PCA Beyond The Concept of Manifolds: Principal Trees, Metro Maps, and Elastic Cubic Complexes

Multidimensional data distributions can have complex topologies and variable local dimensions. To approximate complex data, we propose a new type of low-dimensional ``principal object'': a principal cubic complex. This complex is a generalization of linear and non-linear principal manifolds and includes them as a particular case. To construct such an object, we combine a method of topological grammars with the minimization of an elastic energy defined for its embedment into multidimensional data space. The whole complex is presented as a system of nodes and springs and as a product of one-dimensional continua (represented by graphs), and the grammars describe how these continua transform during the process of optimal complex construction. The simplest case of a topological grammar (``add a node'', ``bisect an edge'') is equivalent to the construction of ``principal trees'', an object useful in many practical applications. We demonstrate how it can be applied to the analysis of bacterial genomes and for visualization of cDNA microarray data using the ``metro map'' representation. The preprint is supplemented by animation: ``How the topological grammar constructs branching principal components (AnimatedBranchingPCA.gif)''.Comment: 19 pages, 8 figure

Elastic Maps and Nets for Approximating Principal Manifolds and Their Application to Microarray Data Visualization

Principal manifolds are defined as lines or surfaces passing through ``the middle'' of data distribution. Linear principal manifolds (Principal Components Analysis) are routinely used for dimension reduction, noise filtering and data visualization. Recently, methods for constructing non-linear principal manifolds were proposed, including our elastic maps approach which is based on a physical analogy with elastic membranes. We have developed a general geometric framework for constructing ``principal objects'' of various dimensions and topologies with the simplest quadratic form of the smoothness penalty which allows very effective parallel implementations. Our approach is implemented in three programming languages (C++, Java and Delphi) with two graphical user interfaces (VidaExpert http://bioinfo.curie.fr/projects/vidaexpert and ViMiDa http://bioinfo-out.curie.fr/projects/vimida applications). In this paper we overview the method of elastic maps and present in detail one of its major applications: the visualization of microarray data in bioinformatics. We show that the method of elastic maps outperforms linear PCA in terms of data approximation, representation of between-point distance structure, preservation of local point neighborhood and representing point classes in low-dimensional spaces.Comment: 35 pages 10 figure

Self-ordering of a Ge island single layer induced by Si overgrowth

We provide a direct experimental proof and the related modeling of the role played by Si overgrowth in promoting the lateral ordering of Ge islands grown by chemical vapor deposition on Si(001). The deposition of silicon induces a shape transformation, from domes to truncated pyramids with a larger base, generating an array of closely spaced interacting islands. By modeling, we show that the resulting gradient in the chemical potential across the island should be the driving force for a selective flow of both Ge and Si atoms at the surface and, in turn, to a real motion of the dots, favoring the lateral order

Spontaneous Ge island ordering promoted by partial silicon capping

In this paper, we show that lateral arrangement of Ge/Si(00 1) self-assembled islands in a square array oriented along the [100]-[0 10] directions can be obtained through the lateral displacement of the islands themselves. We found that when the deposited islands are exposed to an external silicon flux, the impinging silicon atoms induce Ge-Si intermixing resulting in island shape transformation from domes to large pyramids. This transformation leads to an array of closely spaced islands interacting elastically among themselves through the substrate. By means of atomistic simulations we demonstrate that this elastic repulsion drives a net flux of Ge and Si atoms from one side to the other side of the islands, leading to a lateral displacement of the whole island. This displacement ends when the two islands are sufficiently far away, or when another island is approached during the motion. In a dense ensemble of islands, this mechanism drives the tendency to order observed experimentally. (C) 2006 Elsevier Ltd. All rights reserved