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There has been continued interest in seeking a theorem describing optimal low-rank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rank-r… (More)

Recently proposed algorithms for nonlinear dimensionality reduction fall broadly into two categories which have different advantages and disadvantages: global (Isomap [1,2]), and local (Locally Linear Embedding [3], Laplacian Eigenmaps [4]). In this paper we describe variants of the Isomap algorithm which overcome two of the apparent disadvantages of the… (More)

This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object. It is now widely recognised that this kind of topological analysis can give qualitative information about data sets which is not readily available by other means. In particular, it can be an aid to… (More)

In this paper, we discuss a computationally efficient approximation to the classical multidimensional scaling (MDS) algorithm, called Landmark MDS (LMDS), for use when the number of data points is very large. The first step of the algorithm is to run classical MDS to embed a chosen subset of the data, referred to as the 'landmark points', in a… (More)

— We present Gradient Landmark-Based Distributed Routing (GLIDER), a novel naming/addressing scheme and associated routing algorithm, for a network of wireless communicating nodes. We assume that the nodes are fixed (though their geographic locations are not necessarily known), and that each node can communicate wirelessly with some of its geographic… (More)

0 Introduction In [1] Tenenbaum, de Silva and Langford consider the problem of non-linear dimensionality reduction: discovering intrinsically low-dimensional structures embedded in high-dimensional data sets. They describe an algorithm, called Isomap, and demonstrate its successful application to several real and synthetic data sets. In this paper, we… (More)

In this study we concentrate on qualitative topological analysis of the local behavior of the space of natural images. To this end, we use a space of 3 by 3 high-contrast patches M. We develop a theoretical model for the high-density 2-dimensional submanifold of M showing that it has the topology of the Klein bottle. Using our topological software package… (More)

We introduce tools from computational homology to verify coverage in a sensor network. Our methods are unique in that, while they are coordinate-free and assume no localization or orientation capabilities for the nodes, there are also no probabilistic assumptions. We demonstrate the robustness of the techniques by adapting them to a variety of settings,… (More)

We study the problem of computing zigzag persistence of a sequence of homology groups and study a particular sequence derived from the levelsets of a real-valued function on a topological space. The result is a local, symmetric interval descriptor of the function. Our structural results establish a connection between the zigzag pairs in this sequence and… (More)

— We consider coverage problems in robot sensor networks with minimal sensing capabilities. In particular, we demonstrate that a " blind " swarm of robots with no localization and only a weak form of distance estimation can rigorously determine coverage in a bounded planar domain of unknown size and shape. The methods we introduce come from algebraic… (More)