Spectral Perturbation Meets Incomplete Multi-view Data

Beyond existing multi-view clustering, this paper studies a more realistic clustering scenario, referred to as incomplete multi-view clustering, where a number of data instances are missing in certain views. To tackle this problem, we explore spectral perturbation theory. In this work, we show a strong link between perturbation risk bounds and incomplete multi-view clustering. That is, as the similarity matrix fed into spectral clustering is a quantity bounded in magnitude O(1), we transfer the missing problem from data to similarity and tailor a matrix completion method for incomplete similarity matrix. Moreover, we show that the minimization of perturbation risk bounds among different views maximizes the final fusion result across all views. This provides a solid fusion criteria for multi-view data. We motivate and propose a Perturbation-oriented Incomplete multi-view Clustering (PIC) method. Experimental results demonstrate the effectiveness of the proposed method.

VIBRATIONS OF A MEMBRANE WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

We consider the asymptotic behavior of the vibrations of a membrane occupying a domain Ω ⊂ ℝ2. The density, which depends on a small parameter ε, is of order O(1) out of certain regions where it is O(ε−m) with m>0. These regions, the concentrated masses with diameter O(ε), are located near the boundary, at mutual distances O(η), with η=η(ε)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. Depending on the value of the parameter m(m>2, m=2 or m<2) we describe the asymptotic behavior of the eigenvalues. Small eigenvalues, of order O(εm−2) for m>2, are approached via those of a local problem obtained from the micro-structure of the problem, while the eigenvalues of order O(1) are approached through those of a homogenized problem, which depend on the relation between ε and η. Techniques of boundary homogenization and spectral perturbation theory are used to study this problem.