We study the multiscale simplicial flat norm (MSFN) problem, which computes
flat norm at various scales of sets defined as oriented subcomplexes of finite
simplicial complexes in arbitrary dimensions. We show that the multiscale
simplicial flat norm is NP-complete when homology is defined over integers. We
cast the multiscale simplicial flat norm as an instance of integer linear
optimization. Following recent results on related problems, the multiscale
simplicial flat norm integer program can be solved in polynomial time by
solving its linear programming relaxation, when the simplicial complex
satisfies a simple topological condition (absence of relative torsion). Our
most significant contribution is the simplicial deformation theorem, which
states that one may approximate a general current with a simplicial current
while bounding the expansion of its mass. We present explicit bounds on the
quality of this approximation, which indicate that the simplicial current gets
closer to the original current as we make the simplicial complex finer. The
multiscale simplicial flat norm opens up the possibilities of using flat norm
to denoise or extract scale information of large data sets in arbitrary
dimensions. On the other hand, it allows one to employ the large body of
algorithmic results on simplicial complexes to address more general problems
related to currents.Comment: To appear in the Journal of Computational Geometry. Since the last
version, the section comparing our bounds to Sullivan's has been expanded. In
particular, we show that our bounds are uniformly better in the case of
boundaries and less sensitive to simplicial irregularit