Optimal ℓ0 Isoperimetric Coefficient for Measurable Sets

แนวคิดหลัก

The ℓ0 isoperimetric coefficient of axis-aligned cubes is Θ(n^(-1/2), and the ℓ0 isoperimetric coefficient of any measurable set is O(n^(-1/2)).

บทคัดย่อ

The paper proves the following key results:

The ℓ0 isoperimetric coefficient of axis-aligned cubes, denoted as ψ_C, is Θ(n^(-1/2)). This is shown by first proving a combinatorial analogue on the discrete hypercube and then extending it to the continuous cube.

The ℓ0 isoperimetric coefficient of any measurable set K, denoted as ψ_K, is O(n^(-1/2)). This is shown by extending the approach used for the cube to general measurable sets. The key idea is to partition the set K into axis-disjoint subsets using a random splitting plane, and then bound the ℓ0 boundary of the resulting subsets.

As a corollary, the author shows that axis-aligned cubes essentially "maximize" the ℓ0 isoperimetric coefficient: There exists a positive constant q > 0 such that ψ_K ≤ q * ψ_C for any axis-aligned cube C and any measurable set K.

The improved bounds on the ℓ0 isoperimetry immediately imply improved mixing time bounds for the Coordinate-Hit-and-Run Markov chain used for sampling from convex bodies.

สถิติ

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คำพูด

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ข้อมูลเชิงลึกที่สำคัญจาก

On the $\ell_0$ Isoperimetric Coefficient of Measurable Sets

by Manuel Ferna... ที่ arxiv.org 04-05-2024

https://arxiv.org/pdf/2312.00015.pdf

สอบถามเพิ่มเติม

How do the ℓ0 isoperimetric properties of other geometric shapes, such as spheres or simplices, compare to those of axis-aligned cubes

The ℓ0 isoperimetric properties of other geometric shapes, such as spheres or simplices, differ from those of axis-aligned cubes. In the context of the ℓ0 isoperimetric coefficient, which measures the size of the ℓ0 boundaries of sets relative to their volume, axis-aligned cubes exhibit unique characteristics. For spheres, the ℓ0 isoperimetric coefficient is related to the Hamming distance between points on the sphere, which may not align with the concept of axis-disjointness as in the case of cubes. Similarly, for simplices, the structure and boundary properties differ significantly from cubes, leading to distinct ℓ0 isoperimetric coefficients. The comparison of these properties across different geometric shapes would require a tailored analysis for each shape to determine how they fare in terms of the ℓ0 isoperimetric coefficient.

Can the techniques used in this paper be extended to analyze the ℓ0 isoperimetry of more general classes of measurable sets beyond convex bodies

The techniques used in the paper can potentially be extended to analyze the ℓ0 isoperimetry of more general classes of measurable sets beyond convex bodies. By leveraging concepts such as balanced functions and random splitting planes, it may be possible to develop methods to estimate the ℓ0 isoperimetric coefficient for a wider range of shapes and structures. However, the extension to more general classes of sets would require careful consideration of the specific properties and characteristics of those sets. It may involve adapting the random splitting plane approach or developing new techniques tailored to the unique features of the sets under consideration.

What are the implications of the ℓ0 isoperimetric properties established in this work for other computational problems beyond sampling, such as optimization or integration over high-dimensional spaces

The ℓ0 isoperimetric properties established in this work have significant implications for various computational problems beyond sampling, such as optimization or integration over high-dimensional spaces. The ℓ0 isoperimetric coefficient provides insights into the boundary structure and complexity of sets in high-dimensional spaces, which can impact the efficiency and performance of algorithms in optimization and integration tasks. Understanding the ℓ0 isoperimetric properties of sets can help in designing more effective algorithms that take into account the geometric characteristics of the sets involved. This knowledge can lead to improvements in computational efficiency, accuracy, and convergence rates in a wide range of high-dimensional computational problems.