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Convergence rates for Poisson learning to a Poisson equation with measure data

Leon Bungert, Jeff Calder, Max Mihailescu, Kodjo Houssou, Amber Yuan

TL;DR

This work establishes rigorous, high-probability discrete-to-continuum convergence rates for Poisson learning, a graph-based semi-supervised method that propagates labels via a graph Poisson equation with measure-valued data on labeled points. The authors couple three core ideas—convolving measure data to mollify singular data, variational stability to compare discrete and continuum energies, and graph heat-kernel regularization to bridge nonlocal and local descriptions—to prove $L^1$-type convergence rates like $O(\varepsilon^{\frac{1}{d+2}})$ for general data and improved rates for uniform density, under suitable scaling $\varepsilon$ with the number of points $n$ and dimension $d$. They develop a comprehensive framework for Poisson equations with measure data, including Green's functions, stability, mollification, and improved rates at constant density, and then extend the analysis to the graph setting through transportation maps, mollified graph operators, and meticulous nonlocal-to-local energy comparisons. A fine analysis of the graph heat kernel yields precise nonlocal averaging asymptotics and justifies the mollification/remormalization approach, enabling robust label propagation even at very low label rates. The results provide quantitative guarantees for the effectiveness of Poisson learning on random geometric graphs and lay the groundwork for the PWLL method and related reweighting strategies in data-driven PDE-inspired learning.

Abstract

In this paper we prove discrete to continuum convergence rates for Poisson Learning, a graph-based semi-supervised learning algorithm that is based on solving the graph Poisson equation with a source term consisting of a linear combination of Dirac deltas located at labeled points and carrying label information. The corresponding continuum equation is a Poisson equation with measure data in a Euclidean domain $Ω\subset \mathbb{R}^d$. The singular nature of these equations is challenging and requires an approach with several distinct parts: (1) We prove quantitative error estimates when convolving the measure data of a Poisson equation with (approximately) radial function supported on balls. (2) We use quantitative variational techniques to prove discrete to continuum convergence rates on random geometric graphs with bandwidth $\varepsilon>0$ for bounded source terms. (3) We show how to regularize the graph Poisson equation via mollification with the graph heat kernel, and we study fine asymptotics of the heat kernel on random geometric graphs. Combining these three pillars we obtain $L^1$ convergence rates that scale, up to logarithmic factors, like $O(\varepsilon^{\frac{1}{d+2}})$ for general data distributions, and $O(\varepsilon^{\frac{2-σ}{d+4}})$ for uniformly distributed data, where $σ>0$. These rates are valid with high probability if $\varepsilon\gg\left({\log n}/{n}\right)^q$ where $n$ denotes the number of vertices of the graph and $q \approx \frac{1}{3d}$.

Convergence rates for Poisson learning to a Poisson equation with measure data

TL;DR

This work establishes rigorous, high-probability discrete-to-continuum convergence rates for Poisson learning, a graph-based semi-supervised method that propagates labels via a graph Poisson equation with measure-valued data on labeled points. The authors couple three core ideas—convolving measure data to mollify singular data, variational stability to compare discrete and continuum energies, and graph heat-kernel regularization to bridge nonlocal and local descriptions—to prove -type convergence rates like for general data and improved rates for uniform density, under suitable scaling with the number of points and dimension . They develop a comprehensive framework for Poisson equations with measure data, including Green's functions, stability, mollification, and improved rates at constant density, and then extend the analysis to the graph setting through transportation maps, mollified graph operators, and meticulous nonlocal-to-local energy comparisons. A fine analysis of the graph heat kernel yields precise nonlocal averaging asymptotics and justifies the mollification/remormalization approach, enabling robust label propagation even at very low label rates. The results provide quantitative guarantees for the effectiveness of Poisson learning on random geometric graphs and lay the groundwork for the PWLL method and related reweighting strategies in data-driven PDE-inspired learning.

Abstract

In this paper we prove discrete to continuum convergence rates for Poisson Learning, a graph-based semi-supervised learning algorithm that is based on solving the graph Poisson equation with a source term consisting of a linear combination of Dirac deltas located at labeled points and carrying label information. The corresponding continuum equation is a Poisson equation with measure data in a Euclidean domain . The singular nature of these equations is challenging and requires an approach with several distinct parts: (1) We prove quantitative error estimates when convolving the measure data of a Poisson equation with (approximately) radial function supported on balls. (2) We use quantitative variational techniques to prove discrete to continuum convergence rates on random geometric graphs with bandwidth for bounded source terms. (3) We show how to regularize the graph Poisson equation via mollification with the graph heat kernel, and we study fine asymptotics of the heat kernel on random geometric graphs. Combining these three pillars we obtain convergence rates that scale, up to logarithmic factors, like for general data distributions, and for uniformly distributed data, where . These rates are valid with high probability if where denotes the number of vertices of the graph and .
Paper Structure (38 sections, 77 theorems, 556 equations, 1 figure)

This paper contains 38 sections, 77 theorems, 556 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Setting and main results
  4. Calculus on general graphs
  5. Calculus on random geometric graphs
  6. Notation
  7. Poisson equations with measure data and their approximation
  8. Existence of solutions
  9. Green's functions
  10. Regularity for regular data
  11. Stability
  12. Convergence rates for mollified data
  13. Improved rates for constant density
  14. Convergence rates of Poisson learning for smooth data
  15. Discrete to local convergence rate
  16. ...and 23 more sections

Key Result

Proposition 2.2

Let $\Omega$ satisfy ass:omega_lipschitz and $\varrho$ satisfy ass:rho_bounded. Moreover, let $f\in\mathcal{M}\left(\overline\Omega\right)$ be a finite, real-valued Radon measure with $f({\overline\Omega})=0$. Assume there exists a family $\left\{f_n\right\} \subset W^{-1,2}(\Omega)\cap L^1(\Omega)$ Then, there exist unique functions $u_n \in W^{1,2}(\Omega)$ which are weak solutions to eq:continu

Figures (1)

  • Figure 1: Comparisons of (a) Laplace Learning (b) Poisson Learning and (c) Poisson Weighted Laplace Learning (PWLL) for a problem with two points in opposite classes, with labels of $+1$ and $-1$. Laplace learning develops spikes, while Poisson Learning approximates the fundamental solution of Laplace's equation, and PWLL smoothly interpolates the labels.

Theorems & Definitions (183)

  • Remark 1.5: The symbol $\lesssim$
  • Remark 1.6: The big-${\mathcal{O}}$ notation
  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3: Related results
  • Remark 2.4: Uniqueness
  • Remark 2.5: Larger class of test functions
  • Remark 2.6: Approximating sequences
  • proof : Proof of \ref{['prop:existence_solution_continuum_pde']}
  • Definition 2.7
  • ...and 173 more