Estimating Density Ridges by Direct Estimation of DensityDerivativeRatios
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Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:204212, 2017.
Abstract
Estimation of \emphdensity ridges has been gathering a great deal of attention since it enables us to reveal lowerdimensional structures hidden in data. Recently, \emphsubspace constrained mean shift (SCMS) was proposed as a practical algorithm for density ridge estimation. A key technical ingredient in SCMS is to accurately estimate the ratios of the density derivatives to the density. SCMS takes a threestep approach for this purpose — first estimating the data density, then computing its derivatives, and finally taking their ratios. However, this threestep approach can be unreliable because a good density estimator does not necessarily mean a good density derivative estimator and division by an estimated density could significantly magnify the estimation error. To overcome these problems, we propose a novel method that directly estimates the ratios without going through density estimation and division. Our proposed estimator has an analyticform solution and it can be computed efficiently. We further establish a nonparametric convergence bound for the proposed ratio estimator. Finally, based on this direct ratio estimator, we develop a practical algorithm for density ridge estimation and experimentally demonstrate its usefulness on a variety of datasets.
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