Rank Reduction Research Articles

Sparse low rank approximation of potential energy surfaces with applications in estimation of anharmonic zero point energies and frequencies

We propose a method that exploits sparse representation of potential energy surfaces (PES) on a polynomial basis set selected by compressed sensing. The method is useful for studies involving large numbers of PES evaluations, such as the search for local minima, transition states, or integration. We apply this method for estimating zero point energies and frequencies of molecules using a three step approach. In the first step, we interpret the PES as a sparse tensor on polynomial basis and determine its entries by a compressed sensing based algorithm using only a few PES evaluations. Then, we implement a rank reduction strategy to compress this tensor in a suitable low-rank canonical tensor format using standard tensor compression tools. This allows representing a high dimensional PES as a small sum of products of one dimensional functions. Finally, a low dimensional Gauss–Hermite quadrature rule is used to integrate the product of sparse canonical low-rank representation of PES and Green’s function in the second-order diagrammatic vibrational many-body Green’s function theory (XVH2) for estimation of zero-point energies and frequencies. Numerical tests on molecules considered in this work suggest a more efficient scaling of computational cost with molecular size as compared to other methods.

Two generalized bivariate FGM distributions and rank reduction

The Farlie-Gumbel-Morgensten (FGM) family of bivariate distributions with given marginals, is frequently used in theory and applications and has been generalized in many ways. With the help of two auxiliary distributions, we propose another generalization and study its properties. After defining the rank of a distribution as the cardinal of the set of canonical correlations, we prove that some well-known distributions have practically rank two. Consequently we introduce several extended FGM families of rank two and study how to approximate any bivariate distribution to a simpler one belonging to this family.

Rank reduction for high-dimensional generalized additive models

When a regression problem contains multiple predictors, additive models avoid the difficulty of fitting multivariate functions and at the same time retain some nonlinearity of the model. When the dimension is high, the necessity to estimate a large number of functions, even though univariate, can cause concerns regarding statistical efficiency. We propose a rank reduction approach that assumes that all functions share a small common set of latent functions, which allows borrowing information from a large number of functions. The idea is general and could be used in any model with a large number of functions to estimate, but here we restrict our attention to generalized additive models, especially logistic models, that can deal with discrete responses and is useful for classification. Numerical results are reported to illustrate the finite sample performance of the estimator. We also establish an improved convergence rate of the rank reduction approach compared to the standard estimator and extend it to sparse modeling to deal with an even larger number of predictors.

The roles of a Grandmother in African societies - please do not send them to old people's homes.

October 1 is the International Day of Older Persons, a recognition that has been in place for over 20 years now, but I must say very few stop to commemorate this day – myself included. The day is supposed to be celebrated by raising awareness about issues affecting the elderly and to appreciate the contributions that older people make to society. What is the one thing that most of us today are going to become? – get older. Populations around the world are rapidly ageing. Ageing presents both challenges and opportunities. Societies that adapt to this changing demographic and invest in Healthy Ageing can enable individuals to live both longer and healthier lives and for societies to reap the dividends. 80% of people over 60 will live in low- and middle-income countries by 2050. In Africa, old age is a social category experienced in relation to other generations, especially to youth while as in Europe old peoples’ homes have been established and more or less accepted as the living arrangement for the elderly. This is however without draw backs. Though moving into a nursing home is an individual experience, people who move into a nursing home experience different types of changes which they feel to a greater or lesser degree as stressful. The change in social status, the impact on autonomy, the feeling of having no place to call home, the change in social contacts, and the reduction of habitual activities rank first in the presentation of the results and endanger the people’s identity which they had before. Nursing home residents have experiences which they perceive as compulsive and degrading.

Joint Space and Time Processing for Unknown Mutual Coupling Blind Calibration and Mixed Sources Identification Using Uniform Circular Array

In classification and localization of mixed far-field and near-field sources, the unknown mutual coupling degrades the performance of most high-resolution algorithms. In practice, the assumption of an ideal receiving sensor array is rarely satisfied. This paper proposes an effective algorithm of mixed sources identification using uniform circular array under unknown mutual coupling. Firstly, according to rank reduction and joint space–time processing, the directions of arrival of far-field sources is estimated directly without mutual coupling elimination. Addition, the joint space–time processing can improve the estimation results in the case of low signal noise ratio of incoming signal sources and small number of snapshots. Then, these estimates are adopted to reconstruct the mutual coupling matrix. Finally, both direction and range parameters of near-field sources are obtained through spatial search after mutual coupling effects and far-field components elimination. The proposed algorithm is described in detail, and its behavior is illustrated by numerical examples.

