Abstract-In many applications, we are given a finite set of data points sampled from a data manifold and represented as a graph with edge weights determined by pairwise similarities of the samples. Often the pairwise similarities (which are also called affinities) are unreliable due to noise or due to intrinsic difficulties in estimating similarity values of the samples. As observed in several recent approaches, more reliable similarities can be obtained if the original similarities are diffused in the context of other data points, where the context of each point is a set of points most similar to it. Compared to the existing methods, our approach differs in two main aspects. First, instead of diffusing the similarity information on the original graph, we propose to utilize the tensor product graph (TPG) obtained by the tensor product of the original graph with itself. Since TPG takes into account higher order information, it is not a surprise that we obtain more reliable similarities. However, it comes at the price of higher order computational complexity and storage requirement. The key contribution of the proposed approach is that the information propagation on TPG can be computed with the same computational complexity and the same amount of storage as the propagation on the original graph. We prove that a graph diffusion process on TPG is equivalent to a novel iterative algorithm on the original graph, which is guaranteed to converge. After its convergence we obtain new edge weights that can be interpreted as new, learned affinities. We stress that the affinities are learned in an unsupervised setting. We illustrate the benefits of the proposed approach for data manifolds composed of shapes, images, and image patches on two very different tasks of image retrieval and image segmentation. With learned affinities, we achieve the bull's eye retrieval score of 99.99 percent on the MPEG-7 shape dataset, which is much higher than the state-of-the-art algorithms. When the data points are image patches, the NCut with the learned affinities not only significantly outperforms the NCut with the original affinities, but it also outperforms state-of-the-art image segmentation methods.

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A novel and efficient invertible transform for shape segmentation is defined that serves to localize and extract shape characteristics. This transform-the chordal axis transform (CAT}-remedies the deficiencies of the well-known medial axis transform (MAT). The CAT is applicable to shapes with discretized boundaries without restriction on the sparsity or regularity of the discretization. Using Delaunay triangulations of shape interiors, the CAT induces structural segmentation of shapes into limb and torso chain complexes of triangles. This enables the localization, extraction, and characterization of the morphological features of shapes. It also yields a pruning scheme for excising morphologically insignificant features and simplifying shape boundaries and descriptions. Furthermore, it enables the explicit characterization and exhaustive enumeration of primary, semantically salient, shape features. Finally, a process to characterize and represent a shape in terms of its morphological features is presented. This results in the migration of a shape from its affine description to an invariant, and semantically salient feature-based representation in the form of attributed planar graphs. The research described here is part of a larger effort aimed at automating image understanding and computer vision , 6

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As quantum computers become available to the general public, the need has arisen to train a cohort of quantum programmers, many of whom have been developing classical computer programs for most of their careers. While currently available quantum computers have less than 100 qubits, quantum computing hardware is widely expected to grow in terms of qubit count, quality, and connectivity. This review aims to explain the principles of quantum programming, which are quite different from classical programming, with straightforward algebra that makes understanding of the underlying fascinating quantum mechanical principles optional. We give an introduction to quantum computing algorithms and their implementation on real quantum hardware. We survey 20 different quantum algorithms, attempting to describe each in a succinct and self-contained fashion. We show how these algorithms can be implemented on IBM’s quantum computer, and in each case, we discuss the results of the implementation with respect to differences between the simulator and the actual hardware runs. This article introduces computer scientists, physicists, and engineers to quantum algorithms and provides a blueprint for their implementations.

Fault-tolerance is an important issue in network design because sensor networks must function in a dynamic, uncertain world. In this paper, we propose a functional characterization of the fault-tolerant integration of abstract interval estimates. This model provides a test bed for a general framework which we hope to develop to address the general problem of faulttolerant integration of abstract sensor estimates. We further propose a scheme for narrowing the width of the sensor output in a specific failure model and give it a functional representation.The main distinguishing feature of our model over the original Marzullo's model is in reducing the width of the output interval estimate significantly in most cases where the number of sensors involved is large.