A low rank ODE for spectral clustering stability
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Spectral clustering is a well-known technique which identifies k clusters in an undirected graph with weight matrix W∈ℝ^n× n by exploiting its graph Laplacian L(W), whose eigenvalues 0=λ_1≤λ_2 ≤…≤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_⋆ with vanishing k-th spectral gap. A more appropriate structured distance to ambiguity such that L_⋆ 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(L(W+Δ))-λ_k(L(W+Δ)), 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_⋆ of smallest Frobenius norm such that F(Δ_⋆)=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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