Dictionary learning based on dip patch selection training for random noise attenuation

In recent years, sparse representation is seeing increasing application to fundamental signal and image-processing tasks. In sparse representation, a signal can be expressed as a linear combination of a dictionary (atom signals) and sparse coefficients. Dictionary learning has a critical role in obtaining a state-of-the-art sparse representation. A good dictionary should capture the representative features of the data. The whole signal can be used as training patches to learn a dictionary. However, this approach suffers from high computational costs, especially for a 3D cube. A common method is to randomly select some patches from given data as training patches to accelerate the learning process. However, the random selection method without any prior information will damage the signal if the selected patches for training are inappropriately chosen from a simple structure (e.g., training patches are chosen from a simple structure to recover the complex structure). We have developed a dip-oriented dictionary learning method, which incorporates an estimation of the dip field into the selection procedure of training patches. In the proposed approach, patches with a large dip value are selected for the training. However, it is not easy to estimate an accurate dip field from the noisy data directly. Hence, we first apply a curvelet-transform noise reduction method to remove some fine-scale components that presumably contain mostly random noise, and we then calculate a more reliable dip field from the preprocessed data to guide the patch selection. Numerical tests on synthetic shot records and field seismic image examples demonstrate that the proposed method can obtain a similar result compared with the method trained on the entire data set and obtain a better denoised result compared with the random selection method. We also compare the performance using of the proposed method and those methods based on curvelet thresholding and rank reduction on a synthetic shot record.

Detection of atrial fibrillation and other abnormal rhythms from ECG using a multi-layer classifier architecture

Objective: Atrial fibrillation (AF) and other types of abnormal heart rhythm are related to multiple fatal cardiovascular diseases that affect the quality of human life. Hence the development of an automated robust method that can reliably detect AF, in addition to other non-sinus and sinus rhythms, would be a valuable addition to medicine. The present study focuses on developing an algorithm for the classification of short, single-lead electrocardiogram (ECG) recordings into normal, AF, other abnormal rhythms and noisy classes. Approach: The proposed classification framework presents a two-layer, three-node architecture comprising binary classifiers. PQRST markers are detected on each ECG recording, followed by noise removal using a spectrogram power based novel adaptive thresholding scheme. Next, a feature pool comprising time, frequency, morphological and statistical domain ECG features is extracted for the classification task. At each node of the classification framework, suitable feature subsets, identified through feature ranking and dimension reduction, are selected for use. Adaptive boosting is selected as the classifier for the present case. The training data comprises 8528 ECG recordings provided under the PhysioNet 2017 Challenge. F1 scores averaged across the three non-noisy classes are taken as the performance metric. Main result: The final five-fold cross-validation score achieved by the proposed framework on the training data has high accuracy with low variance (0.8254 0.0043). Significance: Further, the proposed algorithm has achieved joint first place in the PhysioNet/Computing in Cardiology Challenge 2017 with a score of 0.83 computed on a hidden test dataset.

Extended reduced‐rank joint estimation of direction of arrival with mutual coupling for coherent signals

AbstractThis paper proposes an extended method for direction‐of‐arrival (DOA) estimation with unknown mutual coupling for coherent signals. The proposed method employs the forward/backward spatial smoothing to decorrelate the coherent signals in the process of rank reduction with the joint iterative subspace optimization approach. Then, the autocalibration of mutual coupling is performed based on the Capon's minimum variance criterion. Performance of the proposed method is compared with the state of the art algorithms in terms of DOA root‐mean‐square error and in terms of probability of successful estimation, where successful estimation is the case when the average of absolute DOA estimation error is within 2.5°. It is shown that the proposed method has a similar performance as the state of the art autocalibration methods while offering a lower computational complexity.

A New Broyden rank reduction method to solve large systems of nonlinear equations

We propose a modification of limited memory Broyden methods, called dynamical Broyden rank reduction method, to solve high dimensional systems of nonlinear equations. Based on a thresholding process of singular values, the proposed method determines a priori the rank of the reduced update matrix. It significantly reduces the number of singular values decomposition calls of the update matrix during the iterations. Local superlinear convergence of the method is proved and some numerical examples are displayed.

