|
Given a Morse function on a manifold, we may consider the critical points and use topological persistence to pair them up. We may then map each feature to the point in R2 whose coordinates are the function values of the corresponding two critical points, always ordered such that the first coordinate is less than the second. We refer to the resulting set of points above the diagonal in R2 as the persistence diagram of the Morse function. The main result in this work is the stability of this diagram. Specifically, we prove that if f and g are two functions with |f(x)-g(x)| < ? for all points x on the manifold, then there is a bijection between the points of the two diagrams, b: P(f)?P(g), such that ||p-b(p)||8 = ? for all points p in the persistence diagram of f.
There is one complication that needs to be explained else the result as stated is misleading. Small variation of a function can destroy or generate arbitrarily many critical point pairs through cancellations or anti-cancellations. These points correspond to points above but near the diagonal. We thus view the diagonal as an infinite supply of points in the persistence diagrams, and each point in P(f) or P(g) that is close enough to the diagonal may correspond to a point on the diagonal.
The stability of the persistence diagram is a fundamental result in algebraic topology. It may be used to understand the stability or instability of geometric concepts (e.g., the medial axis) and to extract their stable features.