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Data perturbation is a technique for generating synthetic data by adding ‘noise’ to raw data, which has an array of applications in science and engineering, primarily in data security and privacy. One challenge for data perturbation is that it usually produces synthetic data resulting in information loss at the expense of privacy protection. The information loss, in turn, renders the accuracy loss for any statistical or machine learning method based on the synthetic data, weakening downstream analysis and deteriorating in machine learning. In this article, we introduce and advocate a fundamental principle of data perturbation, which requires the preservation of the distribution of raw data. To achieve this, we propose a new scheme, named data flush, which ascertains the validity of the downstream analysis and maintains the predictive accuracy of a learning task. It perturbs data nonlinearly while accommodating the requirement of strict privacy protection, for instance, differential privacy. We highlight multiple facets of data flush through examples. 

Recommender systems have been extensively used by the entertainment industry, business marketing and the biomedical industry. In addition to its capacity of providing preference-based recommendations as an unsupervised learning methodology, it has been also proven useful in sales forecasting, product introduction and other production related businesses. Since some consumers and companies need a recommendation or prediction for future budget, labor and supply chain coordination, dynamic recommender systems for precise forecasting have become extremely necessary. In this article, we propose a new recommendation method, namely the dynamic tensor recommender system (DTRS), which aims particularly at forecasting future recommendation. The proposed method utilizes a tensor-valued function of time to integrate time and contextual information, and creates a time-varying coefficient model for temporal tensor factorization through a polynomial spline approximation. Major advantages of the proposed method include competitive future recommendation predictions and effective prediction interval estimations. In theory, we establish the convergence rate of the proposed tensor factorization and asymptotic normality of the spline coefficient estimator. The proposed method is applied to simulations, IRI marketing data and Last.fm data. Numerical studies demonstrate that the proposed method outperforms existing methods in terms of future time forecasting. 

This article provides an overview of tensors, their properties, and their applications in statistics. Tensors, also known as multidimensional arrays, are generalizations of matrices to higher orders and are useful data representation architectures. We first review basic tensor concepts and decompositions, and then we elaborate traditional and recent applications of tensors in the fields of recommender systems and imaging analysis. We also illustrate tensors for network data and explore the relations among interacting units in a complex network system. Some canonical tensor computational algorithms and available software libraries are provided for various tensor decompositions. Future research directions, including tensors in deep learning, are also discussed. 

Abstract In this third paper of the series reporting on the reverberation mapping campaign of active galactic nuclei with asymmetric H β emission-line profiles, we present results for 15 Palomar–Green quasars using spectra obtained between the end of 2016–2021 May. This campaign combines long time spans with relatively high cadence. For eight objects, both the time lags obtained from the entire light curves and the measurements from individual observing seasons are provided. Reverberation mapping of nine of our targets has been attempted for the first time, while the results for six others can be compared with previous campaigns. We measure the H β time lags over periods of years and estimate their black hole masses. The long duration of the campaign enables us to investigate their broad-line region (BLR) geometry and kinematics for different years by using velocity-resolved lags, which demonstrate signatures of diverse BLR geometry and kinematics. The BLR geometry and kinematics of individual objects are discussed. In this sample, the BLR kinematics of Keplerian/virialized motion and inflow is more common than that of outflow.