Persistence homology is a tool used to measure topological features that are present in data sets and functions. Persistence pairs births and deaths of these features as we iterate through the sublevel sets of the data or function of interest. I am concerned with using persistence to characterize the difference between two functions f, g : M -> R, where M is a topological space. Furthermore, I formulate a homotopy from g to f by applying the heat equation to the difference function g-f. By stacking the persistence diagrams associated with this homotopy, we create a vineyard of curves that connect the points in the diagram for f with the points in the diagram for g. I look at the diagrams where M is a square, a sphere, a torus, and a Klein bottle. Looking at these four topologies, we notice trends (and differences) as the persistence diagrams change with respect to time.
Deep Dive into Persistence Diagrams and the Heat Equation Homotopy.
Persistence homology is a tool used to measure topological features that are present in data sets and functions. Persistence pairs births and deaths of these features as we iterate through the sublevel sets of the data or function of interest. I am concerned with using persistence to characterize the difference between two functions f, g : M -> R, where M is a topological space. Furthermore, I formulate a homotopy from g to f by applying the heat equation to the difference function g-f. By stacking the persistence diagrams associated with this homotopy, we create a vineyard of curves that connect the points in the diagram for f with the points in the diagram for g. I look at the diagrams where M is a square, a sphere, a torus, and a Klein bottle. Looking at these four topologies, we notice trends (and differences) as the persistence diagrams change with respect to time.
Given two functions, we are interested in comparing the respective persistence diagrams. The persistence diagram is a set of points in the Cartesian plane used to describe births and deaths of homology groups as we iterate through sublevel sets of a function. There are many ways that we can compare two diagrams, including matching points between the diagrams. However, finding a meaningful matching (see Section 3) is a difficult task, especially if we are interested in capturing the relationship between the underlying functions.
As stated in the proposal, the goal of this Research Initiation Project is to gain an intuition for persistence and homology, as well as to understand the current state of research in these fields. I accomplished this goal by investigating the following problem: For a 2-manifold M , suppose we have two continuous functions f, g : M → R. We can create the persistence diagrams for f and for g. If we know the relationship between f and g, we can make informed decisions when matching points in the persistence diagrams. In particular, we are interested in the case where f and g are homotopic. Given a homotopy between f and g, we can create a vineyard of the persistence diagrams. Then, we use the vines in the vineyard to help make an informed matching of the points in the persistence diagrams.
In this paper, we discuss two known methods of matching persistence diagrams, by measuring the bottleneck and the Wasserstein distances. Although stability results exist for matching persistence diagrams by minimizing either the bottleneck or the Wasserstein distance, these matchings are made without consideration of the underlying functions f and g. If we are able to create a continuous deformation of the function f to g, then we can use this additional information to aid in the matching of the points in the persistence diagrams. As a result, the matching obtained will be based on the underlying functions. We look into alternate way of measuring the distance between the persistence diagrams for the functions by assuming that there exists a homotopy between the functions. We create a homotopy that we call the heat equation homotopy and measure distances between the persistence diagrams for f and g by using these homotopies to aid in the pairing of points in the persistence diagrams of f and g. Then, we turn to analyzing an example of the heat equation homotopy and discuss various interesting patterns.
Observing patterns and features in data sets is a common goal in many disciplines, including biology. Extracting the key features from a noisy data set can be an ambiguous task, and often involves simplifying and finding the best view of the data. Computational topology, and more specifically persistent homology, is a tool used for data analysis. Here, we give a brief review of the necessary background of computational topology, but refer you to [10], [11] and [13] for more details.
2.1. Homology. Let X be a simplicial complex of dimension d. For p ∈ N and p ≤ d, the symbol X p will denote the power set of all p-simplices in X. Each set of X p is called a p-chain. The chain group C p is defined by the set X p under the disjoint union, or symmetric difference, operation. This operation can be interpreted as addition modulo two. The group C p is therefore isomorphic to Z 2 to some non-negative integer power. All algebriac groups in this paper are vector spaces over Z 2 . Consider the boundary homomorphism :
that maps the p-chain α ∈ C p to the boundary of α, a chain in C p-1 [13].
The p th homology group of X, denoted H p (X), is defined as the kernel of ∂ p modulo the image of ∂ p+1 : H p (X) = Ker(∂ p )/Im(∂ p+1 ). The kernel of the homomorphism ∂ p is the set of elements in the domain that are evaluated to zero (the empty set) and the image of ∂ p+1 is the set of elements of the form ∂ p+1 (x), where x is in the domain C p+1 :
The p th Betti number, β p , is the rank of the p th homology group of X. By definition, the rank of a group is the (smallest) number of generators needed to define the group up to isomorphism. Since we are concerned with groups with Z 2 coefficients, the rank uniquely defines the group up to isomorphism. For example, the group with three generators is Z 3 2 = Z 2 ⊕ Z 2 ⊕ Z 2 and the group with n generators is Z n 2 . 2.2. Persistent Homology. Now, we define persistent homology for functions from R to R. The complete discussion of the extension of these ideas to higher dimensions is found in [10] and [11]. We present a simplified setting to focus on the relevant concepts, while avoiding the complications that arise in the general setting.
Suppose C is the graph of f : R → R. We can think of f as the height function on C. Now, we characterize the topology of the sublevel set R f s = f -1 ((-∞, s]), and we monitor how the homology groups change as s goes from negative infinity to infinity. The zeroth persistent homology group, denoted H 0 (R f s ), will change whenever s is
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