68 research outputs found
Coordinate shadows of semi-definite and Euclidean distance matrices
We consider the projected semi-definite and Euclidean distance cones onto a
subset of the matrix entries. These two sets are precisely the input data
defining feasible semi-definite and Euclidean distance completion problems. We
classify when these sets are closed, and use the boundary structure of these
two sets to elucidate the Krislock-Wolkowicz facial reduction algorithm. In
particular, we show that under a chordality assumption, the "minimal cones" of
these problems admit combinatorial characterizations. As a byproduct, we record
a striking relationship between the complexity of the general facial reduction
algorithm (singularity degree) and facial exposedness of conic images under a
linear mapping.Comment: 21 page
Approximating functions on stratified sets
We investigate smooth approximations of functions, with prescribed gradient
behavior on a distinguished stratified subset of the domain. As an application,
we outline how our results yield important consequences for a recently
introduced class of stochastic processes, called the matrix-valued Bessel
processes.Comment: This is the version appearing in Trans. Amer. Math. Soc. 367 (2015),
725-74
- …