# Modeling high dimensional point clouds with the spherical cluster model

arXiv: https://arxiv.org/abs/2512.21960 · HTML: https://mlsanity.com/paper/2512.21960/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 43 references that are in the map.
- First posted on arXiv 26 Dec 2025.

## Builds on (most cited first, 7 of 7)

- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Geometric Median in Nearly Linear Time](https://mlsanity.com/paper/1606.05225/) (2016)
- [On Geometric Analysis of Affine Sparse Subspace Clustering](https://mlsanity.com/paper/1808.05965/) (2018)
- [A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis](https://mlsanity.com/paper/2109.00530/) (2021)
- [Global convergence of the gradient method for functions definable in o-minimal structures](https://mlsanity.com/paper/2303.03534/) (2023)
- [Minimum message length estimation of mixtures of multivariate Gaussian and von Mises-Fisher distributions](https://mlsanity.com/paper/1502.07813/) (2015)
- [Improved seeding strategies for k-means and k-GMM](https://mlsanity.com/paper/2506.21291/) (2025)

## Abstract (arXiv)

A parametric cluster model is a statistical model providing geometric insights onto the points defining a cluster. The em spherical cluster model (SC) approximates a finite point set Psubset ℝd by a sphere S(c,r) as follows. Taking r as a fraction η∈(0,1) (hyper-parameter) of the std deviation of distances between the center c and the data points, the cost of the SC model is the sum over all data points lying outside the sphere S of their power distance with respect to S. The center c of the SC model is the point minimizing this cost. Note that η=0 yields the celebrated center of mass used in KMeans clustering. We make three contributions. First, we show fitting a spherical cluster yields a strictly convex but not smooth combinatorial optimization problem. Second, we present an exact solver using the Clarke gradient on a suitable stratified cell complex defined from an arrangement of hyper-spheres. Finally, we present experiments on a variety of datasets ranging in dimension from d=9 to d=10,000, with two main observations. First, the exact algorithm is orders of magnitude faster than BFGS based heuristics for datasets of small/intermediate dimension and small values of η, and for high dimensional datasets (say d>100) whatever the value of η. Second, the center of the SC model behave as a parameterized high-dimensional median. The SC model is of direct interest for high dimensional multivariate data analysis, and the application to the design of mixtures of SC will be reported in a companion paper.