Understanding Big Data Spectral Clustering

Romain Couillet (1) , Florent Benaych-Georges (2)

1 L2S - Laboratoire des signaux et systèmes
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

This article introduces an original approach to understand the behavior of standard kernel spectral clustering algorithms (such as the Ng–Jordan–Weiss method) for large dimensional datasets. Precisely, using advanced methods from the field of random matrix theory and assuming Gaussian data vectors, we show that the Laplacian of the kernel matrix can asymptotically be well approximated by an analytically tractable equivalent random matrix. The analysis of the former allows one to understand deeply the mechanism into play and in particular the impact of the choice of the kernel function and some theoretical limits of the method. Despite our Gaussian assumption, we also observe that the predicted theoretical behavior is a close match to that experienced on real datasets (taken from the MNIST database).

Domaines: Mathématiques [math]; Sciences de l'ingénieur [physics]; Statistiques [stat]

Romain Couillet, Florent Benaych-Georges. Understanding Big Data Spectral Clustering. IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP 2015) , Dec 2015, Cancun, Mexico. ⟨10.1109/camsap.2015.7383728⟩. ⟨hal-01242494⟩