Conference Paper

Rates of convergence for the cluster tree.

Conference: Advances in Neural Information Processing Systems 23: 24th Annual Conference on Neural Information Processing Systems 2010. Proceedings of a meeting held 6-9 December 2010, Vancouver, British Columbia, Canada.
Source: DBLP
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    ABSTRACT: Based on the work of Hartigan, the clusters of a distribution are often defined to be the connected components of a density level set. Unfortunately, this definition depends on the user-specified level, and in general finding a reasonable level is a difficult task. In addition, the definition is not rigorous for discontinuous densities, since the topological structure of a density level set may be changed by modifying the density on a set of measure zero. In this work, we address these issues by first modifying the notion of density level sets in a way that makes the level sets independent of the actual choice of the density. We then propose a simple algorithm for estimating the smallest level at which the modified level sets have more than one connected component. For this algorithm we provide a finite sample analysis, which is then used to show that the algorithm consistently estimates both the smallest level and the corresponding connected components. We further establish rates of convergence for the two estimation problems, and last but not least, we present a simple strategy for determining the width-parameter of the involved density estimator in a data-depending way. The resulting algorithm turns out to be adaptive, that is, it achieves the optimal rates achievable by our analysis without knowing characteristics of the underlying distribution.
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    ABSTRACT: Mode clustering is a nonparametric method for clustering that defines clusters using the basins of attraction of a density estimator's modes. We provide several enhancements to mode clustering: (i) a soft variant of cluster assignment, (ii) a measure of connectivity between clusters, (iii) a technique for choosing the bandwidth, (iv) a method for denoising small clusters, and (v) an approach to visualizing the clusters. Combining all these enhancements gives us a useful procedure for clustering in multivariate problems.
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    ABSTRACT: Fiber tractography on diffusion imaging data offers rich potential for describing white matter pathways in the human brain, but characterizing the spatial organization in these large and complex data sets remains a challenge. We show that level set trees-which provide a concise representation of the hierarchical mode structure of probability density functions-offer a statistically-principled framework for visualizing and analyzing topography in fiber streamlines. Using diffusion spectrum imaging data collected on neurologically healthy controls (N = 30), we mapped white matter pathways from the cortex into the striatum using a deterministic tractography algorithm that estimates fiber bundles as dimensionless streamlines. Level set trees were used for interactive exploration of patterns in the endpoint distributions of the mapped fiber pathways and an efficient segmentation of the pathways that had empirical accuracy comparable to standard nonparametric clustering techniques. We show that level set trees can also be generalized to model pseudo-density functions in order to analyze a broader array of data types, including entire fiber streamlines. Finally, resampling methods show the reliability of the level set tree as a descriptive measure of topographic structure, illustrating its potential as a statistical descriptor in brain imaging analysis. These results highlight the broad applicability of level set trees for visualizing and analyzing high-dimensional data like fiber tractography output.
    PLoS ONE 01/2014; 9(4):e93344. · 3.53 Impact Factor


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