Reconstructing undirected graphs from eigenspaces - Archive ouverte HAL Access content directly
Journal Articles Journal of Machine Learning Research Year : 2017

## Reconstructing undirected graphs from eigenspaces

Yohann de Castro
Thibault Espinasse
Paul Rochet
• Function : Author

#### Abstract

In this paper, we aim at recovering an undirected weighted graph of $N$ vertices from the knowledge of a perturbed version of the eigenspaces of its adjacency matrix $W$. For instance, this situation arises for stationary signals on graphs or for Markov chains observed at random times. Our approach is based on minimizing a cost function given by the Frobenius norm of the commutator $\mathsf{A} \mathsf{B}-\mathsf{B} \mathsf{A}$ between symmetric matrices $\mathsf{A}$ and $\mathsf{B}$. In the Erd\H{o}s-R\'enyi model with no self-loops, we show that identifiability (i.e., the ability to reconstruct $W$ from the knowledge of its eigenspaces) follows a sharp phase transition on the expected number of edges with threshold function $N\log N/2$. Given an estimation of the eigenspaces based on a $n$-sample, we provide support selection procedures from theoretical and practical point of views. In particular, when deleting an edge from the active support, our study unveils that our test statistic is the order of $\mathcal O(1/n)$ when we overestimate the true support and lower bounded by a positive constant when the estimated support is smaller than the true support. This feature leads to a powerful practical support estimation procedure. Simulated and real life numerical experiments assert our new methodology.

### Dates and versions

hal-01588125 , version 1 (15-09-2017)

### Identifiers

• HAL Id : hal-01588125 , version 1
• ARXIV :

### Cite

Yohann de Castro, Thibault Espinasse, Paul Rochet. Reconstructing undirected graphs from eigenspaces. Journal of Machine Learning Research, 2017. ⟨hal-01588125⟩

### Export

BibTeX TEI Dublin Core DC Terms EndNote Datacite

176 View