Crackle: The Homology of Noise

Abstract

We study the homology of simplicial complexes built via deterministic rules from a random set of vertices. In particular, we show that, depending on the randomness that generates the vertices, the homology of these complexes can either become trivial as the number n of vertices grows, or can contain more and more complex structures. The different behaviours are consequences of different underlying distributions for the generation of vertices, and we consider three illustrative examples, when the vertices are sampled from Gaussian, exponential, and power-law distributions in R d .We also discuss consequences of our results for manifold learning with noisy data, describing the topological phenomena that arise in this scenario as 'crackle', in analogy to audio crackle in temporal signal analysis.

Random Čech complexes on Riemannian manifolds

“…The study of random simplicial complexes originated in the seminal result of Erdős and Rényi [14] on the phase transition for connectivity in random graphs G(n, p) (with n vertices, and where edges are included independently and with probability p) . In their paper, Erdős and Rényi studied these graphs in the limit when n → ∞ and p = p(n) → 0, and showed that the phase transition for connectivity occurs around p = log n∕n, when the expected degree is approximately log n. 1 Over the past process on M with intensity n (see definition in Section 2.3). With no loss of generality, and to shorten notation, we will assume that Vol(M) = 1 as this will only affect the results by an overall scaling constant.…”

“…The proof of Theorem 1.1, has a similar outline to the one in [7], but with considerable geometric adjustments required for the Riemannian case. In fact, the approach we use here for addressing the general setting turns out to be powerful by allowing us to (1) weaken some of the conditions required in [7], and (2) prove many other statements for randomČech complexes in the Riemannian setting.…”

“…For example, for the -dimensional sphere S we have 0 (S ) = (S ) = 1, while k (S ) = 0 for k ≠ 0, . Another example is the 2-dimensional torus T 2 that has 0 (T 2 ) = 1 (a single component), 1 (T ) = 2 (two essential "loops"), and 2 (T 2 ) = 1 (the boundary of the 3D solid) (see Figure 1).…”

“…Given a point p ∈ M one can consider the geodesics that pass through p at time t = 0 with velocity vector as a given vector v ∈ T p M. The geodesic equations are a system of second order differential equations and a simple application of Picard's existence and uniqueness theorem shows that these geodesic are unique, and we denote them by v (t). We can then consider a map exp p ∶ T p M → M, called the exponential map at p, which assigns to each vector v ∈ T p M the position at which the unique geodesic starting at p with velocity v is at time 1, that is, exp p (v) = v (1). The derivative of exp p at the origin in T p M is the identity and so, by the inverse function theorem, exp p is a local diffeomorphism of a small ball around 0 ∈ T p M to a small neighborhood of p ∈ M. As a consequence, one can use the exponential map to define local coordinates around p. For instance, fixing an orthonormal basis for T p M one obtains coordinates on T p M, which can be regarded as local coordinates (x 1 , … , x ) around p. These are the so called geodesic normal coordinates and have the following properties.…”

“…The main idea is to use topological features (eg, homology, Euler characteristic, persistent homology) as a "signature" for various types of complex high-dimensional data.Geometric complexes are used often in TDA to translate data points into a combinatorial-topological space, which in turn can be fed into a software algorithm that calculates its relevant topological properties. It is therefore desired to develop a solid statistical theory for geometric complexes (see eg, [2,6,10,35]), and an imperative part of this effort is to develop its probabilistic foundations (see eg, [1,4,12,37]). Most of the results on random geometric complexes and graphs to date have been studied for point processes in a Euclidean space.…”