Dimension of Marginals of Kronecker Product Models

In this talk I present a complete solution to the dimension question for restricted Boltzmann machines (RBMs) that was first conjectured by Cueto, Morton, and Sturmfels in 2010. The RBM always has the expected dimension. This result comes from recent work with Jason Morton on the dimension and other properties of a much larger class of models called Kronecker product models: Take two sufficient statistics matrices A,B (in the RBM case, these are independence models, so Segre matrices). Take the Kronecker product of these matrices to obtain a new matrix F=A⊗BWhy care about the standard conjectures? (representing a new sufficient statistic), then marginalize the hidden variables corresponding to one of the original matrices. As this construction suggests, one can allow arbitrary exponential families (toric varieties) in place of the two independence models (Segre varieties) from the RBM.