Remark 2.3.5. Components of variables x $^{+}$, x $^{−}$, s $^{+}$, s − in the multiscale simplicial flat norm ILP (Equation 2.8) could assume values other than {− 1 , 0 , 1 } , indicating integer multiplicities higher than 1 for the corresponding simplices in the optimal decomposition. The definition of multiscale simplicial flat norm (Equation 2.5) does allow such larger multiplicities. At the same time, if one insists on using each ( d +1) -simplex at most once when calculating the multiscale simplicial flat norm, and insists on similar restrictions on d -simplices in the optimal decomposition, we can modify the ILP such that Theorem 2.3.3 still holds. Denoting the entire variable vector by x = ( x $^{+}$, x $^{−}$, s $^{+}$, s $^{−}$) ∈ Z 2 m +2 $^{n}$, we add the up- per bound constraints x ≤ 1 , where 1 is the (2 m +2 n ) -vector of ones. These inequalities could be converted to the set of equations x + y = 1 , where y is the (2 m + 2 n ) -vector of slack variables that are nonnegative. These modifications give an ILP whose polyhe- dron is in the same form as described in Theorem 2.3.1, with the equations denoted as A $^{′}$x′ = b ′ for the variable vector x ′ = ( x , y ) . The new constraint matrix A ′ is related to the constraint matrix A of the original multiscale simplicial flat norm ILP given in Lemma 2.3.2 as where I is the 2 m + 2 n identity matrix, and O is the m × (2 m + 2 n ) zero matrix. Hence A ′ is obtained from A by first adding 2 m + 2 n rows with a single nonzero entry of +1 , and then adding to the resulting matrix 2 n + 2 m more columns with a single nonzero entry of +1 . These operations preserve total unimodularity [37, pg. 280], and hence the new constraint matrix A ′ is totally unimodular when [ ∂$_{d}$$_{+1}$ ] is so. The new right-hand side vector b ′ ∈ Z 3 m +2 n consists of the input chain t and the vector of ones from the new upper bound constraints. 26 A ′ =    A O I I$^{}$  $_{}$,