Clustering Tree-Structured Data on Manifold

Abstract

Tree-structured data usually contain both topological and geometrical information, and are necessarily considered on manifold instead of Euclidean space for appropriate data parameterization and analysis. In this study, we propose a novel tree-structured data parameterization, called Topology-Attribute matrix (T-A matrix), so the data clustering task can be conducted on matrix manifold. We incorporate the structure constraints embedded in data into the non-negative matrix factorization method to determine meta-trees from the T-A matrix, and the signature vector of each single tree can then be extracted by meta-tree decomposition. The meta-tree space turns out to be a cone space, in which we explore the distance metric and implement the clustering algorithm based on the concepts like Fréchet mean. Finally, the T-A matrix based clustering (TAMBAC) framework is evaluated and compared using both simulated data and real retinal images to illus trate its efficiency and accuracy.

Integration of Cancer Genomics Data for Tree‐based Dimensionality Reduction and Cancer Outcome Prediction

“…Secondly, tree structure based optimized parameters with cross‐validation was investigated to reason the rich genotype information embedded in tree‐structured data. Specifically, tree‐structured data usually contain both topological information and certain attributes associated with each tree node or edge . Thirdly, the tree‐based dimensionality reduction method was developed to obtain the compact feature representation for clustering problems .…”

“…Specifically, tree-struc-tured data usually contain both topological information and certain attributes associated with each tree node or edge. [30] Thirdly, the tree-based dimensionality reduction method was developed to obtain the compact feature representation for clustering problems. [35] By classifying patterns into different groups in an unsupervised manner, tree-structured data clustering could predict clinical outcome of cancer.…”

“…Among them, the Tree Bisection and Reconnection (TBR) distance, the Tree Edit (TED) distance, and the Quotient Euclidean (QED) distance have been applied to tree‐structured data parameterization. More specifically, the T‐A matrix based clustering (TAMBAC) framework is developed for tree‐structured data clustering based on a novel tree data parameterization (called TAMP) and the structure‐constrained nonnegative matrix factorization (SCNMF) . TAMP represents both the topological and geometric information of trees in forms of matrices so that the tree‐structured data analysis problem could be solved in the matrix manifold realm.…”

“…More specifically, the T-A matrix based clustering (TAMBAC) framework is developed for treestructured data clustering based on a novel tree data parameterization (called TAMP) and the structure-constrained nonnegative matrix factorization (SCNMF). [30] TAMP represents both the topological and geometric information of trees in forms of matrices so that the tree-structured data analysis problem could be solved in the matrix manifold realm. The performance demonstrates the efficacy and accuracy of the TAMBAC framework.…”

Hypothesis Testing for Two Sample Comparison of Network Data

“…The definition of QED is consistent with the long standing definition of graph edit distance (Sanfeliu and Fu, 1983); however, the calculation of QED is very challenging because there may exist numerous paths between two network data points such that it becomes impractical to enumerate all of them to find the shortest one (Lu and Miao, 2016). Therefore, an efficient approximation to QED is necessary.…”