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Consistency is a key property of all statistical procedures analyzing randomly sampled data. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of…
In this article, the joint fluctuations of the extreme eigenvalues and eigenvectors of a large dimensional sample covariance matrix are analyzed when the associated population covariance matrix is a finite-rank perturbation of the identity…
Spectral clustering refers to a family of unsupervised learning algorithms that compute a spectral embedding of the original data based on the eigenvectors of a similarity graph. This non-linear transformation of the data is both the key of…
We consider the single eigenvalue fluctuations of random matrices of general Wigner-type, under a one-cut assumption on the density of states. For eigenvalues in the bulk, we prove that the asymptotic fluctuations of a single eigenvalue…
Estimating eigenvectors and low-dimensional subspaces is of central importance for numerous problems in statistics, computer science, and applied mathematics. This paper characterizes the behavior of perturbed eigenvectors for a range of…
For random matrices with block correlation structure we show that the fluctuations of linear eigenvalue statistics are Gaussian on all mesoscopic scales with universal variance which coincides with that of the Gaussian unitary or Gaussian…
Based on a Gaussian mixture type model , we derive an eigen selection procedure that improves the usual spectral clustering in high-dimensional settings. Concretely, we derive the asymptotic expansion of the spiked eigenvalues under…
Spectral clustering is a technique that clusters elements using the top few eigenvectors of their (possibly normalized) similarity matrix. The quality of spectral clustering is closely tied to the convergence properties of these principal…
Spectral clustering is a powerful method for finding structure in a dataset through the eigenvectors of a similarity matrix. It often outperforms traditional clustering algorithms such as $k$-means when the structure of the individual…
Clustering in image analysis is a central technique that allows to classify elements of an image. We describe a simple clustering technique that uses the method of similarity matrices. We expand upon recent results in spectral analysis for…
Spectral clustering is a popular algorithm that clusters points using the eigenvalues and eigenvectors of Laplacian matrices derived from the data. For years, spectral clustering has been working mysteriously. This paper explains spectral…
Spectral clustering is one of the most widely used techniques for extracting the underlying global structure of a data set. Compressed sensing and matrix completion have emerged as prevailing methods for efficiently recovering sparse and…
This article proposes a first analysis of kernel spectral clustering methods in the regime where the dimension $p$ of the data vectors to be clustered and their number $n$ grow large at the same rate. We demonstrate, under a $k$-class…
In this article we study the fluctuation of linear statistics of eigenvalues of circulant, symmetric circulant, reverse circulant and Hankel matrices. We show that the linear spectral statistics of these matrices converges to the Gaussian…
This paper is to study a signal-plus-noise model in high dimensional settings when the dimension and the sample size are comparable. Specifically, we assume that the noise has a general covariance matrix that allows for heteroskedasticity,…
We evaluate the misclustering probability of a spectral clustering algorithm under a Gaussian mixture model with a general covariance structure. The algorithm partitions the data into two groups based on the sign of the first principal…
The spectral clustering algorithm is often used as a binary clustering method for unclassified data by applying the principal component analysis. To study theoretical properties of the algorithm, the assumption of conditional…
We study linear spectral statistics of high dimensional sample covariance matrices in a regime where the empirical spectral distribution remains governed by the classical sample covariance law but the fluctuation theory is nonclassical. Our…
Spectral clustering is one of the most popular algorithms to group high dimensional data. It is easy to implement and computationally efficient. Despite its popularity and successful applications, its theoretical properties have not been…
Smooth linear statistics of random permutation matrices, sampled under a general Ewens distribution, exhibit an interesting non-universality phenomenon. Though they have bounded variance, their fluctuations are asymptotically non-Gaussian…