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

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.

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BibTeX RIS

BibTeX

@misc{trosset2026learning,
  title = {{Learning Submanifolds for Subsequent Inference on Random Dot Product Graphs, Part 1: Theory}},
  author = {Michael W. Trosset and Carey E. Priebe},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19357},
  eprint = {2609.19357},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19357},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Learning Submanifolds for Subsequent Inference on Random Dot Product Graphs, Part 1: Theory
AU  - Michael W. Trosset
AU  - Carey E. Priebe
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.19357
UR  - https://arxiv.org/abs/2609.19357
AB  - 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.
ER  -

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