Principal manifolds and graphs in practice: from molecular biology to dynamical systems

We present several applications of non-linear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen's self-organizing maps, a class of artificial neural networks. On several examples we show advantages of using non-linear objects for data approximation in comparison to the linear ones. We propose four numerical criteria for comparing linear and non-linear mappings of datasets into the spaces of lower dimension. The examples are taken from comparative political science, from analysis of high-throughput data in molecular biology, from analysis of dynamical systems.Comment: 12 pages, 9 figure

Angle Tree: Nearest Neighbor Search in High Dimensions with Low Intrinsic Dimensionality

We propose an extension of tree-based space-partitioning indexing structures for data with low intrinsic dimensionality embedded in a high dimensional space. We call this extension an Angle Tree. Our extension can be applied to both classical kd-trees as well as the more recent rp-trees. The key idea of our approach is to store the angle (the "dihedral angle") between the data region (which is a low dimensional manifold) and the random hyperplane that splits the region (the "splitter"). We show that the dihedral angle can be used to obtain a tight lower bound on the distance between the query point and any point on the opposite side of the splitter. This in turn can be used to efficiently prune the search space. We introduce a novel randomized strategy to efficiently calculate the dihedral angle with a high degree of accuracy. Experiments and analysis on real and synthetic data sets shows that the Angle Tree is the most efficient known indexing structure for nearest neighbor queries in terms of preprocessing and space usage while achieving high accuracy and fast search time.Comment: To be submitted to IEEE Transactions on Pattern Analysis and Machine Intelligenc

Intrinsic Inference on the Mean Geodesic of Planar Shapes and Tree Discrimination by Leaf Growth

For planar landmark based shapes, taking into account the non-Euclidean geometry of the shape space, a statistical test for a common mean first geodesic principal component (GPC) is devised. It rests on one of two asymptotic scenarios, both of which are identical in a Euclidean geometry. For both scenarios, strong consistency and central limit theorems are established, along with an algorithm for the computation of a Ziezold mean geodesic. In application, this allows to verify the geodesic hypothesis for leaf growth of Canadian black poplars and to discriminate genetically different trees by observations of leaf shape growth over brief time intervals. With a test based on Procrustes tangent space coordinates, not involving the shape space's curvature, neither can be achieved.Comment: 28 pages, 4 figure

On the Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages

We extend our study of Motion Planning via Manifold Samples (MMS), a general algorithmic framework that combines geometric methods for the exact and complete analysis of low-dimensional configuration spaces with sampling-based approaches that are appropriate for higher dimensions. The framework explores the configuration space by taking samples that are entire low-dimensional manifolds of the configuration space capturing its connectivity much better than isolated point samples. The contributions of this paper are as follows: (i) We present a recursive application of MMS in a six-dimensional configuration space, enabling the coordination of two polygonal robots translating and rotating amidst polygonal obstacles. In the adduced experiments for the more demanding test cases MMS clearly outperforms PRM, with over 20-fold speedup in a coordination-tight setting. (ii) A probabilistic completeness proof for the most prevalent case, namely MMS with samples that are affine subspaces. (iii) A closer examination of the test cases reveals that MMS has, in comparison to standard sampling-based algorithms, a significant advantage in scenarios containing high-dimensional narrow passages. This provokes a novel characterization of narrow passages which attempts to capture their dimensionality, an attribute that had been (to a large extent) unattended in previous definitions.Comment: 20 page