The formulation of noncommutative geometry (NCG) beyond the classical setting of the Chamseddine-Connes spectral action for spectral triples (towards, say, quantum or random NCG) is a well-known problem. In physics, it could help to geometrically understand quantum gauge theories; in mathematics, it is related to random matrix theory. While a formulation of ensembles of (or "path integrals over") Dirac operators on spin manifolds is too challenging, using fuzzy geometries instead (certain finite-dimensional spectral triples that can be thought of as truncations of smooth geometries) leads to a well-defined matrix ensemble whose measure has the peculiarity of presenting products of traces of noncommutative polynomials. In this talk we quickly review how these ensembles are determined using chord diagrams, and provide a renormalization theory of more general matrix ensembles. The renormalization flow is computed by certain matrix algebra with entries on a certain (bigger) cousin of the free algebra. We give a proof in terms of the of ribbon graphs and cyclic and noncommutative derivatives that are well-known in free probability. Based on 2111.02858 [Lett. Math. Phys., to appear] and 2007.10914 [Ann. Henri Poincaré 22 (2021)]