  Low-rank inducing unitarily invariant norms have been introduced to convexify
problems with low-rank/sparsity constraint. They are the convex envelope of a
unitary invariant norm and the indicator function of an upper bounding rank
constraint. The most well-known member of this family is the so-called nuclear
norm. To solve optimization problems involving such norms with proximal
splitting methods, efficient ways of evaluating the proximal mapping of the
low-rank inducing norms are needed. This is known for the nuclear norm, but not
for most other members of the low-rank inducing family. This work supplies a
framework that reduces the proximal mapping evaluation into a nested binary
search, in which each iteration requires the solution of a much simpler
problem. This simpler problem can often be solved analytically as it is
demonstrated for the so-called low-rank inducing Frobenius and spectral norms.
Moreover, the framework allows to compute the proximal mapping of compositions
of these norms with increasing convex functions and the projections onto their
epigraphs. This has the additional advantage that we can also deal with
compositions of increasing convex functions and low-rank inducing norms in
proximal splitting methods.
