Estimating Density Ridges by Direct Estimation of Density-Derivative-Ratios

Hiroaki Sasaki, Takafumi Kanamori, Masashi Sugiyama
Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:204-212, 2017.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v54-sasaki17a, title = {{Estimating Density Ridges by Direct Estimation of Density-Derivative-Ratios}}, author = {Sasaki, Hiroaki and Kanamori, Takafumi and Sugiyama, Masashi}, booktitle = {Proceedings of the 20th International Conference on Artificial Intelligence and Statistics}, pages = {204--212}, year = {2017}, editor = {Singh, Aarti and Zhu, Jerry}, volume = {54}, series = {Proceedings of Machine Learning Research}, month = {20--22 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v54/sasaki17a/sasaki17a.pdf}, url = {https://proceedings.mlr.press/v54/sasaki17a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Estimating Density Ridges by Direct Estimation of Density-Derivative-Ratios %A Hiroaki Sasaki %A Takafumi Kanamori %A Masashi Sugiyama %B Proceedings of the 20th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2017 %E Aarti Singh %E Jerry Zhu %F pmlr-v54-sasaki17a %I PMLR %P 204--212 %U https://proceedings.mlr.press/v54/sasaki17a.html %V 54 %X 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.
APA
Sasaki, H., Kanamori, T. & Sugiyama, M.. (2017). Estimating Density Ridges by Direct Estimation of Density-Derivative-Ratios. Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 54:204-212 Available from https://proceedings.mlr.press/v54/sasaki17a.html.

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