Fixed-Rank Approximation of a Positive-Semidefinite Matrix from Streaming Data

Several important applications, such as streaming PCA and semidefinite programming, involve a large-scale positive-semidefinite (psd) matrix that is presented as a sequence of linear updates. Because of storage limitations, it may only be possible to retain a sketch of the psd matrix. This paper develops a new algorithm for fixed-rank psd approximation from a sketch. The approach combines the Nystrom approximation with a novel mechanism for rank truncation. Theoretical analysis establishes that the proposed method can achieve any prescribed relative error in the Schatten 1-norm and that it exploits the spectral decay of the input matrix. Computer experiments show that the proposed method dominates alternative techniques for fixed-rank psd matrix approximation across a wide range of examples

Practical sketching algorithms for low-rank matrix approximation

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data

Results of a novel screening tool measuring dietary sodium knowledge in patients with chronic kidney disease.

BackgroundReducing dietary sodium has potential to benefit patients with chronic kidney disease (CKD). Little research is available defining dietary sodium knowledge gaps in patients with pre-dialysis CKD. We designed a brief screening tool to rapidly identify patient knowledge gaps related to dietary sodium for patients with CKD not yet on dialysis.MethodsA Short Sodium Knowledge Survey (SSKS) was developed and administered to patients with pre-dialysis CKD. We also asked patients if they received counseling on dietary sodium reduction and about recommended intake limits. We performed logistic regression to examine the association between sodium knowledge and patient characteristics. Characteristics of patients who answered all SSKS questions correctly were compared to those who did not.ResultsOne-hundred fifty-five patients were surveyed. The mean (SD) age was 56.6 (15.1) years, 84 (54%) were men, and 119 (77%) were white. Sixty-seven patients (43.2%) correctly identified their daily intake sodium limit. Fifty-eight (37.4%) were unable to answer all survey questions correctly. In analysis adjusted for age, sex, race, education, health literacy, CKD stage, self-reported hypertension and attendance in a kidney education class, women and patients of non-white race had lower odds of correctly answering survey questions (0.36 [0.16,0.81]; p = 0.01 women versus men and 0.33 [0.14,0.76]; p = 0.01 non-white versus white, respectively).ConclusionsOur survey provides a mechanism to quickly identify dietary sodium knowledge gaps in patients with CKD. Women and patients of non-white race may have knowledge barriers impeding adherence to sodium reduction advice

Atom clusters and vibrational excitations in chemically-disordered Pt357Fe

Inelastic nuclear resonant scattering spectra of Fe-57 atoms were measured on crystalline alloys of Pt3Fe-57 that were chemically disordered, partially ordered, and L1(2) ordered. Phonon partial density of states curves for Fe-57 were obtained from these spectra. Upon disordering, about 10% of the spectral intensity underwent a distinct shift from 25 to 19 meV. This change in optical modes accounted for most of the change of the vibrational entropy of disordering contributed by Fe atoms, which was (+0.10 +/- 0.03) k(B) (Fe atom)(-1). Prospects for parametrizing the vibrational entropy with low-order cluster variables were assessed. To calculate the difference in vibrational entropy of the disordered and ordered alloys, the clusters must be large enough to account for the abundances of several of the atom configurations of the first-nearest-neighbor shell about the Fe-57 atoms

Local Chemical Environments and the Phonon Partial Densities of States of 57Fe in 57Fe3Al

Inelastic nuclear resonant scattering spectra were measured on alloys of Fe3Al that were chemically disordered, partially ordered, and D03 ordered. The features in the phonon partial density of states of 57Fe were found to change systematically with chemical short-range order in the alloy. Changes in the phonon partial density of states were modeled successfully by assigning vibrational spectra to 57Fe atoms in different first-nearest-neighbor chemical environments

Ant colony optimization for the single model U-type assembly line balancing problem

Cataloged from PDF version of article.An assembly line is a production line in which units move continuously through a sequence of stations. The assembly line balancing problem is defined as the allocation of tasks to an ordered sequence of stations subject to precedence constraints with the objective of optimizing a performance measure. In this paper, we propose ant colony algorithms to solve the single-model U-type assembly line balancing problem. We conduct an extensive experimental study in which the performance of the proposed algorithm is compared against best known algorithms reported in the literature. The results indicate that the proposed algorithms display very competitive performance against them. & 2009 Elsevier B.V. All rights reserved

Randomized Single-View Algorithms for Low-Rank Matrix Approximation

This paper develops a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by computer experiments

Quantum Algorithms for Learning and Testing Juntas

In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: - whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; - with no access to any classical or quantum membership ("black-box") queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; - which require only a few quantum examples but possibly many classical random examples (which are considered quite "cheap" relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: - We give an algorithm for testing k-juntas to accuracy $\epsilon$ that uses $O(k/\epsilon)$ quantum examples. This improves on the number of examples used by the best known classical algorithm. - We establish the following lower bound: any FS-based k-junta testing algorithm requires $\Omega(\sqrt{k})$ queries. - We give an algorithm for learning $k$-juntas to accuracy $\epsilon$ that uses $O(\epsilon^{-1} k\log k)$ quantum examples and $O(2^k \log(1/\epsilon))$ random examples. We show that this learning algorithms is close to optimal by giving a related lower bound.Comment: 15 pages, 1 figure. Uses synttree package. To appear in Quantum Information Processin