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

   Little, Anna V ; Maggioni, Mauro ; Murphy, James M ( January 2020 , Journal of machine learning research)

   We consider the problem of clustering with the longest-leg path distance (LLPD) metric, which is informative for elongated and irregularly shaped clusters. We prove 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, in the presence of a large number of high-dimensional outliers. By combining these results with spectral clustering with respect to LLPD, we provide conditions under which the Laplacian eigengap statistic correctly determines the number of clusters for a large class of data sets, and prove guarantees on the labeling accuracy of the proposed algorithm. Our methods are quite general and provide performance guarantees for spectral clustering with any ultrametric. We also introduce an efficient, easy to implement approximation algorithm for the LLPD based on a multiscale analysis of adjacency graphs, which allows for the runtime of LLPD spectral clustering to be quasilinear in the number of data points. 

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

   Little, Anna ; Maggioni, Mauro ; Murphy, James M. ( January 2020 , Journal of machine learning research)

   null (Ed.)

   We consider the problem of clustering with the longest-leg path distance (LLPD) metric, which is informative for elongated and irregularly shaped clusters. We prove 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, in the presence of a large number of high-dimensional outliers. By combining these results with spectral clustering with respect to LLPD, we provide conditions under which the Laplacian eigengap statistic correctly determines the number of clusters for a large class of data sets, and prove guarantees on the labeling accuracy of the proposed algorithm. Our methods are quite general and provide performance guarantees for spectral clustering with any ultrametric. We also introduce an efficient, easy to implement approximation algorithm for the LLPD based on a multiscale analysis of adjacency graphs, which allows for the runtime of LLPD spectral clustering to be quasilinear in the number of data points. 

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3. Hyperspectral Image Clustering with Spatially-Regularized Ultrametrics

   https://doi.org/10.3390/rs13050955  

   Zhang, Shukun ; Murphy, James M. ( March 2021 , Remote Sensing)

   null (Ed.)

   We propose a method for the unsupervised clustering of hyperspectral images based on spatially regularized spectral clustering with ultrametric path distances. The proposed method efficiently combines data density and spectral-spatial geometry to distinguish between material classes in the data, without the need for training labels. The proposed method is efficient, with quasilinear scaling in the number of data points, and enjoys robust theoretical performance guarantees. Extensive experiments on synthetic and real HSI data demonstrate its strong performance compared to benchmark and state-of-the-art methods. Indeed, the proposed method not only achieves excellent labeling accuracy, but also efficiently estimates the number of clusters. Thus, unlike almost all existing hyperspectral clustering methods, the proposed algorithm is essentially parameter-free. 

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4. Subspace clustering using ensembles of K -subspaces

   https://doi.org/10.1093/imaiai/iaaa031  

   Lipor, John ; Hong, David ; Tan, Yan Shuo ; Balzano, Laura ( November 2020 , Information and Inference: A Journal of the IMA)

   null (Ed.)

   Abstract Subspace clustering is the unsupervised grouping of points lying near a union of low-dimensional linear subspaces. Algorithms based directly on geometric properties of such data tend to either provide poor empirical performance, lack theoretical guarantees or depend heavily on their initialization. We present a novel geometric approach to the subspace clustering problem that leverages ensembles of the $K$-subspace (KSS) algorithm via the evidence accumulation clustering framework. Our algorithm, referred to as ensemble $K$-subspaces (EKSSs), forms a co-association matrix whose $(i,j)$th entry is the number of times points $i$ and $j$ are clustered together by several runs of KSS with random initializations. We prove general recovery guarantees for any algorithm that forms an affinity matrix with entries close to a monotonic transformation of pairwise absolute inner products. We then show that a specific instance of EKSS results in an affinity matrix with entries of this form, and hence our proposed algorithm can provably recover subspaces under similar conditions to state-of-the-art algorithms. The finding is, to the best of our knowledge, the first recovery guarantee for evidence accumulation clustering and for KSS variants. We show on synthetic data that our method performs well in the traditionally challenging settings of subspaces with large intersection, subspaces with small principal angles and noisy data. Finally, we evaluate our algorithm on six common benchmark datasets and show that unlike existing methods, EKSS achieves excellent empirical performance when there are both a small and large number of points per subspace. 

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5. Learning by Unsupervised Nonlinear Diffusion

   Maggioni, Mauro ; Murphy, James M. ( January 2019 , Journal of machine learning research)

   This paper proposes and analyzes a novel clustering algorithm, called \emph{learning by unsupervised nonlinear diffusion (LUND)}, that combines graph-based diffusion geometry with techniques based on density and mode estimation. LUND is suitable for data generated from mixtures of distributions with densities that are both multimodal and supported near nonlinear sets. A crucial aspect of this algorithm is the use of time of a data-adapted diffusion process, and associated diffusion distances, as a scale parameter that is different from the local spatial scale parameter used in many clustering algorithms. We prove estimates for the behavior of diffusion distances with respect to this time parameter under a flexible nonparametric data model, identifying a range of times in which the mesoscopic equilibria of the underlying process are revealed, corresponding to a gap between within-cluster and between-cluster diffusion distances. These structures may be missed by the top eigenvectors of the graph Laplacian, commonly used in spectral clustering. This analysis is leveraged to prove sufficient conditions guaranteeing the accuracy of LUND. We implement LUND and confirm its theoretical properties on illustrative data sets, demonstrating its theoretical and empirical advantages over both spectral and density-based clustering. 

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