a rank-formula_57 tensor formula_69 is called a border tensor if there exists a sequence of tensors of rank at most formula_70 whose limit is formula_69. if formula_57 is the least value for which such a convergent sequence exists, then it is called the border rank of formula_69. for order-2 tensors, i.e., matrices, rank and border rank "always" coincide, however, for tensors of order formula_74 they may differ. border tensors were first studied in the context of fast "approximate" matrix multiplication algorithms by bini, lotti, and romani in 1980.