A sparsified Christoffel function for high-dimensional inference

Abstract: Christoffel polynomials are classical tools from approximation theory. They can be used to estimate the (compact) support of a measure $\mu$ on $\mathbb{R}^d$ based on its low-degree moments. Recently, they have been applied to problems in data science, including outlier detection and support inference. A major downside of Christoffel polynomials in such applications is the fact that, in order to compute their coefficients, one must invert a matrix whose size grows rapidly with the dimension $d$. In this paper, we propose a modification of the Christoffel polynomial which is significantly cheaper to compute, but retains many of its desirable properties. Our approach relies on sparsity of the underlying measure $\mu$, described by a graphical model. The complexity of our modification depends on the treewidth of this model.