Pinch Ratio Clustering from a Topologically Intrinsic Lexicographic Ordering

Abstract

This paper introduces an algorithm for determining data clusters called TILO/PRC (Topologically Intrinsic Lexicographic Ordering/Pinch Ratio Clustering). The theoretical foundation for this algorithm, developed in [14], uses ideas from topology (particularly knot theory) suggesting that it should be very flexible and robust with respect to noise.The TILO portion of the algorithm progressively improves a linear ordering of the points in a data set until the ordering satisfies a topological condition called strongly irreducible. The PRC algorithm then divides the data set based on this ordering and a heuristic metric called the pinch ratio.We demonstrate the effectiveness of TILO/PRC for finding clusters in a wide variety of real and synthetic data sets and compare the results to existing clustering methods. Moreover, because the output of TILO depends on the initial ordering, we consider the effects of different random orderings on the final clusters defined by PRC, and show that choosing an initial ordering based on a different clustering algorithm can improve the final clusters. These results verify that both the theoretical foundations of TILO and the heuristic notion of pinch ratio are reasonable.

HMGB1 coordinates SASP‐related chromatin folding and RNA homeostasis on the path to senescence

“…In brief, topologically intrinsic ordering was used to permutate the linear order of TADs (as the starting organization level in the Hi-C matrices) until a certain "robustly irreducible" topological condition is satisfied. Then, the "pinch ratio" algorithm (Heisterkamp & Johnson, 2013) was applied to heuristically slice the network at connections between TADs exhibiting local interaction minima, while also considering noise in the matrices. Finally, this analysis returns a list of TADs grouped into multiple clusters in cis, also via its built-in measure for network robustness defining the end-point.…”

HMGB1 coordinates SASP-related chromatin folding and RNA homeostasis on the path to senescence

“…In brief, topologically-intrinsic ordering was used to permutate the linear order of TADs (as a starting organization level in the Hi-C matrices) until a certain "robustly irreducible" topological condition is satisfied. Then, the "pinch ratio" algorithm is used (Heisterkamp and Johnson, 2013) is applied to heuristically slice the network at connections between TADs were local interaction minima are, while also considering noise in the matrices. Finally, this analysis returns a list of TADs grouped into multiple clusters in cis, also via its built-in measure for network robustness defining the end-point.…”

An application of topological graph clustering to protein function prediction

“…First we describe the TILO/PRC algorithm, which clusters the vertices of a graph by finding a linear order on the vertices with nice properties. See [4,5] for more details. Let G = (V, E) be a graph, where V is the set of vertices and E is a set of weighted edges.…”

“…Note that we are free to pick the index of any local minimum of the boundary vector in order to partition the data. The pinch ratio cut, defined in [4], is one heuristic for choosing such an index which has been shown to work well in practice. However, in the present work we will always make cuts at every index corresponding to a local minimum.…”

“…In this paper, we apply a novel semisupervised topological data analysis approach to the same problem. This algorithm is based on the TILO/PRC (Topologically Intrinsic Lexicographic Ordering/Pinch Cluster Ratio) algorithm developed by the second and third author [4,5] and described in the next section.…”