Gaussian Approximation of Convex Sets by Intersections of Halfspaces (2311.08575v1)

Abstract: We study the approximability of general convex sets in $\mathbb{R}^n$ by intersections of halfspaces, where the approximation quality is measured with respect to the standard Gaussian distribution $N(0,I_n)$ and the complexity of an approximation is the number of halfspaces used. While a large body of research has considered the approximation of convex sets by intersections of halfspaces under distance metrics such as the Lebesgue measure and Hausdorff distance, prior to our work there has not been a systematic study of convex approximation under the Gaussian distribution. We establish a range of upper and lower bounds, both for general convex sets and for specific natural convex sets that are of particular interest. Our results demonstrate that the landscape of approximation is intriguingly different under the Gaussian distribution versus previously studied distance measures. For example, we show that $2^{\Theta(\sqrt{n})}$ halfspaces are both necessary and sufficient to approximate the origin-centered $\ell_2$ ball of Gaussian volume 1/2 to any constant accuracy, and that for $1 \leq p < 2$, the origin-centered $\ell_p$ ball of Gaussian volume 1/2 can be approximated to any constant accuracy as an intersection of $2^{\widetilde{O}(n^{3/4})}$ many halfspaces. These bounds are quite different from known approximation results under more commonly studied distance measures. Our results are proved using techniques from many different areas. These include classical results on convex polyhedral approximation, Cram\'er-type bounds on large deviations from probability theory, and -- perhaps surprisingly -- a range of topics from computational complexity, including computational learning theory, unconditional pseudorandomness, and the study of influences and noise sensitivity in the analysis of Boolean functions.

• The paper demonstrates that 2^(Θ(√n)) halfspaces are necessary and sufficient to approximate an origin-centered ℓ2 ball of Gaussian volume 1/2 with constant accuracy.
• The study extends its analysis to approximating ℓp balls for 1 ≤ p < 2 using 2^(Õ(n^(3/4))) halfspaces, revealing unique behaviors under Gaussian measures.
• Methodologically, it integrates convex approximation techniques, probability theory, and computational complexity to inform efficient algorithms in Gaussian settings.

An Expert Overview of "Gaussian Approximation of Convex Sets by Intersections of Halfspaces"

The paper presented in the paper "Gaussian Approximation of Convex Sets by Intersections of Halfspaces" explores the complex area of approximating convex sets within the field of $\R^n$. The measurable approximation quality, evaluated against the standard Gaussian distribution, marks a key differentiator from previous studies that primarily focused on distance metrics like the Lebesgue measure and the Hausdorff distance.

Core Contributions

1. Bounding Complexity with Gaussian Measures: A significant segment of the paper focuses on establishing both upper and lower bounds for the approximation of convex sets when considering the Gaussian measure. One of the standout results is the demonstration that $2^{\Theta(\sqrt{n})}$ halfspaces are necessary and sufficient for approximating an origin-centered $\ell_2$ ball of Gaussian volume 1/2 to any constant accuracy. This forms a central result given the discrepancy observed when comparing to previously established distance measures.
2. Approximation of Specific Convex Bodies in Gaussian Space: Further to the general bounds provided, the research also extends to the approximation of specific convex sets of natural interest, such as the $\ell_p$ balls. Notably, for $1 \leq p < 2$, the origin-centered $\ell_p$ ball can be approximated using an intersection of $2^{\tilde{O}(n^{3/4})}$ halfspaces to any constant accuracy—another distinct result in comparison to prior methods.
3. Methodological Advances: A combination of techniques from various domains underpin the results. Insights are drawn from traditional convex polyhedral approximation methods, complemented by probability theory (in particular, Cramer-type bounds) and computational complexity perspectives such as learning theory and pseudorandomness.

Implications and Further Research

The implications of this paper are mainly twofold: theoretical and practical. Theoretically, it establishes a divergent behavior between Gaussian measures and other metric spaces when it comes to polyhedral approximation, thereby enriching the understanding of Gaussian space's unique properties in regard to convex bodies. Practically, the results hint at more efficient algorithm designs for applications in optimization and learning theory within Gaussian-distributed datasets.

Future avenues for research could explore finer-grained results that delineate other specific convex bodies' approximation behaviors under the Gaussian distribution, or the applicability of these results to general computational processes involving large-scale Gaussian datasets.

Conclusion

In summary, this paper provides a groundbreaking analysis of convex set approximation in high-dimensional Gaussian spaces using intersections of halfspaces. The outcomes reveal nuanced behaviors not previously seen in classical convex approximation laws, opening up novel pathways and considerations both theoretically and practically for future work within the intersection of geometry, probability, and computational complexity.