# Uncovering Locally Low-dimensional Structure in Networks by Locally Optimal Spectral Embedding

arXiv: https://arxiv.org/abs/2603.11965 · HTML: https://mlsanity.com/paper/2603.11965/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 69 references that are in the map.
- First posted on arXiv 12 Mar 2026.

## Builds on (most cited first, 28 of 28)

- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [DeepWalk: Online Learning of Social Representations](https://mlsanity.com/paper/1403.6652/) (2014)
- [Consistency of spectral clustering in stochastic block models](https://mlsanity.com/paper/1312.2050/) (2013)
- [Knowledge Graphs: Opportunities and Challenges](https://mlsanity.com/paper/2303.13948/) (2023)
- [A statistical interpretation of spectral embedding: the generalised random dot product graph](https://mlsanity.com/paper/1709.05506/) (2017)
- [Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel](https://mlsanity.com/paper/1309.4111/) (2013)
- [Optimality of Spectral Clustering in the Gaussian Mixture Model](https://mlsanity.com/paper/1911.00538/) (2019)
- [The impossibility of low rank representations for triangle-rich complex networks](https://mlsanity.com/paper/2003.12635/) (2020)
- [Statistical exploration of the Manifold Hypothesis](https://mlsanity.com/paper/2208.11665/) (2022)
- [Perfect Clustering for Stochastic Blockmodel Graphs via Adjacency Spectral Embedding](https://mlsanity.com/paper/1310.0532/) (2013)
- [Manifold structure in graph embeddings](https://mlsanity.com/paper/2006.05168/) (2020)
- [On the Power of Edge Independent Graph Models](https://mlsanity.com/paper/2111.00048/) (2021)
- [Spectral Clustering Based on Local PCA](https://mlsanity.com/paper/1301.2007/) (2013)
- [A Short Survey on Importance Weighting for Machine Learning](https://mlsanity.com/paper/2403.10175/) (2024)
- [Network Representation Using Graph Root Distributions](https://mlsanity.com/paper/1802.09684/) (2018)
- [Graph analysis of functional brain networks: practical issues in translational neuroscience](https://mlsanity.com/paper/1406.7391/) (2014)
- [Asymptotically efficient estimators for stochastic blockmodels: the naive MLE, the rank-constrained MLE, and the spectral](https://mlsanity.com/paper/1710.10936/) (2017)
- [Spectral clustering on spherical coordinates under the degree-corrected stochastic blockmodel](https://mlsanity.com/paper/2011.04558/) (2020)
- [Matrix factorisation and the interpretation of geodesic distance](https://mlsanity.com/paper/2106.01260/) (2021)
- [Popularity Adjusted Block Models are Generalized Random Dot Product Graphs](https://mlsanity.com/paper/2109.04010/) (2021)
- [Consistency of adjacency spectral embedding for the mixed membership stochastic blockmodel](https://mlsanity.com/paper/1705.04518/) (2017)
- [Weighted Spectral Embedding of Graphs](https://mlsanity.com/paper/1809.11115/) (2018)
- [Local2Global: A distributed approach for scaling representation learning on graphs](https://mlsanity.com/paper/2201.04729/) (2022)
- [Fine-Grained Network Analysis for Modern Software Ecosystems](https://mlsanity.com/paper/2012.04760/) (2020)
- [Implications of sparsity and high triangle density for graph representation learning](https://mlsanity.com/paper/2210.15277/) (2022)
- [T-Stochastic Graphs](https://mlsanity.com/paper/2309.01301/) (2023)

## Abstract (arXiv)

Standard Adjacency Spectral Embedding (ASE) relies on a global low-rank assumption often incompatible with the sparse, transitive structure of real-world networks, causing local geometric features to be 'smeared'. To address this, we introduce Local Adjacency Spectral Embedding (LASE), which uncovers locally low-dimensional structure via weighted spectral decomposition. Under a latent position model with a kernel feature map, we treat the image of latent positions as a locally low-dimensional set in infinite-dimensional feature space. We establish finite-sample bounds quantifying the trade-off between the statistical cost of localisation and the reduced truncation error achieved by targeting a locally low-dimensional region of the embedding. Furthermore, we prove that sufficient localisation induces rapid spectral decay and the emergence of a distinct spectral gap, theoretically justifying low-dimensional local embeddings. Experiments on synthetic and real networks show that LASE improves local reconstruction and visualisation over global and subgraph baselines, and we introduce UMAP-LASE for assembling overlapping local embeddings into high-fidelity global visualisations.