RIPless compressed sensing from anisotropic measurements

Compressed sensing is the art of reconstructing a sparse vector from its inner products with respect to a small set of randomly chosen measurement vectors. It is usually assumed that the ensemble of measurement vectors is in isotropic position in the sense that the associated covariance matrix is proportional to the identity matrix. In this paper, we establish bounds on the number of required measurements in the anisotropic case, where the ensemble of measurement vectors possesses a non-trivial covariance matrix. Essentially, we find that the required sampling rate grows proportionally to the condition number of the covariance matrix. In contrast to other recent contributions to this problem, our arguments do not rely on any restricted isometry properties (RIP's), but rather on ideas from convex geometry which have been systematically studied in the theory of low-rank matrix recovery. This allows for a simple argument and slightly improved bounds, but may lead to a worse dependency on noise (which we do not consider in the present paper).Comment: 19 pages. To appear in Linear Algebra and its Applications, Special Issue on Sparse Approximate Solution of Linear System

Methods and means used in programming intelligent searches of technical documents

In order to meet the data research requirements of the Safety, Reliability & Quality Assurance activities at Kennedy Space Center (KSC), a new computer search method for technical data documents was developed. By their very nature, technical documents are partially encrypted because of the author's use of acronyms, abbreviations, and shortcut notations. This problem of computerized searching is compounded at KSC by the volume of documentation that is produced during normal Space Shuttle operations. The Centralized Document Database (CDD) is designed to solve this problem. It provides a common interface to an unlimited number of files of various sizes, with the capability to perform any diversified types and levels of data searches. The heart of the CDD is the nature and capability of its search algorithms. The most complex form of search that the program uses is with the use of a domain-specific database of acronyms, abbreviations, synonyms, and word frequency tables. This database, along with basic sentence parsing, is used to convert a request for information into a relational network. This network is used as a filter on the original document file to determine the most likely locations for the data requested. This type of search will locate information that traditional techniques, (i.e., Boolean structured key-word searching), would not find

An exact prediction of [script N] = 4 supersymmetric Yang–Mills theory for string theory

We propose that the expectation value of a circular BPS-Wilson loop in [script N] = 4 supersymmetric Yang–Mills can be calculated exactly, to all orders in a 1/N expansion and to all orders in g2N. Using the AdS/CFT duality, this result yields a prediction of the value of the string amplitude with a circular boundary to all orders in alpha[prime] and to all orders in gs. We then compare this result with string theory. We find that the gauge theory calculation, for large g2N and to all orders in the 1/N2 expansion, does agree with the leading string theory calculation, to all orders in gs and to lowest order in alpha[prime]. We also find a relation between the expectation value of any closed smooth Wilson loop and the loop related to it by an inversion that takes a point along the loop to infinity, and compare this result, again successfully, with string theory

A line of CFTs: from generalized free fields to SYK

We point out that there is a simple variant of the SYK model, which we call cSYK, that is $SL(2,R)$ invariant for all values of the coupling. The modification consists of replacing the UV part of the SYK action with a quadratic bilocal term. The corresponding bulk dual is a non-gravitational theory in a rigid AdS$_2$ background. At weak coupling cSYK is a generalized free field theory; at strong coupling, it approaches the infrared of SYK. The existence of this line of fixed points explains the previously found connection between the three-point function of bilinears in these two theories at large $q$.Comment: 26 pages, v