Title: On the behavior of 1-Laplacian ratio cuts on nearly rectangular domains
Abstract
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 $\varOmega _0 \subset \mathbb{R}^2$, define a sequence of sets $(\varOmega _n)_{n=0}^{\infty }$ where $\varOmega _{n+1}$ is the subset of $\varOmega _n$ where the first eigenfunction of the (properly normalized) Neumann $p$-Laplacian $ -\varDelta ^{(p)} \phi = \lambda _1 |\phi |^{p-2} \phi $ 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 $\partial \varOmega _0$. 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.
Chan, William; Jackson, Stephen; Trang, Nam(
, Forum of Mathematics, Sigma)
Abstract
This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals.
The following summarizes the main results proved under suitable partition hypotheses.
If$\kappa $is a cardinal,$\epsilon < \kappa $,${\mathrm {cof}}(\epsilon ) = \omega $,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and a$\delta < \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if$f \upharpoonright \delta = g \upharpoonright \delta $and$\sup (f) = \sup (g)$, then$\Phi (f) = \Phi (g)$.
If$\kappa $is a cardinal,$\epsilon $is countable,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$holds and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the strong almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and finitely many ordinals$\delta _0, ..., \delta _k \leq \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if for all$0 \leq i \leq k$,$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$, then$\Phi (f) = \Phi (g)$.
If$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\kappa _2$,$\epsilon \leq \kappa $and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere monotonicity property: There is a club$C \subseteq \kappa $so that for all$f,g \in [C]^\epsilon _*$, if for all$\alpha < \epsilon $,$f(\alpha ) \leq g(\alpha )$, then$\Phi (f) \leq \Phi (g)$.
Suppose dependent choice ($\mathsf {DC}$),${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$and the almost everywhere short length club uniformization principle for${\omega _1}$hold. Then every function$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$satisfies a finite continuity property with respect to closure points: Let$\mathfrak {C}_f$be the club of$\alpha < {\omega _1}$so that$\sup (f \upharpoonright \alpha ) = \alpha $. There is a club$C \subseteq {\omega _1}$and finitely many functions$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$so that for all$f \in [C]^{\omega _1}_*$, for all$g \in [C]^{\omega _1}_*$, if$\mathfrak {C}_g = \mathfrak {C}_f$and for all$i < n$,$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$, then$\Phi (g) = \Phi (f)$.
Suppose$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\epsilon _2$for all$\epsilon < \kappa $. For all$\chi < \kappa $,$[\kappa ]^{<\kappa }$does not inject into${}^\chi \mathrm {ON}$, the class of$\chi $-length sequences of ordinals, and therefore,$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$. As a consequence, under the axiom of determinacy$(\mathsf {AD})$, these two cardinality results hold when$\kappa $is one of the following weak or strong partition cardinals of determinacy:${\omega _1}$,$\omega _2$,$\boldsymbol {\delta }_n^1$(for all$1 \leq n < \omega $) and$\boldsymbol {\delta }^2_1$(assuming in addition$\mathsf {DC}_{\mathbb {R}}$).
It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$associated to a domain$$\Omega \subset {\mathbb {R}}^n$$with a uniformly rectifiable boundary$$\Gamma $$of dimension$$d < n-1$$, the now usual distance to the boundary$$D = D_\beta $$given by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$for$$X \in \Omega $$, where$$\beta >0$$and$$\gamma \in (-1,1)$$. In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$satisfies a Carleson measure estimate on$$\Omega $$. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).
Chekuri, Chandra; Quanrud, Kent; Torres, Manuel(
, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms)
The densest subgraph problem in a graph (\dsg), in the simplest
form, is the following. Given an undirected graph $G=(V,E)$ find a
subset $S \subseteq V$ of vertices that maximizes the ratio
$|E(S)|/|S|$ where $E(S)$ is the set of edges with both endpoints
in $S$. \dsg and several of its variants are well-studied in theory
and practice and have many applications in data mining and network
analysis. In this paper we study fast algorithms and structural
aspects of \dsg via the lens of \emph{supermodularity}.
For this we consider the densest supermodular subset problem (\dssp):
given a non-negative supermodular function $f: 2^V
\rightarrow \mathbb{R}_+$, maximize $f(S)/|S|$.
For \dsg we describe a simple flow-based algorithm that outputs a
$(1-\eps)$-approximation in deterministic $\tilde{O}(m/\eps)$ time
where $m$ is the number of edges. Our algorithm is the first to have
a near-linear dependence on $m$ and $1/\eps$ and improves previous
methods based on an LP relaxation. It generalizes to hypergraphs, and
also yields a faster algorithm for directed \dsg.
Greedy peeling algorithms have been very popular for \dsg and
several variants due to their efficiency, empirical performance, and
worst-case approximation guarantees. We describe a simple peeling
algorithm for \dssp and analyze its approximation guarantee in a
fashion that unifies several existing results. Boob et
al.\ \cite{bgpstww-20} developed an \emph{iterative} peeling
algorithm for \dsg which appears to work very well in practice, and
made a conjecture about its convergence to optimality. We
affirmatively answer their conjecture, and in fact prove that a
natural generalization of their algorithm converges to a
$(1-\eps)$-approximation for \emph{any} supermodular function $f$;
the key to our proof is to consider an LP formulation that is
derived via the \Lovasz extension of a supermodular function. For
\dsg the bound on the number of iterations we prove is $O(\frac{\Delta
\ln |V|}{\lambda^*}\cdot \frac{1}{\eps^2})$ where $\Delta$ is
the maximum degree and $\lambda^*$ is the optimum value. Our work suggests
that iterative peeling can be an effective heuristic for
several objectives considered in the literature.
