Scientists working with large volumes of high-dimensional data, such as global climate patterns, stellar spectra, or human gene distributions, regularly confront the problem of dimensionality reduction: finding meaningful low-dimensional structures hidden in their high-dimensional observations. The human brain confronts the same problem in everyday perception, extracting from its high-dimensional sensory inputs-30,000 auditory nerve fibers or 10(6) optic nerve fibers-a manageably small number of perceptually relevant features. Here we describe an approach to solving dimensionality reduction problems that uses easily measured local metric information to learn the underlying global geometry of a data set. Unlike classical techniques such as principal component analysis (PCA) and multidimensional scaling (MDS), our approach is capable of discovering the nonlinear degrees of freedom that underlie complex natural observations, such as human handwriting or images of a face under different viewing conditions. In contrast to previous algorithms for nonlinear dimensionality reduction, ours efficiently computes a globally optimal solution, and, for an important class of data manifolds, is guaranteed to converge asymptotically to the true structure.

Information Dynamics Lab, HP Labs. I.zhang@hp.com Ahtract-We present Gradient Landmark-Based Distributed Routing (GLIDER), a novel naminghddressing scheme and associated routing algorithm, for a network of wireless communicating nodes. We assume that the nodes are fixed (though their geographic Iocations are not necessarily known), and that each node can communicate wirelessly with some of its geographic neighbors-a common scenario in sensor networks. We develop a protocol which in a preprocessing phase discovers the glohal topology of the sensor field and, as a byproduct, partitions the nodes into routable tiles-regions where the node placement is sufficiently dense and regular that Iocal greedy methods can work well. Such gIobal topology includes not just connectivity but also higher order topological features, such as the presence of holes. We address each node by the name of the tile containing it and a set of local coordinates derived from connectivity graph distances between the node and certain landmark nodes associated with its own and neighboring tiles. We use the tile adjacency graph for global route planning and the local coordinates for realizing actual inter-and intra-tile routes. We show that efficient loadbalanced global routing can be implemented quite simply using such a scheme. BACKGROUNDTechniques for routing information are central to all communication networks. Routing algorithms are intimately coupled to the way that nodes in the network are addressed or named. Such algorithms fall somewhere in the spectrum from proactive to reactive [151, according to the extent of precomputation done to facilitate route discovery. In stable networks with powerful nodes, such as the Internet. routing tables in special router nodes are proactively maintained and take advantage of the hierarchical structure of IP addresses to enable route discovery. At the other end, in ad hoc sensor and communication networks, where topology changes are frequent and node hardware less powerful, reactwe protocols that discover a route on-demand become desirable. Unfortunately, in the absence of auxiliary data structures, reactive protocols such as AODV [12] or DSR [XI, may resort to flooding the network in order to discover the desired route.In this paper we are primarily interested in routing on wireless sensor networks. Such networks are often deployed in settings where the nodes operate untethered; thus power conservation becomes a serious concem and flooding is undesirabte. Early uses of sensor networks were primarily data collection applications, requiring the one-time construction of aggregation or broadcast trees. As the sophistication of sensor network applications increases, however, there is more drmand for point-to-point routing of information to support data centric storage [ 141 and more complex database-like queries and operations. Examples include multi-resolution storage, range searching, and the like. A survey of networking and data storage techniques for sensor networks is given in [20]. While the fragile ...