Chapter 5: Function-Induced Persistence Topics in Computational Topology: An Algorithmic View Given a domain X, its topology provides a rather coarse description of it. However, functions or maps defined on X give us powerful ways to encode various both geometric and other type of information of X. For example, the Gaussian curvature function κ : X IR describes the ! local geometric shape of X. Or imagine that X is a terrain on earth, and h : X IR is the elevation of each point; see the right figure for an ! illustration. Or imagine that X models the hidden space where a social network is sampled from, and a point represents a person. We can use a function I : X IR that sends each p X to I(p), which is the person ! 2 Figure 1: Red color in- p's influence over her/his friends. These two examples are real-valued dicates higher elevation. functions defined over X, also called a scalar function or a scalar field. One can also have more complex maps, such as multivariate functions f : X IRd, which encodes multi-dimensional properties of X. ! In this section, we first focus on real-valued functions. In Section 1, we introduce one type of topological information of X w.r.to a scalar field on a smooth function. We then talk about its analog in the discrete setting in Section 1. Next, in Section 3, we introduce an important family of topological description of a domain induced by a function defined on it, called function-induced persistence. Last in Section 5, we give a simple and efficient algorithm for computing the 0-th dimensional function-induced persistence which takes only near linear running time (in constrast the O(n3) running time in the general case). 1 Morse functions and critical points 1.1 Gradients and critical points Let us first consider a simple scalar function defined on the real line: f : IR IR; the graph of such a function is shown in Figure 2 on the ! right. Recall from calculus the definition of the derivative of a function at any point x IR is defined as: 2 d f(x + t) f(x) Df(x) = f(x) = lim − (1) dx t!1 t Intuitively, Df(x) gives the rate of change of the value of f at x. This can be visualized as the slope of the tangent line of the graph of the function f at x. The critical points of f are the set of points x such that Df(x) = 0 { for the function defined on the real line, there are two types of critical points in the generic case: maxima and minima. These points Figure 2: The graph of are \critical" because they mark where the behavior of f changes. a function f : IR IR ! defined on the real line 1 Now suppose we have a real-valued function f : IRd IR defined on ! IRd. We can draw the graph of this function in IRd+1; recall Figure 1 where we show the graph of a function h : IR2 IR. Imagine that we are at a point x IRd. As we ! 2 move a little around x, the rate of change of f differs depending on which direction we move. This d gives rise to the directional derivative Dvf(x) at x in direction (unit vector) v IR , defined as: 2 f(x + t v) f(x) Dvf(x) = lim · − (2) t!1 t Definition 1.1 (Gradient and critical points) Given a differentiable function f : IRd IR ! and x IRd, the gradient f of f at x is defined as: 2 r @f @f @f f(x) = [ ]T ; (3) r @x1 @x2 ··· @xd d where < x1; x2; : : : ; xd > represents a coordinate system for IR . Equivalently, f(x) is along the direction v where Dvf(x) is maximized, and the magnititude r f(x) of f(x) is the value of this maximal directional derivative. kr k r A point x IRd is a critical point if f(x) = [0 0 0]T . If x is critical, then f(x) is called 2 r ··· a critical value for f. In other words, the gradient of f at x specifies the steepest descendin direction of f with its magnititude being the rate of change along that direction. The critical points of f are those points where the directional derivative vanishes in all directions { locally, the rate of change for f is zero no matter which direction one deviates from x. See Figure 3 for the three types of critical points in a generic setting for a function f : IR2 IR: mimina, saddle points, and maxima. ! Finally, given a differentiable function f : M IR defined on a smooth manifold, we can ! extend the above