Learning Submanifolds for Subsequent Inference on Random Dot Product Graphs, Part 1: Theory

2026-09-16

AI Summary

本文提出了一种框架，通过Isomap流形学习方法来推断随机点积图的低维支撑流形，并使用半监督决策规则进行推理。

Abstract

We propose a framework for restricted inference on random dot product graphs whose latent positions lie on an unknown low-dimensional support manifold. For general decision problems, we propose semisupervised decision rules that use auxiliary data to learn the support manifold. Specifically, our rules use the Isomap manifold learning procedure to construct a low-dimensional Euclidean representation of the observed graph, in which space an isometrically invariant function maps configurations of points to actions. We study the behavior of the proposed rules as the quantity of auxiliary data sampled from the unknown support manifold increases. We show that, as the auxiliary sample size increases, the risk of the semisupervised rule converges to the risk of an oracle rule that relies on the maximal amount of low-dimensional Euclidean structure that can be extracted from the support manifold. Examples, applications, and simulation studies are deferred to a sequel.

Problem

• random dot product graphs
• latent positions
• low-dimensional support manifold
• restricted inference
• semisupervised decision rules

Innovation

• random dot product graphs
• low-dimensional support manifold
• semisupervised decision rules
• Isomap manifold learning
• auxiliary data