Power weighted shortest paths for clustering Euclidean data

Abstract

We study the use of power weighted shortest path metrics for clustering high dimensional Euclidean data, under the assumption that the data is drawn from a collection of disjoint low dimensional manifolds. We argue, theoretically and experimentally, that this leads to higher clustering accuracy. We also present a fast algorithm for computing these distances.1. We prove that p-wspm's behave as expected for data satisfying the manifold hypothesis.That is, we show that the maximum distance between points in the same cluster is small with high probability, and tends to zero as the number of data points tends to infinity. On the other hand, the maximum distance between points in different clusters remains bounded away from zero.2. We show how p-wspm's can be thought of as interpolants between the Euclidean metric and the longest leg path distance (defined in §2.3), which we shall abbreviate to LLPD.3. We introduce a novel modified version of Dijkstra's algorithm that computes the k nearest neighbors, with respect to any p-wspm or the LLPD, of any x α in X in O(k 2 T Enn ) time, where T Enn is the cost of a Euclidean nearest-neighbor query. Hence one can construct a p-wspm k-NN graph in O(nk 2 T Enn ). As we typically have k n, i.e. k = O(log(n)) or even k = O(1), this means that constructing a p-wspm k-NN graph requires only marginally more time than constructing a Euclidean k-NN graph (which requires O(nkT Enn )).4. We verify experimentally that using a p-wspm in lieu of the Euclidean metric results in an appreciable increase in clustering accuracy, at the cost of a small increase in run time, for a wide range of real and synthetic data sets.After establishing notation and surveying the literature in §2, we prove our main results in §3 and §4. In §5 we present our algorithm for computing k nearest neighbors in any p-wspm, while in §6 we report the results of our numerical experiments.

Hyperspectral Image Clustering with Spatially-Regularized Ultrametrics

“…Compared to the 1 geodesic, which is density-agnostic, the UPD prefers paths that avoid low-density regions. Like classical shortest paths, it may be computed efficiently using a Dijkstra-type algorithm [35]. A comparison of Euclidean distances and UPD is in Figure 1.…”

Balancing Geometry and Density: Path Distances on High-Dimensional Data

“…Early uses of density-based distances for interpolation [45] led to the formulation of PWSPD in the context of unsupervised and semisupervised learning and applications [24,50,13,44,14,11,39,38,34,53,12]. It will occasionally be useful to think of p p (•, •) as the path distance in the complete graph on X with edge weights x i − x j p , which we shall denote G p X .…”

“…, where C d is a constant that depends exponentially on the intrinsic dimensionality of the data [38]. When p is large, the PWSPDs derived from a complete graph are known in some cases to be the same as the PWSPDs derived from a kNN graph under particular scalings of k n. This provides a significant computational advantage, since kNN graphs are much sparser, and reduces the complexity of computing all-pairs PWSPD to O(kn 2 ) [33].…”

Clustering and visualization of single-cell RNA-seq data using path metrics

“…To achieve both of these goals, we propose an embedding method based on the power weighted path metric. These metrics balance density and geometry considerations in the data, making them useful for many machine learning tasks such as clustering and semi-supervised learning (18)(19)(20)(21)(22)(23)(24)(25)(26). They have performed well in numerous applications especially imaging (24,25,27,28), but their usefulness for the analysis of single cell RNA sequence data remains unexplored.…”