definitions of gradients and critical points to it. We will not define it formally here. Intuitively, at a point x M, we now consider the tangent space of M at x, denoted by 2 TMx. Roughly speaking, within an infinitesmally small neighborhood of x, one can imagine that m the function f is defined on TMx. If the manifold M is of dimension m, then TMx is IR . We can then define the gradient of f with respect to TMx { that is, the gradient f is a vector field r f : M TM, and f(x) TMx represents the steepest descending direction of f among all r ! r 2 directions v TMx with its magnititude being the rate of change along this direction. A point x 2 is a critical point if its gradient f(x) vanishes at x. r 1.2 Morse functions and Morse Lemma From the first derivative of a function we can determine critical points. We can say much about the \type" of the critical points by inspecting the second derivatives of f around a point x. Specifically, consider the Hessian Matrix. Definition 1.2 (Hessian Matrix) A Hessian Matrix of a second order differentiable function f : IRd IR at x is the matrix of second derivatives, ! @2f @2f @2f (x) (x) (x) @x1@x1 @x1@x2 @x1@xd 2 2 ··· 2 @ f (x) @ f 2(x) @ f (x) @x2@x1 @x2@x2 @x2@xd Hessian(x) = ··· . .. @2f @2f @2f (x) 2(x) (x) @xd@x1 @xd@x2 ··· @xd@xd 2 Local quadratic approximation Gradient flow -neighborhood Lower link Figure 3: Local neighborhoods of critical points. Top row: minimum; middle row: saddle; bottom row: maximum. Definition 1.3 (Degenerate critical points) A critical point x of f is degenerate if Det(Hessian(x)) = 0 (Hessian(x) is not full rank). Otherwise, the critical point x is considered non-degenerate. For example, Letf = x2 + 1 (4) f 0(x) = 2x (5) f 0(0) = 0 (criticalpoint) (6) @(2x) f 00(x) = (7) @x f 00(0) = 2 (notdegenerate) (8) Letf = x3 (9) f 0(x) = 3x2 (10) f 0(0) = 0 (criticalpoint) (11) @(3x2) f 00(x) = = 6x (12) @x f 00(0) = 0 (degenerate) (13) 3 Generally, the behavior around degenerate critical points are hard to manage. So, we'll only consider functions such that no critical points are degenerate. This brings us (finally) to the definition of Morse functions. Definition 1.4 (Morse Function) A function f : M IR is a Morse function iff the following conditions are met: ! 1. None of f's critical points are degenerate. 2. No two of f's critical points share the same function value. In other words, Morse functions are nicely behaving functions. Limiting our study only to Morse functions is not too restrictive as it turns out that the Morse functions form an open and dense subset of the space of all smooth functions C1(M) on M. So in this sense, a generic function is a Morse function. More importantly, considering such nice family of functions give us clean characterization of the topology induced by the function. Lemma 1.5 (Morse Lemma) Given a Morse function f : M IR, let p be a non-degenerate ! critical point of f, then there are local coordinate chart of a sufficiently small neighborhood of x such that (i) the coordinate of p is (0; 0; 0) (the origin) in this chart, and (ii) locally the function f can be represented as f(x) = f(p) x2 :::x2 + x2 :::x2; for s [0; d] − 1 − s s+1 d 2 for every point x = (x1; x2:::xd) in a small neighborhood of p. The index of a critical point p is s. There are d+1 types of critical points for a Morse function. Index-0 critical points are also called minima, while index-d critical points are also called maxima. Consider the example in Figure 3, where for a function f defined on a 2-manifold, there are only three types of critical points, with index-0 (minima), index-1 (saddle), and index-2 (maxima). 2 2 1. Local Minima: f(x) = f(p) + x1 + x2, when s = 0. 2. Saddle Point: f(x) = f(p) x2 + x2, when s = 1. − 1 2 3. Local Maxima: f(x) = f(p) x2 x2, when s = 2. − 1 − 2 1.3 Connection to topology First, a couple of definitions: Definition 1.6 (Level Set) Let f : M IR. Then all real numbers a have a preimage, f −1(a), known as a level set. ! M a = f −1(a) = x M f(x) = a f 2 j g Informally, a level set is a set of all the points in M that result in the same function value. Definition 1.7 (Interval-Level Set) Let f : M IR and let I IR.