We investigate the inverse scale space flow as a decomposition method for
decomposing data into generalised singular vectors. We show that the inverse
scale space flow, based on convex and absolutely one-homogeneous regularisation
functionals, can decompose data represented by the application of a forward
operator to a linear combination of generalised singular vectors into its
individual singular vectors. We verify that for this decomposition to hold
true, two additional conditions on the singular vectors are sufficient:
orthogonality in the data space and inclusion of partial sums of the
subgradients of the singular vectors in the subdifferential of the
regularisation functional at zero. We also address the converse question of
when the inverse scale space flow returns a generalised singular vector given
that the initial data is arbitrary (and therefore not necessarily in the range
of the forward operator). We prove that the inverse scale space flow is
guaranteed to return a singular vector if the data satisfies a novel dual
singular vector condition. We conclude the paper with numerical results that
validate the theoretical results and that demonstrate the importance of the
additional conditions required to guarantee the decomposition result