Much of the recent work in statistics and machine learning has centered around understanding the extent to which different notions of structure can benefit statistical inference in noisy and high-dimensional learning problems. Various forms of structure, from smoothness and sparsity, to cluster and manifold structures underlie many of the recent successful applications of high-dimensional data analysis.

In my talk, I will briefly discuss several examples of structured inference that we have recently investigated. I will also present in detail one line of research investigating the minimax rates for inferring the homology of a manifold. I will also describe natural extensions of these ideas to problems related to the topology of the cluster tree.