Finally, we show that the $2$-approximation for densest-at-least-$k$
subgraph \cite{ks-09} extends to the supermodular setting. We also
give a unified analysis of the peeling algorithm for this problem, and
via this analysis derive an approximation guarantee for a
generalization of \dssp to maximize $f(S)/g(|S|)$ for a concave
function $g$.
Carlen, Eric; Müller-Hermes, Alexander(
, Letters in Mathematical Physics)
Abstract
Let$$\phi $$be a positive map from the$$n\times n$$matrices$$\mathcal {M}_n$$to the$$m\times m$$matrices$$\mathcal {M}_m$$. It is known that$$\phi $$is 2-positive if and only if for all$$K\in \mathcal {M}_n$$and all strictly positive$$X\in \mathcal {M}_n$$,$$\phi (K^*X^{-1}K) \geqslant \phi (K)^*\phi (X)^{-1}\phi (K)$$. This inequality is not generally true if$$\phi $$is merely a Schwarz map. We show that the corresponding tracial inequality$${{\,\textrm{Tr}\,}}[\phi (K^*X^{-1}K)] \geqslant {{\,\textrm{Tr}\,}}[\phi (K)^*\phi (X)^{-1}\phi (K)]$$holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results.
Henzinger, Monika; Jin, Billy; Peng, Richard; Williamson, David P.(
, Leibniz international proceedings in informatics)
Tauman Kalai, Yael
(Ed.)
Over the last two decades, a significant line of work in theoretical algorithms has made progress in solving linear systems of the form 𝐋𝐱 = 𝐛, where 𝐋 is the Laplacian matrix of a weighted graph with weights w(i,j) > 0 on the edges. The solution 𝐱 of the linear system can be interpreted as the potentials of an electrical flow in which the resistance on edge (i,j) is 1/w(i,j). Kelner, Orrechia, Sidford, and Zhu [Kelner et al., 2013] give a combinatorial, near-linear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law by updating flows around cycles (cycle toggling).
In this paper, we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law by cut toggling: each iteration updates all potentials on one side of a fundamental cut of a spanning tree by the same amount. We prove that this dual algorithm also runs in a near-linear number of iterations.
We show, however, that if we abstract cut toggling as a natural data structure problem, this problem can be reduced to the online vector-matrix-vector problem (OMv), which has been conjectured to be difficult for dynamic algorithms [Henzinger et al., 2015]. The conjecture implies that the data structure does not have an O(n^{1-ε}) time algorithm for any ε > 0, and thus a straightforward implementation of the cut-toggling algorithm requires essentially linear time per iteration.
To circumvent the lower bound, we batch update steps, and perform them simultaneously instead of sequentially. An appropriate choice of batching leads to an Õ(m^{1.5}) time cut-toggling algorithm for solving Laplacian systems. Furthermore, we show that if we sparsify the graph and call our algorithm recursively on the Laplacian system implied by batching and sparsifying, we can reduce the running time to O(m^{1 + ε}) for any ε > 0. Thus, the dual cut-toggling algorithm can achieve (almost) the same running time as its primal cycle-toggling counterpart.
Hamilton, Wesley, Marzuola, Jeremy L., and Wu, Hau-tieng. On the behavior of 1-Laplacian ratio cuts on nearly rectangular domains. Information and Inference: A Journal of the IMA . Web. doi:10.1093/imaiai/iaaa034.
Hamilton, Wesley, Marzuola, Jeremy L., & Wu, Hau-tieng. On the behavior of 1-Laplacian ratio cuts on nearly rectangular domains. Information and Inference: A Journal of the IMA, (). https://doi.org/10.1093/imaiai/iaaa034
Hamilton, Wesley, Marzuola, Jeremy L., and Wu, Hau-tieng.
"On the behavior of 1-Laplacian ratio cuts on nearly rectangular domains". Information and Inference: A Journal of the IMA (). Country unknown/Code not available: Oxford University Press. https://doi.org/10.1093/imaiai/iaaa034.https://par.nsf.gov/biblio/10205611.
@article{osti_10205611,
place = {Country unknown/Code not available},
title = {On the behavior of 1-Laplacian ratio cuts on nearly rectangular domains},
url = {https://par.nsf.gov/biblio/10205611},
DOI = {10.1093/imaiai/iaaa034},
abstractNote = {Abstract 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 $\varOmega _0 \subset \mathbb{R}^2$, define a sequence of sets $(\varOmega _n)_{n=0}^{\infty }$ where $\varOmega _{n+1}$ is the subset of $\varOmega _n$ where the first eigenfunction of the (properly normalized) Neumann $p$-Laplacian $ -\varDelta ^{(p)} \phi = \lambda _1 |\phi |^{p-2} \phi $ 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 $\partial \varOmega _0$. 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.},
journal = {Information and Inference: A Journal of the IMA},
publisher = {Oxford University Press},
author = {Hamilton, Wesley and Marzuola, Jeremy L. and Wu, Hau-tieng},
}
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