Bottom-up self-assembly offers a means to synthesize materials with desirable structural and functional properties that cannot easily be fabricated by other techniques. An improved understanding of the structural pathways and mechanisms by which self-assembling materials spontaneously form from their constituent building blocks is of value in understanding the fundamental principles of assembly and in guiding inverse building block design. We present an approach to infer systematically assembly pathways and mechanisms by nonlinear data mining of molecular simulation trajectories using diffusion maps. We have validated our methodology in applications to Brownian dynamics simulations of the assembly of anisotropic "patchy colloids" into polyhedral aggregates. For particles designed to form tetrahedral aggregates, we identify two divergent assembly pathways leading to chains of interlocking dimers and tetramers and chains of interlocking trigonal planar trimers. For particles designed to assemble icosahedral aggregates, our approach recovers two distinct assembly pathways corresponding to monomeric addition and budding from a disordered liquid phase. These assembly routes were previously reported by inspection of simulation trajectories by Wilber et al. (J. Chem. Phys. 2007, 127, 085106; J. Chem. Phys. 2009, 131, 175102), validating the capacity of our approach to systematically recover assembly mechanisms previously discernible only by trajectory visualization.