用于谱聚类稳定性的低秩 ODE
A low rank ODE for spectral clustering stability
AI总结：
本文研究谱聚类谱间隙消失的结构化扰动问题，提出秩-4对称矩阵 ODE 来求解最小 Frobenius 范数容许扰动，并通过数值算例验证其低计算与内存成本。
摘要：
谱聚类是一种著名技术，它通过利用无向图的图 Laplacian 矩阵 $L(W)$，在权重矩阵为 $W\in\mathbb{R}^{n\times n}$ 的无向图中识别 $k$ 个簇；该矩阵的特征值 $0=λ_1\leq λ_2 \leq \dots \leq λ_n$ 及特征向量与这 $k$ 个簇相关。由于 $λ_{k+1}$ 和 $λ_k$ 的计算会影响该方法的可靠性，第 $k$ 个谱间隙 $λ_{k+1}-λ_k$ 常被视为稳定性指标。这一差值可看作 $L(W)$ 与任意第 $k$ 个谱间隙消失的对称矩阵 $L_\star$ 之间的非结构化距离。Andreotti 等人（2021）提出了一种更合适的、面向歧义的结构化距离，使 $L_\star$ 表示某个图的 Laplacian 矩阵。略有不同的是，我们考虑目标泛函 $F(Δ)=λ_{k+1}(L(W+Δ))-λ_k(L(W+Δ))$，其中 $Δ$ 是一种扰动，使 $W+Δ$ 具有非负元素并与 $W$ 具有相同模式。我们寻找 Frobenius 范数最小且满足 $F(Δ_\star)=0$ 的容许扰动 $Δ_\star$。为求解该优化问题，我们利用其潜在的低秩结构，构造了一个秩为 4 的对称矩阵 ODE，其驻点正是所寻求的优化器。如若干说明性数值算例所示，该方程的积分受益于低秩结构，只需适中的计算量和内存需求。
英文摘要：
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 $0=λ_1\leq λ_2 \leq \dots \leq λ_n$ and eigenvectors are related to the $k$ clusters. Since the computation of $λ_{k+1}$ and $λ_k$ affects the reliability of this method, the $k$-th spectral gap $λ_{k+1}-λ_k$ 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(Δ)=λ_{k+1}\left(L(W+Δ)\right)-λ_k\left(L(W+Δ)\right)$, where $Δ$ is a perturbation such that $W+Δ$ has non-negative entries and the same pattern of $W$. We look for an admissible perturbation $Δ_\star$ of smallest Frobenius norm such that $F(Δ_\star)=0$. 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.
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