Entropy-Rate Clustering: Cluster Analysis via Maximizing a Submodular Function Subject to a Matroid Constraint

• Published in: IEEE transactions on pattern analysis and machine intelligence Vol. 36; no. 1; pp. 99 - 112
• Main Authors: Ming-Yu Liu, Tuzel, Oncel, Ramalingam, Srikumar, Chellappa, Rama
• Format: Journal Article
• Language: English
• Published: Los Alamitos, CA IEEE 01.01.2014; IEEE Computer Society; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; Applied sciences; Artificial intelligence; Clustering; Clustering algorithms; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Data processing. List processing. Character string processing; discrete optimization; Entropy; Exact sciences and technology; Graph theory; Heuristic; Image segmentation; Information retrieval. Graph; information theory; Linear programming; Mathematics; Memory organisation. Data processing; Pattern recognition. Digital image processing. Computational geometry; Sciences and techniques of general use; Software; submodular function; superpixel segmentation; Theoretical computing; Topology; Uncertainty; Cluster analysis; Data analysis; Graph construction; Combinatorial problem; Random walk; Matroid; Group size; Cluster; Entropy; Graph theory; Topology; Clustering; Random graph; superpixel segmentation; Submodular function; Image segmentation; Vector space; Greedy algorithm; discrete optimization; Discrete programming; Metric; Objective function; Linear space; Information theory
• ISSN: 0162-8828; 1939-3539; 2160-9292; 1939-3539
• DOI: 10.1109/TPAMI.2013.107

We propose a new objective function for clustering. This objective function consists of two components: the entropy rate of a random walk on a graph and a balancing term. The entropy rate favors formation of compact and homogeneous clusters, while the balancing function encourages clusters with similar sizes and penalizes larger clusters that aggressively group samples. We present a novel graph construction for the graph associated with the data and show that this construction induces a matroid--a combinatorial structure that generalizes the concept of linear independence in vector spaces. The clustering result is given by the graph topology that maximizes the objective function under the matroid constraint. By exploiting the submodular and monotonic properties of the objective function, we develop an efficient greedy algorithm. Furthermore, we prove an approximation bound of 1/2 for the optimality of the greedy solution. We validate the proposed algorithm on various benchmarks and show its competitive performances with respect to popular clustering algorithms. We further apply it for the task of superpixel segmentation. Experiments on the Berkeley segmentation data set reveal its superior performances over the state-of-the-art superpixel segmentation algorithms in all the standard evaluation metrics.