Persistent magnitude

In this paper we introduce the persistent magnitude, a new numerical invariant of (sufficiently nice) graded persistence modules. It is weighted and signed count bars module, in which bar form [a,b) degree d counted with weight (e−a−e−b) sign (−1)d. Persistent magnitude has good formal properties, such as additivity respect to exact sequences compatibility tensor products, interpretations terms both associated functor, Laplace transform. Our definition inspired by Otter's notion blurred homology: show that finite metric space precisely its homology. Turning result on head, obtain strategy for turning existing homology theories into invariants applying magnitude. We explore detail case Morse functions, Rips