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 $0=\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_n$ and eigenvectors are related to the $k$ clusters. Since the computation of $\lambda_{k+1}$ and $\lambda_k$ affects the reliability of this method, the $k$-th spectral gap $\lambda_{k+1}-\lambda_k$ 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)-\lambda_k\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.

翻译：谱聚类是一种众所周知的技术，通过利用图拉普拉斯矩阵 $L(W)$（其特征值 $0=\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_n$ 及特征向量与 $k$ 个聚类相关）来识别无向图（其权重矩阵为 $W\in\mathbb{R}^{n\times n}$）中的 $k$ 个聚类。由于 $\lambda_{k+1}$ 和 $\lambda_k$ 的计算会影响该方法的可靠性，第 $k$ 个谱间隙 $\lambda_{k+1}-\lambda_k$ 常被视为稳定性指标。这个差值可视为 $L(W)$ 与任意具有消失的第 $k$ 个谱间隙的对称矩阵 $L_\star$ 之间的非结构化距离。Andreotti 等人（2021）提出了一种更合适的、考虑 $L_\star$ 表示图的拉普拉斯矩阵的结构化模糊距离。略有不同地，我们考虑目标泛函 $F(\Delta)=\lambda_{k+1}\left(L(W+\Delta)\right)-\lambda_k\left(L(W+\Delta)\right)$，其中 $\Delta$ 是一个扰动，使得 $W+\Delta$ 具有非负元素且与 $W$ 保持相同模式。我们寻找最小 Frobenius 范数的可容许扰动 $\Delta_\star$，使得 $F(\Delta_\star)=0$。为了解决这个优化问题，我们利用其底层低秩结构。我们构造了一个秩为4的对称矩阵常微分方程，其驻点即为所求的优化值。该方程的积分受益于低秩结构，计算开销和内存需求适中，这在一些说明性数值示例中得到了展示。