  We show how to sketch semidefinite programs (SDPs) using positive maps in
order to reduce their dimension. More precisely, we use
Johnson\hyp{}Lindenstrauss transforms to produce a smaller SDP whose solution
preserves feasibility or approximates the value of the original problem with
high probability. These techniques allow to improve both complexity and storage
space requirements. They apply to problems in which the Schatten 1-norm of the
matrices specifying the SDP and also of a solution to the problem is constant
in the problem size. Furthermore, we provide some results which clarify the
limitations of positive, linear sketches in this setting.
