# Mode and Ridge Estimation in Euclidean and Directional Product Spaces: A Mean Shift Approach

arXiv: https://arxiv.org/abs/2110.08505 · HTML: https://mlsanity.com/paper/2110.08505/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 102 references that are in the map.
- First posted on arXiv 16 Oct 2021.

## Builds on (most cited first, 18 of 18)

- [Ray: A Distributed Framework for Emerging AI Applications](https://mlsanity.com/paper/1712.05889/) (2017)
- [Coordinate Descent Algorithms](https://mlsanity.com/paper/1502.04759/) (2015)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [A Tutorial on Kernel Density Estimation and Recent Advances](https://mlsanity.com/paper/1704.03924/) (2017)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [A review of mean-shift algorithms for clustering](https://mlsanity.com/paper/1503.00687/) (2015)
- [Product Manifold Learning](https://mlsanity.com/paper/2010.09908/) (2020)
- [Asymptotic theory for density ridges](https://mlsanity.com/paper/1406.5663/) (2014)
- [The modal age of Statistics](https://mlsanity.com/paper/1807.02789/) (2018)
- [A Comprehensive Approach to Mode Clustering](https://mlsanity.com/paper/1406.1780/) (2014)
- [Kernel Smoothing, Mean Shift, and Their Learning Theory with Directional Data](https://mlsanity.com/paper/2010.13523/) (2020)
- [Exact risk improvement of bandwidth selectors for kernel density estimation with directional data](https://mlsanity.com/paper/1306.0517/) (2013)
- [Algorithms for ridge estimation with convergence guarantees](https://mlsanity.com/paper/2104.12314/) (2021)
- [A comparison of bandwidth selectors for mean shift clustering](https://mlsanity.com/paper/1310.7855/) (2013)
- [Central limit theorems for directional and linear random variables with applications](https://mlsanity.com/paper/1402.6836/) (2014)
- [Generalized Mode and Ridge Estimation](https://mlsanity.com/paper/1406.1803/) (2014)
- [Linear Convergence of the Subspace Constrained Mean Shift Algorithm: From Euclidean to Directional Data](https://mlsanity.com/paper/2104.14977/) (2021)
- [The EM Perspective of Directional Mean Shift Algorithm](https://mlsanity.com/paper/2101.10058/) (2021)

## Built on by (most cited first, 1 of 1)

- [Stable Density Ridges: Consistency and Convergence of Subspace Constrained Mean Shift](https://mlsanity.com/paper/2608.05112/) (2026)

## Abstract (arXiv)

The set of local modes and density ridge lines are important summary characteristics of the data-generating distribution. In this work, we focus on estimating local modes and density ridges from point cloud data in a product space combining two or more Euclidean and/or directional metric spaces. Specifically, our approach extends the (subspace constrained) mean shift algorithm to such product spaces, addressing potential challenges in the generalization process. We establish the algorithmic convergence of the proposed methods, along with practical implementation guidelines. Experiments on simulated and real-world datasets demonstrate the effectiveness of our proposed methods.