Modeling high dimensional point clouds with the spherical cluster model

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.