# Hypothesis testing on invariant subspaces of non-diagonalizable matrices with applications to network statistics

arXiv: https://arxiv.org/abs/2303.18233 · HTML: https://mlsanity.com/paper/2303.18233/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 42 references that are in the map.
- First posted on arXiv 31 Mar 2023.

## Builds on (most cited first, 8 of 8)

- [Statistical inference on random dot product graphs: a survey](https://mlsanity.com/paper/1709.05454/) (2017)
- [backShift: Learning causal cyclic graphs from unknown shift interventions](https://mlsanity.com/paper/1506.02494/) (2015)
- [On the folded normal distribution](https://mlsanity.com/paper/1402.3559/) (2014)
- [Covariate-assisted spectral clustering](https://mlsanity.com/paper/1411.2158/) (2014)
- [Subsampling Sparse Graphons Under Minimal Assumptions](https://mlsanity.com/paper/1907.12528/) (2019)
- [Optimal Bayesian Estimation for Random Dot Product Graphs](https://mlsanity.com/paper/1904.12070/) (2019)
- [Testing Simultaneous Diagonalizability](https://mlsanity.com/paper/2101.07776/) (2021)
- [Network regression and supervised centrality estimation](https://mlsanity.com/paper/2111.12921/) (2021)

## Abstract (arXiv)

We generalise the inference procedure for eigenvectors of symmetrizable matrices of Tyler (1981) to that of invariant and singular subspaces of non-diagonalizable matrices. Wald tests for invariant vectors and t-tests for their individual coefficients perform well in simulations, despite the matrix being not symmetric. Using these results, it is now possible to perform inference on network statistics that depend on eigenvectors of non-symmetric adjacency matrices as they arise in empirical applications from directed networks. Further, we find that statisticians only need control over the first-order Davis-Kahan bound to control convergence rates of invariant subspace estimators to higher-orders. For general invariant subspaces, the minimal eigenvalue separation dominates the first-order bound potentially slowing convergence rates considerably. In an example, we find that accounting for uncertainty in network estimates changes empirical conclusions about the ranking of nodes' popularity.