A low rank ODE for spectral clustering stability

Spectral clustering is a well-known technique which identifies $k$ clusters in an undirected graph with weight matrix $W\in\mathbb{R}^{n\times n}$ by exploiting its graph Laplacian $L(W)$, whose eigenvalues $0=\lambda1\leq \lambda2 \leq \dots \leq \lambdan$ and eigenvectors are related to the $k$ clusters. Since the computation of $\lambda{k+1}$ and $\lambdak$ affects the reliability of this method, the $k$-th spectral gap $\lambda{k+1}-\lambdak$ is often considered as a stability indicator. This difference can be seen as an unstructured distance between $L(W)$ and an arbitrary symmetric matrix $L\star$ with vanishing $k$-th spectral gap. A more appropriate structured distance to ambiguity such that $L\star$ represents the Laplacian of a graph has been proposed by Andreotti et al. (2021). Slightly differently, we consider the objective functional $ F(\Delta)=\lambda{k+1}\left(L(W+\Delta)\right)-\lambdak\left(L(W+\Delta)\right)$, where $\Delta$ is a perturbation such that $W+\Delta$ has non-negative entries and the same pattern of $W$. We look for an admissible perturbation $\Delta\star$ of smallest Frobenius norm such that $F(\Delta_\star)=0$. In order to solve this optimization problem, we exploit its low rank underlying structure. We formulate a rank-4 symmetric matrix ODE whose stationary points are the optimizers sought. The integration of this equation benefits from the low rank structure with a moderate computational effort and memory requirement, as it is shown in some illustrative numerical examples.