Free convex sets are closed under a rich class of convex combinations called matrix convex combinations. Roughly speaking, a matrix convex combination is sum of tuples of matrices where contraction matrices summing to the identity play the role of the convex coefficients. A prototypical example of such a set is a free spectrahedron; a set which arises as the positivity domain of a linear matrix inequality. I am particularly interested in understanding extreme points of free convex sets.
My research on tensor decomposition is currently focused on existence of best low rank tensor approximations. It is well known that higher order tensors can fail to have a best low rank approximation, and that low rank approximations fail to exist with positive probability when working over the reals. It is my goal to develop deterministic methods which may be used to guarantee that a given tensor has a best low rank approximation. A closely related research theme is development of algebraic methods for computation of tensor decompositions which are better able to handle numerically ill-conditioned tensors. For a basic introduction to tensors, see my communication article Tensors and multilinear algebra: what and why
Extreme points (at level one) of the free spectrahedron defined by the following linear matrix inequality in four variables.
