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alignment cost

Alignment cost is a quantitative metric used in spectral clustering to evaluate how closely a rotated set of leading eigenvectors aligns with an ideal, discrete cluster indicator structure. In this framework, the continuous representations derived from the spectral decomposition of an affinity or Laplacian matrix are rotated to match canonical coordinate axes, where each data point ideally exhibits a non-zero value along only a single cluster dimension. The alignment cost measures the deviation or residual error of the rotated embedding from this orthogonal partition structure. Minimizing this cost enables the direct assignment of data points to clusters without relying on iterative post-processing heuristics such as k-means, while also providing a criterion to automatically determine the optimal number of clusters by selecting the candidate dimension with the minimal cost.

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Self-Tuning Spectral Clustering

Self-Tuning Spectral Clustering

Lihi Zelnik-Manor, P. Perona

OrganizationsCalifornia Institute of Technology

Why you should read this

Introduces a self-tuning spectral clustering method that computes local scaling for each data point and infers the number of groups directly from eigenvector structure, effectively clustering multi-scale, cluttered datasets without manual parameter selection or randomized k-means initialization.

We study a number of open issues in spectral clustering: (i) Selecting the appropriate scale of analysis, (ii) Handling multi-scale data, (iii) Clustering with irregular background clutter, and, (iv) Finding automatically the number of groups. We first propose that a ‘local’ scale should be used to compute the affinity between each pair of points. This local scaling leads to better clustering especially when the data includes multiple scales and when the clusters are placed within a cluttered background. We further suggest exploiting the structure of the eigenvectors to infer automatically the number of groups. This leads to a new algorithm in which the final randomly initialized k-means stage is eliminated.

Added

2026-09-14