deformation (constrained to the complex) used to get it. For instance, in a triangulation of a manifold, we constrain ourselves to only use deformations on the manifold. We generalize the definition of SFN to define a multiscale simplicial flat norm (MSFN) of T in the simplicial complex K by including a scale parameter λ ∈ [0 , ∞ ). This definition is the simplicial version of the multiscale flat norm defined in Equation (2.3). The default, or nonscale, simplicial flat norm in Equation (2.4) is a special case of the multiscale simplicial flat norm with the default value of λ = 1. The (non-simplicial) flat norm with scale λ > 0 of a d -dimensional current T can be rewritten as F$_{λ}$ ( T ) = λ d · F$_{1}$ ( T /λ ). Thus the flat norm with scale can be thought of as the traditional flat norm applied to a scaled copy of the input current. An equivalent statement can be made for the simplicial flat norm, but crucially requires that the simplicial complex be similarly scaled. To avoid this complex scaling issue especially when considering all possible scales, and to simplify our notation, we henceforth study the more general multiscale simplicial flat norm (which also allows us to consider the λ = 0 case). We assume the d - and ( d +1)-dimensional volumes of simplices to be any nonnegative values. For example, when σ$_{i}$ is a 1-simplex, i.e., edge, V$_{1}$( σ$_{i}$ ) could be taken as its Euclidean length. Similarly, V$_{2}$( τ$_{j}$ ) for a triangle τ$_{j}$ could be its area. For ease of notation, we denote V$_{d}$( σ$_{i}$ ) by w$_{i}$ and V$_{d}$$_{+1}$( τ$_{j}$ ) by v$_{j}$ , with the dimensions d and d + 1 evident from the context. Remark 2.2.1. The minimum in the definition of the multiscale simplicial flat norm 19 F λ $_{S}$( T ) = min s ∈$_{Z}$ n    m ∑ i =1 V$_{d}$( σ$_{i}$ ) | x$_{i}$ | + λ$^{}$  n ∑ j =1 V$_{d}$$_{+1}$( τ$_{j}$ ) ∣ $_{∣}$s$_{j}$ ∣   ∣ x = t − [ ∂$_{d}$$_{+1}$ ] s , x ∈ Z m    . (2.5)