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

arXiv: https://arxiv.org/abs/2609.19357 · HTML: https://mlsanity.com/paper/2609.19357/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 25 references that are in the map.
- First posted on arXiv 16 Sep 2026.

## Builds on (most cited first, 9 of 9)

- [Statistical inference on random dot product graphs: a survey](https://mlsanity.com/paper/1709.05454/) (2017)
- [A statistical interpretation of spectral embedding: the generalised random dot product graph](https://mlsanity.com/paper/1709.05506/) (2017)
- [Community Detection and Classification in Hierarchical Stochastic Blockmodels](https://mlsanity.com/paper/1503.02115/) (2015)
- [Manifold structure in graph embeddings](https://mlsanity.com/paper/2006.05168/) (2020)
- [On estimation and inference in latent structure random graphs](https://mlsanity.com/paper/1806.01401/) (2018)
- [Learning 1-Dimensional Submanifolds for Subsequent Inference on Random Dot Product Graphs](https://mlsanity.com/paper/2004.07348/) (2020)
- [Matrix factorisation and the interpretation of geodesic distance](https://mlsanity.com/paper/2106.01260/) (2021)
- [Rehabilitating Isomap: Euclidean Representation of Geodesic Structure](https://mlsanity.com/paper/2006.10858/) (2020)
- [Continuous Multidimensional Scaling](https://mlsanity.com/paper/2402.04436/) (2024)

## Abstract (arXiv)

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.