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Finding the convex envelope of a boundary datum using random geometric graphs

Aurelia Deshayes, Nicolás Frevenza, Alfredo Miranda, Julio D. Rossi

TL;DR

This work shows that the convex envelope of a boundary datum inside a bounded domain can be recovered as the uniform limit of a value function from a one-player game on a random geometric graph. By constructing a proximity graph on $n$ i.i.d. points in $[0,1]^d$ with radius $r_n$ and encoding convexity via a discrete second-order scheme that uses an annular neighbor set and a reflected point, the authors connect graph-based dynamics to the continuous convex envelope problem. Under a superconnectivity scaling $nr_n^d/\log n\to\infty$ and carefully chosen thickness parameters $δ_n$, they prove that the extended discrete value $\tilde{u}_n$ converges uniformly to the unique viscosity solution of the continuous problem $λ1[D^2u](x)=0$ in $D$ with boundary data on $∂D$. The results bridge discrete stochastic games on graphs with PDE-based convex analysis, with potential applications in graph-based semi-supervised learning and PDE approximations on random data.

Abstract

In this paper we approximate the convex envelope of a boundary datum inside a bounded domain in the Euclidean space. We work with a random graph that is obtained as random points with uniform distribution that are connected by proximity ($x\sim y$ when $|x-y|<r$). On the graph we solve an equation (that approximate the first eigenvalue of the Hessian of a smooth function) with an exterior datum. Under appropriate assumptions on $r$ we show that the unique solution to the equation in the graph converges to the convex envelope of the boundary datum as the number of points goes to infinity.

Finding the convex envelope of a boundary datum using random geometric graphs

TL;DR

This work shows that the convex envelope of a boundary datum inside a bounded domain can be recovered as the uniform limit of a value function from a one-player game on a random geometric graph. By constructing a proximity graph on i.i.d. points in with radius and encoding convexity via a discrete second-order scheme that uses an annular neighbor set and a reflected point, the authors connect graph-based dynamics to the continuous convex envelope problem. Under a superconnectivity scaling and carefully chosen thickness parameters , they prove that the extended discrete value converges uniformly to the unique viscosity solution of the continuous problem in with boundary data on . The results bridge discrete stochastic games on graphs with PDE-based convex analysis, with potential applications in graph-based semi-supervised learning and PDE approximations on random data.

Abstract

In this paper we approximate the convex envelope of a boundary datum inside a bounded domain in the Euclidean space. We work with a random graph that is obtained as random points with uniform distribution that are connected by proximity ( when ). On the graph we solve an equation (that approximate the first eigenvalue of the Hessian of a smooth function) with an exterior datum. Under appropriate assumptions on we show that the unique solution to the equation in the graph converges to the convex envelope of the boundary datum as the number of points goes to infinity.
Paper Structure (7 sections, 10 theorems, 101 equations)

This paper contains 7 sections, 10 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. General convexity
  3. An informal description of the main results
  4. Setting and statements of the results
  5. Some properties of Gn
  6. Proof of Theorem \ref{['game.value.holds.DPP']}
  7. Proof of Theorem \ref{['game.value.convergence']}

Key Result

Theorem 1

Let $(r_n)_{n\in\mathbb{N}}$ and $(\delta_n)_{n\in\mathbb{N}}$ be sequences satisfying the condition condicion.parametros. Then, $\mathbb{P}$-almost surely, for $n$ sufficiently large, the game value $u_n\colon {\mathcal{C}}_n\to {\mathbb R}$ defined in game.value is the unique solution of the follo

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3: Comparison Principle
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 8 more