  We investigate the problem of finding inner ap-proximations of positive
semidefinite (PSD) cones. We developa novel decomposition framework of the PSD
cone by meansof conical combinations of smaller dimensional sub-cones. Weshow
that many inner approximation techniques could besummarized within this
framework, including the set of (scaled)diagonally dominant matrices,
Factor-widthkmatrices, andChordal Sparse matrices. Furthermore, we provide a
moreflexible family of inner approximations of the PSD cone, wherewe aim to
arrange the sub-cones so that they are maximallyseparated from each other. In
doing so, these approximationstend to occupy large fractions of the volume of
the PSD cone.The proposed approach is connected to a classical packingproblem
in Riemannian Geometry. Precisely, we show thatthe problem of finding maximally
distant sub-cones in anambient PSD cone is equivalent to the problem of
packingsub-spaces in a Grassmannian Manifold. We further leverageexisting
computational method for constructing packings inGrassmannian manifolds to
build tighter approximations ofthe PSD cone. Numerical experiments show how the
proposedframework can balance between accuracy and computationalcomplexity, to
efficiently solve positive-semidefinite programs.
