On the behavior of 1-Laplacian ratio cuts on nearly rectangular domains

The p-Laplacian has attracted more and more attention in data analysis disciplines in the past decade. However, there is still a knowledge gap about its behavior, which limits its practical application. In this paper, we are interested in its iterative behavior in domains contained in two-dimensional Euclidean space. Given a connected set Ω0 ⊂ R2, define a sequence of sets (Ωn)∞n=0 where Ωn+1 is the subset of Ωn where the first eigenfunction of the (properly normalized) Neumann p-Laplacian −Δ(p)φ = λ1|φ|p−2φ is positive (or negative). For p = 1, this is also referred to as the ratio cut of the domain. We conjecture that these sets converge to the set of rectangles with eccentricity bounded by 2 in the Gromov–Hausdorff distance as long as they have a certain distance to the boundary ∂Ω0. We establish some aspects of this conjecture for p = 1 where we prove that (1) the 1-Laplacian spectral cut of domains sufficiently close to rectangles is a circular arc that is closer to flat than the original domain (leading eventually to quadrilaterals) and (2) quadrilaterals close to a rectangle of aspect ratio 2 stay close to quadrilaterals and move closer to rectangles in a suitable metric. We also discuss some numerical aspects and pose many open questions.

Duke Scholars

Author Hau-Tieng Wu Mathematics