Convergence of a Steepest Descent Algorithm for Ratio Cut Clustering

AbstractUnsupervised clustering of scattered, noisy and high-dimensional data points is an important anddifficult problem. Tight continuous relaxations of balanced cut problems have recently been shown toprovide excellent clustering results. In this paper, we present an explicit-implicit gradient flow schemefor the relaxed ratio cut problem, and prove that the algorithm converges to a critical point of the energy.We also show the efficiency of the proposed algorithm on the two moons dataset. 1 Introduction Partitioning data points into sensible groups is a fundamental problem in machine learning and has a widerange of applications. An efficient approach to deal with this problem is to cast the data partitioningproblem as a graph clustering problem. Given a set of data points V = {x 1 ,...,x n } and similarityweights {w i,j } 1≤i,j≤n , the clustering problem aims at finding a balanced cut of the graph of the data.In this work, we consider the balanced cut of Hagen and Kahng [5] known as ratio cut. The ratio cutproblem isMinimize RatioCut(S) =P