Conversions between barycentric, RKFUN, and Newton representations of rational interpolants

We derive explicit formulas for converting between rational interpolants in barycentric, rational Krylov (RKFUN), and Newton form. We show applications of these conversions when working with rational approximants produced by the AAA algorithm [Y. Nakatsukasa, O. S\`ete, L. N. Trefethen, arXiv preprint 1612.00337, 2016] within the Rational Krylov Toolbox and for the solution of nonlinear eigenvalue problems

Personalized Prediction and Sparsity Pursuit in Latent Factor Models

<p>Personalized information filtering extracts the information specifically relevant to a user, predicting his/her preference over a large number of items, based on the opinions of users who think alike or its content. This problem is cast into the framework of regression and classification, where we integrate additional user-specific and content-specific predictors in partial latent models, for higher predictive accuracy. In particular, we factorize a user-over-item preference matrix into a product of two matrices, each representing a user’s preference and an item preference by users. Then we propose a likelihood method to seek a sparsest latent factorization, from a class of overcomplete factorizations, possibly with a high percentage of missing values. This promotes additional sparsity beyond rank reduction. Computationally, we design methods based on a “decomposition and combination” strategy, to break large-scale optimization into many small subproblems to solve in a recursive and parallel manner. On this basis, we implement the proposed methods through multi-platform shared-memory parallel programming, and through Mahout, a library for scalable machine learning and data mining, for mapReduce computation. For example, our methods are scalable to a dataset consisting of three billions of observations on a single machine with sufficient memory, having good timings. Both theoretical and numerical investigations show that the proposed methods exhibit a significant improvement in accuracy over state-of-the-art scalable methods. Supplementary materials for this article are available online.</p

Significant SNPs and each one's associated phenotype numbers at the significance level of .

<p>Top 3 SNP-phenotype associations are listed with corresponding P-values.</p

The baseline characteristics of 638 subjects, including gender, age, years of education, handedness (R/L) and intracranial volume (ICV).

<p>P-values were calculated to test for differences among the diagnostic groups, HC, MCI and AD.</p

Comparison of the Manhattan plots for genome-wide p-values for phenotype volume of right inferior lateral ventricle from longitudinal analysis (left) and cross-sectional analysis (right); SNP rs429358 is not included.

<p>Comparison of the Manhattan plots for genome-wide p-values for phenotype volume of right inferior lateral ventricle from longitudinal analysis (left) and cross-sectional analysis (right); SNP rs429358 is not included.</p

Trajectories of phenotype left hippocampus volume over time (in months) in three allele groups of SNP rs2075650.

<p>Trajectories of phenotype left hippocampus volume over time (in months) in three allele groups of SNP rs2075650.</p

Comparison of the Manhattan plots for genome-wide p-values for phenotype volume of right inferior lateral ventricle from longitudinal analysis (left) and cross-sectional analysis (right); SNP rs429358 is not included.

<p>Comparison of the Manhattan plots for genome-wide p-values for phenotype volume of right inferior lateral ventricle from longitudinal analysis (left) and cross-sectional analysis (right); SNP rs429358 is not included.</p

Comparison of the Q-Q plots for genome-wide p-values for phenotype volume of right inferior lateral ventricle from longitudinal analysis (left) and from cross-sectional analysis (right); SNP rs429358 is not included.

<p>Comparison of the Q-Q plots for genome-wide p-values for phenotype volume of right inferior lateral ventricle from longitudinal analysis (left) and from cross-sectional analysis (right); SNP rs429358 is not included.</p

The Q-Q plots for genome-wide p-values for phenotype left hippocampus volume from longitudinal analysis based on (a) GEE with the sandwich covariance estimator (left, inflation factor = 1.070), (b) GEE with the model-based covariance estimator (middle, inflation factor = 2.077), and (c) linear mixed model with only a random intercept term (right, inflation factor = 1.976).

<p>The Q-Q plots for genome-wide p-values for phenotype left hippocampus volume from longitudinal analysis based on (a) GEE with the sandwich covariance estimator (left, inflation factor  = 1.070), (b) GEE with the model-based covariance estimator (middle, inflation factor  = 2.077), and (c) linear mixed model with only a random intercept term (right, inflation factor  = 1.976).</p

Comparison of the Q-Q plots without (left) or with (right) top 10 PCs.

<p>Comparison of the Q-Q plots without (left) or with (right) top 10 PCs.</p