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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2512.21960 |
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| _version_ | 1866911340060213248 |
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| author | Cazals, Frédéric Commaret, Antoine Goldenberg, Louis |
| author_facet | Cazals, Frédéric Commaret, Antoine Goldenberg, Louis |
| contents | 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 $P\subset \mathbb{R}^d$ by a sphere $S(c,r)$ as follows. Taking $r$ as a fraction $η\in(0,1)$ (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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21960 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modeling high dimensional point clouds with the spherical cluster model Cazals, Frédéric Commaret, Antoine Goldenberg, Louis Methodology Machine Learning 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 $P\subset \mathbb{R}^d$ by a sphere $S(c,r)$ as follows. Taking $r$ as a fraction $η\in(0,1)$ (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. |
| title | Modeling high dimensional point clouds with the spherical cluster model |
| topic | Methodology Machine Learning |
| url | https://arxiv.org/abs/2512.21960 |