Robust Estimation of Sliced-Exponential Distributions⋆

Sliced distributions enable the characterization of multivariate data as both a vector of continuous and possibly dependent random variables, or as a semi-algebraic, tightly enclosing set. Sliced distributions inject the physical space into a higher-dimensional feature space using a polynomial mapping. This paper introduces the Sliced-Exponential (SE) subclass of distributions, proposes a suitable data-based polynomial basis for it, and compares its performance against that of the Sliced-Normal (SN) subclass. The key advantage of the SEs over the SNs is that their maximum likelihood estimate results from solving a convex optimization program in a number of decision variables that grows linearly with the dimension of feature space. This is in sharp contrast to the SNs which, as all other Sum of Squares (SOS) methods, have a number of decision variables that increases exponentially with such a dimension thereby limiting their applicability. In addition, SEs have greater versatility since they are not restricted to the space of SOS polynomials. However, this enhanced versatility when coupled with an inaccurate estimation of the normalization constant might yield spurious distributions. This paper presents strategies that mitigate these anomalies by restricting the decision space. Furthermore, we use numerical experiments of increasing dimension size to determine practical limitations, and to set good practice guidelines.