• Corpus ID: 236493608

Large sample spectral analysis of graph-based multi-manifold clustering

  title={Large sample spectral analysis of graph-based multi-manifold clustering},
  author={Nicol{\'a}s Garc{\'i}a Trillos and Pengfei He and Chenghui Li},
In this work we study statistical properties of graph-based algorithms for multi-manifold clustering (MMC). In MMC the goal is to retrieve the multi-manifold structure underlying a given Euclidean data set when this one is assumed to be obtained by sampling a distribution on a union of manifolds M = M1 ∪ · · · ∪ MN that may intersect with each other and that may have different dimensions. We investigate sufficient conditions that similarity graphs on data sets must satisfy in order for their… 



Geometric structure of graph Laplacian embeddings

A notion of a well-separated mixture model which only depends on the model itself is introduced, and it is proved that when the model is well separated, with high probability the embedded data set concentrates on cones that are centered around orthogonal vectors.

Spectral Clustering on Multiple Manifolds

This paper proposes a new method, called spectral multi-manifold clustering (SMMC), which is able to handle intersections, and demonstrates the promising performance of this method on synthetic as well as real datasets.

The geometry of kernelized spectral clustering

This work studies the performance of spectral clustering in recovering the latent labels of i.i.d. samples from a finite mixture of nonparametric distributions and controls the fraction of samples mislabeled under finite mixtures with non Parametric components.

Sparse Manifold Clustering and Embedding

An algorithm called Sparse Manifold Clustering and Embedding (SMCE) for simultaneous clustering and dimensionality reduction of data lying in multiple nonlinear manifolds finds a small neighborhood around each data point and connects each point to its neighbors with appropriate weights.

Path-Based Spectral Clustering: Guarantees, Robustness to Outliers, and Fast Algorithms

This work provides conditions under which the Laplacian eigengap statistic correctly determines the number of clusters for a large class of data sets, and proves finite-sample guarantees on the performance of clustering with respect to this metric when random samples are drawn from multiple intrinsically low-dimensional clusters in high-dimensional space.

Lipschitz regularity of graph Laplacians on random data clouds

This paper proves high probability interior and global Lipschitz estimates for solutions of graph Poisson equations, and obtains high probability and approximate convergence rates for the convergence of graph Laplacian eigenvectors towards eigenfunctions of the corresponding weighted Laplace-Beltrami operators.

Graph Laplacians and their Convergence on Random Neighborhood Graphs

This paper determines the pointwise limit of three different graph Laplacians used in the literature as the sample size increases and the neighborhood size approaches zero and shows that for a uniform measure on the submanifold all graph LaPLacians have the same limit up to constants.

Spectral Convergence of the connection Laplacian from random samples

This paper proves that the eigenvectors and eigenvalues of these Laplacians converge in the limit of infinitely many independent random samples, and generalizes the spectral convergence results to the case where the data points are sampled from a non-uniform distribution, and for manifolds with and without boundary.

A variational approach to the consistency of spectral clustering