Estimation of emphdensity ridges has been gathering a great deal of attention since it enables us to reveal lower-dimensional 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 three-step approach for this purpose — first estimating the data density, then computing its derivatives, and finally taking their ratios. However, this three-step 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 analytic-form solution and it can be computed efficiently. We further establish a non-parametric 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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