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

Improving compressed sensing with the diamond norm

In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a minimal number of linear measurements. Within the paradigm of compressed sensing, this is made computationally efficient by minimizing the nuclear norm as a convex surrogate for rank. In this work, we identify an improved regularizer based on the so-called diamond norm, a concept imported from quantum information theory. We show that -for a class of matrices saturating a certain norm inequality- the descent cone of the diamond norm is contained in that of the nuclear norm. This suggests superior reconstruction properties for these matrices. We explicitly characterize this set of matrices. Moreover, we demonstrate numerically that the diamond norm indeed outperforms the nuclear norm in a number of relevant applications: These include signal analysis tasks such as blind matrix deconvolution or the retrieval of certain unitary basis changes, as well as the quantum information problem of process tomography with random measurements. The diamond norm is defined for matrices that can be interpreted as order-4 tensors and it turns out that the above condition depends crucially on that tensorial structure. In this sense, this work touches on an aspect of the notoriously difficult tensor completion problem.Comment: 25 pages + Appendix, 7 Figures, published versio

International Association of Geodesy

History, scope and recent activities of the International Association of Geodesy

Letter from Richard E. Gross of the Lewiston Public Library

September 18, 1974, Letter from Richard E. Gross, director of the Lewiston Public Library, to Charlotte Michaud. Writing excerpt in French on reverse.https://digitalcommons.usm.maine.edu/michaud-historical-notes/1003/thumbnail.jp

A Use of Theory of Constraints Thinking Processes for Improvements in the Merged Beams Experiment at Oak Ridge National Laboratory.

Thinking exercises used in the Theory of Constraints (TOC) were used to find and remove constraints at the Merged Beams Experiment at Oak Ridge National Laboratory. The goal of this project was to significantly reduce the amount of time used to take a certain type of measurement during an experimental cycle. After the TOC exercises were used, a basic plan for change was discovered. Preliminary data were taken to establish a baseline of performance from which changes were made. Post-Modification was analyzed showing the project was a success. The overlying reasoning for this exercise was to prove successfully that continuous improvement techniques used in the manufacturing industry can also be successful in a research environment. After overcoming the differences in the goals between each environment, it can be concluded that this reasoning is justified