Further, computational complexity of calculating flat norm in arbitrary dimensions has not been studied. But this is not a surprising observation, given the continuous, rather than combinatorial, setting in which flat norm computation has been posed so far. Simplicial complexes that triangulate the input space are often used as representa- tions of manifolds. Such representations use triangular or tetrahedral meshes [17] as opposed to the uniform square or cubical grid meshes in R 2 and R $^{3}$. Various simplicial complexes are often used to represent data (in any dimension) that captures interactions in a broad sense, e.g., the Vietoris–Rips complex to capture coverage of coordinate-free sensor networks [12,13]. It is natural to consider flat norm calculations in such settings of simplicial complexes for denoising or regularizing sets, or for other similar tasks. At the same time, requiring that the simplicial complex be embedded in high dimensional space modeled by regular square grids may be cumbersome, and computationally prohibitive in many cases. We define a simplicial flat norm (SFN) for an input set T given as a subcomplex of the finite oriented simplicial complex K triangulating the set, or underlying space Ω. More generally, T is the simplicial representation of a rectifiable current with integer multiplicities. The choices of the higher dimensional sets S are restricted to K as well. We extend this definition to the multiscale simplicial flat norm (MSFN) by including a scale parameter λ . The simplicial flat norm is thus a special case of the multiscale simplicial flat norm with the default value of λ = 1. This discrete setting lets us address the worst case complexity of computing flat norm. Given its combinatorial nature, one would expect the problem to be difficult in arbitrary dimensions. Indeed, we show the problem of computing the multiscale simplicial flat 14 2.1.1 Contributions