee and Meckes to analyze k-dimensional projections of symmetric convex bodies.
• Uses a new abstract normal approximation result from Chatterjee and Meckes to bound the total variation distance between the projected measure and a Gaussian distribution.
• Applies a concentration inequality from [14] to strengthen tail estimates for log-concave measures, improving error control.
• Employs a single smoothing argument to convert Kolmogorov distance bounds into total variation bounds, reducing error loss compared to prior two-step smoothing methods.
• Derives error bounds in terms of frame norms and frame vectors associated with the symmetry group of the body, particularly for 1-unconditional bodies and regular simplices.
• Uses the structure of the symmetry group (e.g., reflections and permutations) to define exchangeable pairs and control moments in the Stein method framework.

Experimental results

Research questions

• RQ1 Can the Stein's method approach be extended from 1-dimensional to k-dimensional marginals of symmetric convex bodies for k ≥ 2?
• RQ2 What are the quantitative total variation error bounds for Gaussian approximation of k-dimensional projections of isotropic, log-concave measures on symmetric convex bodies?
• RQ3 How do the error bounds from this method compare to those from classical Berry-Esseen or Klartag's concentration-based methods in terms of rate and loss?
• RQ4 Can a single smoothing argument achieve tighter total variation bounds than the two-step approach used in earlier works?
• RQ5 To what extent do symmetry structures—such as those of 1-unconditional bodies or regular simplices—enable explicit, constructive identification of nearly Gaussian projections?

Key findings

• For k-dimensional marginals of isotropic, log-concave measures on 1-unconditional bodies, the total variation distance to a Gaussian is bounded by O(n^{-1/4}) times a term involving the ℓ4-norms of the projection direction coefficients.
• For marginals on the simplex, the total variation error is bounded by O(n^{-1/4}) times a sum of ℓ4-norms of the frame vectors, with explicit dependence on the symmetry structure.
• The method achieves tighter total variation bounds than prior approaches by using only one smoothing step, reducing error loss compared to the two-step method in [18].
• The error bounds are quantitatively improved over previous results, particularly for the Kolmogorov distance, and the total variation bounds are of the same order as the Kolmogorov bounds.
• The results demonstrate that for large classes of symmetric convex bodies, specific high-dimensional projections (not just existence) are approximately Gaussian, providing a constructive alternative to Dvoretzky-type theorems.
• The framework applies to both 1-unconditional bodies and regular simplices, with the latter relying on the symmetry group generated by permutations and sign flips of the standard basis vectors.