# Bias-Corrected Joint Spectral Embedding for Multilayer Networks with Invariant Subspace: Entrywise Eigenvector Perturbation and Inference

arXiv: https://arxiv.org/abs/2406.07849 · HTML: https://mlsanity.com/paper/2406.07849/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 112 references that are in the map.
- First posted on arXiv 12 Jun 2024.

## Builds on (most cited first, 13 of 13)

- [Heteroskedastic PCA: Algorithm, Optimality, and Applications](https://mlsanity.com/paper/1810.08316/) (2018)
- [Inference for Heteroskedastic PCA with Missing Data](https://mlsanity.com/paper/2107.12365/) (2021)
- [Network Representation Using Graph Root Distributions](https://mlsanity.com/paper/1802.09684/) (2018)
- [Bootstrapping Networks with Latent Space Structure](https://mlsanity.com/paper/1907.10821/) (2019)
- [Fundamental Limits of Spectral Clustering in Stochastic Block Models](https://mlsanity.com/paper/2301.09289/) (2023)
- [Spectral Clustering for Multiple Sparse Networks: I](https://mlsanity.com/paper/1805.10594/) (2018)
- [Clustering of Diverse Multiplex Networks](https://mlsanity.com/paper/2110.05308/) (2021)
- [Joint Spectral Clustering in Multilayer Degree-Corrected Stochastic Blockmodels](https://mlsanity.com/paper/2212.05053/) (2022)
- [Limit theorems for out-of-sample extensions of the adjacency and Laplacian spectral embeddings](https://mlsanity.com/paper/1910.00423/) (2019)
- [SIMPLE: Statistical Inference on Membership Profiles in Large Networks](https://mlsanity.com/paper/1910.01734/) (2019)
- [The Sup-norm Perturbation of HOSVD and Low Rank Tensor Denoising](https://mlsanity.com/paper/1707.01207/) (2017)
- [Eigenvector-Assisted Statistical Inference for Signal-Plus-Noise Matrix Models](https://mlsanity.com/paper/2203.16688/) (2022)
- [Multilayer random dot product graphs: Estimation and online change point detection](https://mlsanity.com/paper/2306.15286/) (2023)

## Abstract (arXiv)

In this paper, we propose to estimate the invariant subspace across heterogeneous multiple networks using a novel bias-corrected joint spectral embedding algorithm. The proposed algorithm recursively calibrates the diagonal bias of the sum of squared network adjacency matrices by leveraging the closed-form bias formula and iteratively updates the subspace estimator using the most recent estimated bias. Correspondingly, we establish a complete recipe for the entrywise subspace estimation theory for the proposed algorithm, including a sharp entrywise subspace perturbation bound and the entrywise eigenvector central limit theorem. Leveraging these results, we settle two multiple network inference problems: the exact community detection in multilayer stochastic block models and the hypothesis testing of the equality of membership profiles in multilayer mixed membership models. Our proof relies on delicate leave-one-out and leave-two-out analyses that are specifically tailored to block-wise symmetric random matrices and a martingale argument that is of fundamental interest for the entrywise eigenvector central limit theorem.