{"ID":2824978,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.21960","arxiv_id":"2512.21960","title":"Modeling high dimensional point clouds with the spherical cluster model","abstract":"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 $d\u003e100$) 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.","short_abstract":"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 o...","url_abs":"https://arxiv.org/abs/2512.21960","url_pdf":"https://arxiv.org/pdf/2512.21960v1","authors":"[\"Frédéric Cazals\",\"Antoine Commaret\",\"Louis Goldenberg\"]","published":"2025-12-26T10:11:57Z","proceeding":"stat.ME","tasks":"[\"stat.ME\",\"cs.LG\"]","methods":"[]","has_code":false}