Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property

Compressed Sensing aims to capture attributes of $k$-sparse signals using very few measurements. In the standard Compressed Sensing paradigm, the \m\times \n measurement matrix \A is required to act as a near isometry on the set of all $k$-sparse signals (Restricted Isometry Property or RIP). Although it is known that certain probabilistic processes generate \m \times \n matrices that satisfy RIP with high probability, there is no practical algorithm for verifying whether a given sensing matrix \A has this property, crucial for the feasibility of the standard recovery algorithms. In contrast this paper provides simple criteria that guarantee that a deterministic sensing matrix satisfying these criteria acts as a near isometry on an overwhelming majority of $k$-sparse signals; in particular, most such signals have a unique representation in the measurement domain. Probability still plays a critical role, but it enters the signal model rather than the construction of the sensing matrix. We require the columns of the sensing matrix to form a group under pointwise multiplication. The construction allows recovery methods for which the expected performance is sub-linear in \n, and only quadratic in \m; the focus on expected performance is more typical of mainstream signal processing than the worst-case analysis that prevails in standard Compressed Sensing. Our framework encompasses many families of deterministic sensing matrices, including those formed from discrete chirps, Delsarte-Goethals codes, and extended BCH codes.Comment: 16 Pages, 2 figures, to appear in IEEE Journal of Selected Topics in Signal Processing, the special issue on Compressed Sensin

IR divergences and Regge limits of subleading-color contributions to the four-gluon amplitude in N=4 SYM Theory

We derive a compact all-loop-order expression for the IR-divergent part of the N=4 SYM four-gluon amplitude, which includes both planar and all subleading-color contributions, based on the assumption that the higher-loop soft anomalous dimension matrices are proportional to the one-loop soft anomalous dimension matrix, as has been recently conjectured. We also consider the Regge limit of the four-gluon amplitude, and we present evidence that the leading logarithmic growth of the subleading-color amplitudes is less severe than that of the planar amplitudes. We examine possible 1/N^2 corrections to the gluon Regge trajectory, previously obtained in the planar limit from the BDS ansatz. The double-trace amplitudes have Regge behavior as well, with a nonsense-choosing Regge trajectory and a Regge cut which first emerges at three loops.Comment: 29 pages; v2: corrections to sections 5.1 and 5.2, references added; v3: clarification in introduction, references added; v4: published versio

Geometry of the Welch Bounds

A geometric perspective involving Grammian and frame operators is used to derive the entire family of Welch bounds. This perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k-tensors, tight frames, homogeneous polynomials, and t-designs. In particular. a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k-tensors. It is also shown that tightness of the bounds requires tight frames. The lack of tight frames in symmetric k-tensors in many cases, however, leads to consideration of sets that come as close as possible to attaining the bounds. The geometric derivation is then extended in the setting of generalized or continuous frames. The Welch bounds for finite sets and countably infinite sets become special cases of this general setting.Comment: changes from previous version include: correction of typos, additional references added, new Example 3.

The Neurocognitive Process of Digital Radicalization: A Theoretical Model and Analytical Framework

Recent studies suggest that empathy induced by narrative messages can effectively facilitate persuasion and reduce psychological reactance. Although limited, emerging research on the etiology of radical political behavior has begun to explore the role of narratives in shaping an individual’s beliefs, attitudes, and intentions that culminate in radicalization. The existing studies focus exclusively on the influence of narrative persuasion on an individual, but they overlook the necessity of empathy and that in the absence of empathy, persuasion is not salient. We argue that terrorist organizations are strategic in cultivating empathetic-persuasive messages using audiovisual materials, and disseminating their message within the digital medium. Therefore, in this paper we propose a theoretical model and analytical framework capable of helping us better understand the neurocognitive process of digital radicalization

An Information-geometric Approach to Sensor Management

An information-geometric approach to sensor management is introduced that is based on following geodesic curves in a manifold of possible sensor configurations. This perspective arises by observing that, given a parameter estimation problem to be addressed through management of sensor assets, any particular sensor configuration corresponds to a Riemannian metric on the parameter manifold. With this perspective, managing sensors involves navigation on the space of all Riemannian metrics on the parameter manifold, which is itself a Riemannian manifold. Existing work assumes the metric on the parameter manifold is one that, in statistical terms, corresponds to a Jeffreys prior on the parameter to be estimated. It is observed that informative priors, as arise in sensor management, can also be accommodated. Given an initial sensor configuration, the trajectory along which to move in sensor configuration space to gather most information is seen to be locally defined by the geodesic structure of this manifold. Further, divergences based on Fisher and Shannon information lead to the same Riemannian metric and geodesics.Comment: 4 pages, 3 figures, to appear in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, March 201

Applications of Subleading-color Amplitudes in N=4 SYM Theory

A number of features and applications of subleading color amplitudes of N=4 SYM theory are reviewed. Particular attention is given to the IR divergences of the subleading-color amplitudes, the relationships of N=4 SYM theory to N=8 supergravity, and to geometric interpretations of one-loop subleading color and N^k MHV amplitudes of N=4 SYM theory.Comment: 39 pages; v2: minor correction

Subleading-color contributions to gluon-gluon scattering in N=4 SYM theory and relations to N=8 supergravity

We study the subleading-color (nonplanar) contributions to the four-gluon scattering amplitudes in N=4 supersymmetric SU(N) Yang-Mills theory. Using the formalisms of Catani and of Sterman and Tejeda-Yeomans, we develop explicit expressions for the infrared-divergent contributions of all the subleading-color L-loop amplitudes up to three loops, and make some conjectures for the IR behavior for arbitrary L. We also derive several intriguing relations between the subleading-color one- and two-loop four-gluon amplitudes and the four-graviton amplitudes of N=8 supergravity. The exact one- and two-loop N=8 supergravity amplitudes can be expressed in terms of the one- and two-loop N-independent N=4 SYM amplitudes respectively, but the natural generalization to higher loops fails, despite having a simple interpretation in terms of the 't Hooft picture. We also find that, at least through two loops, the subleading-color amplitudes of N=4 SYM theory have uniform transcendentality (as do the leading-color amplitudes). Moreover, the N=4 SYM Catani operators, which express the IR-divergent contributions of loop amplitudes in terms of lower-loop amplitudes, are also shown to have uniform transcendentality, and to be the maximum transcendentality piece of the QCD Catani operators.Comment: 30 pages; v2: corrections in sec. 6, minor addition in sec. 3.