On Fiber Diameters of Continuous Maps

We present a surprisingly short proof that for any continuous map f : Rn → Rm, if n > m, then there exists no bound on the diameter of fibers of f . Moreover, we show that when m = 1, the union of small fibers of f is bounded; when m > 1, the union of small fibers need not be bounded. Applications to data analysis are considered. High-dimensional data sets are often difficult to analyze directly and, consequently, methods of simplifying them are important to modern data-intensive sciences. Continuous mappings f : R → R are frequently used to reduce the dimension of large data sets. Indeed, a classic result of Johnson and Lindenstrauss [6] shows that for N points in any Euclidean space, there exists an injective Lipschitz function which maps these points into R with minimal distortion in pairwise distances. However, while continuous maps enjoy many desirable properties, the following suggests that a measure of caution should be exercised before employing them for high-dimensional data analysis. We present a simple proof that for any continuous map f : R → R, if n > m then there exists no bound on the diameter of fibers of f . Therefore, points can be arbitrarily far apart in R, yet map to the same point under f . Definition. The fibers of a map f : X → Y are the preimages f−1(y) = {x ∈ X : f(x) = y} of points in Y . Definition. The diameter of a set A is the supremum sup{d(x, y) : x, y ∈ A}, where d(x, y) denotes the Euclidean distance between x and y. We begin by considering real-valued functions. Proposition. Let f : R → R be a continuous function where n > 1. Then for any M > 0, there exists y ∈ R whose fiber has diameter greater than M . Proof. Assume that some M > 0 bounds all fiber diameters. Consider three points a, b, c ∈ R such that the distance between any two is 2M , as in Figure 1. As M bounds the fiber diameters, f(a), f(b), and f(c) must be distinct; without loss of generality, suppose f(a) < f(b) < f(c). By the intermediate value theorem, the line segment ac contains a point x such that f(x) = f(b). As the distance from b to any point on ac is greater than M , the fiber containing b must have diameter greater than M , contradicting our assumption that M bounds all fiber diameters. The intermediate value theorem plays a central role in the proof above, and will turn up again several times in what follows. A generalization of this proposition can be established using the Borsuk-Ulam theorem [1], a result about continuous mappings from an n-sphere S to R. Date: September 1, 2015.