\begin{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 $\eta\in(0,1)$ (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 $\eta=0$ yields the celebrated center of mass used in
KMeans clustering.

\toblue 
We show that fitting a spherical cluster leads to a strictly
convex but non-smooth combinatorial optimization problem, and we
develop an exact solver based on the Clarke gradient of non-smooth
functionals over a suitable stratified cell complex induced by an
arrangement of hyperspheres. To the best of our knowledge, our method
is the first practical application of the theory of semiflows of
convex maps, which generalizes the gradient flows of smooth maps.  We
\toblack present experiments on a variety of datasets ranging in
dimension from $d=9$ to $d=10,000$, with two main observations.
First, our exact algorithm is orders of magnitude faster than BFGS
based heuristics for datasets of small/intermediate dimension and
small values of $\eta$, and for high dimensional datasets (say
$d>100$) whatever the value of $\eta$. Second, the center of the SC
model behave as a parameterized high-dimensional median.

\toblue
The SC model is of direct interest for high dimensional multivariate data analysis,
and holds promises for the design of mixtures.
\toblack
\end{abstract}
