Directed complexes, non-linear rank and convex sensing

Convex sensing, i.e. sensing by quasiconvex functions, naturally arises in neuroscience and other contexts. One example of a convex sensing problem is measuring the underlying dimension of data. A related problem is computing the ''nonlinear rank'', i.e. the minimal rank of a matrix modulo the action by the group of row-wise nonlinear monotone transformations. While an exact algorithm for computing the nonlinear rank is unknown, it can be efficiently estimated using topological methods, as it is closely related to the geometry and topology of hyperplane arrangements.

A natural tool that captures the essence of convex sensing problems is directed complexes, which capture much of the relevant geometric information. For example, the nonlinear rank, as well as other geometric properties of data can be estimated from the homology of an associated directed complex. I will present recent results and conjectures about the directed complexes associated to some convex sensing problems.