# Mode-Seeking Clustering and Density Ridge Estimation via Direct Estimation of Density-Derivative-Ratios

arXiv: https://arxiv.org/abs/1707.01711 · HTML: https://mlsanity.com/paper/1707.01711/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jul 2017.

## Built on by (most cited first, 3 of 3)

- [Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networks](https://mlsanity.com/paper/2512.23643/) (2025)
- [Test-Time Compositional Generalization in Diffusion Models via Concept Discovery](https://mlsanity.com/paper/2605.07078/) (2026)
- [Ridge Estimation with Nonlinear Transformations](https://mlsanity.com/paper/2306.05722/) (2023)

## Abstract (arXiv)

Modes and ridges of the probability density function behind observed data are useful geometric features. Mode-seeking clustering assigns cluster labels by associating data samples with the nearest modes, and estimation of density ridges enables us to find lower-dimensional structures hidden in data. A key technical challenge both in mode-seeking clustering and density ridge estimation is accurate estimation of the ratios of the first- and second-order density derivatives to the density. A naive approach takes a three-step approach of 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 the estimated density could significantly magnify the estimation error. To cope with these problems, we propose a novel estimator for the density-derivative-ratios. The proposed estimator does not involve density estimation, but rather directly approximates the ratios of density derivatives of any order. Moreover, we establish a convergence rate of the proposed estimator. Based on the proposed estimator, novel methods both for mode-seeking clustering and density ridge estimation are developed, and the respective convergence rates to the mode and ridge of the underlying density are also established. Finally, we experimentally demonstrate that the developed methods significantly outperform existing methods, particularly for relatively high-dimensional data.