  In this paper, we study the facial structure of the linear image of a cone.
We define the singularity degree of a face of a cone to be the minimum number
of steps it takes to expose it using exposing vectors from the dual cone. We
show that the singularity degree of the linear image of a cone is exactly the
number of facial reduction steps to obtain the minimal face in a corresponding
primal conic optimization problem. This result generalizes the relationship
between the complexity of general facial reduction algorithms and facial
exposedness of conic images under a linear transform by Drusvyatskiy, Pataki
and Wolkowicz to arbitrary singularity degree. We present our results in the
original form and also in its nullspace form. As a by-product, we show that
frameworks underlying a chordal graph have at most one level of stress matrix.