FERLrTc: 2D+3D facial expression recognition via low-rank tensor completion

In this paper, a 4D tensor model is firstly constructed to explore efficient structural information and correlations from multi-modal data (both 2D and 3D face data). As the dimensionality of the generated 4D tensor is high, a tensor dimensionality reduction technique is in need. Since many real-world high-order data often reside in a low dimensional subspace, Tucker decomposition as a powerful technique is utilized to capture multilinear low-rank structure and to extract useful information from the generated 4D tensor data. Our goal is to use Tucker decomposition to obtain a set of core tensors with smaller sizes and factor matrices which are projected into the 4D tensor data for classification prediction. To characterize the involved similarities of the 4D tensor, the low-rank and sparse representation is built in terms of the low-rank structure of factor matrices and the sparsity of the core tensor in the Tucker decomposition of the generated 4D tensor. A tensor completion (TC) framework is embedded to recover the missing information in the 4D tensor modeling process. Thus, a novel tensor dimensionality reduction approach for 2D+3D facial expression recognition via low-rank tensor completion (FERLrTC) is proposed to solve the factor matrices in a majorization–minimization manner by using a rank reduction strategy. Numerical experiments are conducted with a full implementation on the BU-3DFE and Bosphorus databases and synthetic data to illustrate the effectiveness of the proposed approach.

Spatially varying coefficient modeling for large datasets: Eliminating N from spatial regressions

While spatially varying coefficient (SVC) modeling is popular in applied science, its computational burden is substantial. This is especially true if a multiscale property of SVC is considered. Given this background, this study develops a Moran’s eigenvector-based spatially varying coefficients (M-SVC) modeling approach that estimates multiscale SVCs computationally efficiently. This estimation is accelerated through a (i) rank reduction, (ii) pre-compression, and (iii) sequential likelihood maximization. Steps (i) and (ii) eliminate the sample size N from the likelihood function; after these steps, the likelihood maximization cost is independent of N. Step (iii) further accelerates the likelihood maximization so that multiscale SVCs can be estimated even if the number of SVCs, K, is large. The M-SVC approach is compared with geographically weighted regression (GWR) through Monte Carlo simulation experiments. These simulation results show that our approach is far faster than GWR when N is large, despite numerically estimating 2K parameters while GWR numerically estimates only 1 parameter. Then, the proposed approach is applied to a land price analysis as an illustration. The developed SVC estimation approach is implemented in the R package “spmoran.”

The effect of capillary and intermolecular forces on instability of the electrostatically actuated microbeam with T-shaped paddle in the presence of fringing field

• The mechanical behavior of a clamped–clamped microbeam with middle paddle is analyzed. • The pull in voltage declines and the deflection enlarges as the intermolecular forces increase. • Presence of the electrostatic force accelerated the instability resulting from the capillary force . • Fringing field effect considerably affected determination of the bifurcation points . The present study is aimed to investigate pull-in instability, static deformation, natural frequency, primary and subharmonic resonances in the clamped–clamped microbeam with middle T-shaped paddle in presence of electrostatic force, Casimir force, van der Waals force and fringing field effect; besides, it is focused on investigating the effect of capillary force on instability and adhesion considering the effects of middle layer elongation and axial force. The microbeam is modeled as an Euler Bernoulli beam. In order to solve the nonlinear equations, the Galerkin-based rank reduction method was used. The discretized equations are solved by the perturbation theory in the neighborhood of primary and subharmonic resonances. The obtained results showed that the van der Waals and Casimir forces had attractive nature and led to the increased static deformation as well as occurrence of the pull-in at lower voltages; besides, the effects of attraction in Casimir was more than that in van der Waals. Results showed that with increase in the dimensionless length, which is influenced by the capillary force, the maximum static deformation was increased as well, leading to instability in capillary coefficients. Furthermore, presence of the electrostatic force also accelerated the instability resulting from the capillary force. Effects of the axial force and elongation in the clamped state resulted in the reduced static deformation as well as delayed instability. It is shown that changing the Casimir and Van der Waals forces as well as the fringing field effect, in addition to causing changes in the voltage and range, considerably affected determination of the bifurcation points. To validate the analytical results, numerical simulation is performed.

Testing collinearity of vector time series

Summary We investigate the collinearity of vector time series in the frequency domain, by examining the rank of the spectral density matrix at a given frequency of interest. Rank reduction corresponds to collinearity at the given frequency. When the time series is nonstationary and has been differenced to stationarity, collinearity corresponds to co-integration at a particular frequency. We examine rank through the Schur complements of the spectral density matrix, testing for rank reduction via assessing the positivity of these Schur complements, which are obtained from a nonparametric estimator of the spectral density. New asymptotic results for the test statistics are derived under the fixed bandwidth ratio paradigm; they diverge under the alternative, but under the null hypothesis of collinearity the test statistics converge to a non-standard limiting distribution. Subsampling is used to obtain the limiting null quantiles. A simulation study and an empirical illustration for 6-variate time series data are provided.

Probabilistic inequalities and measurements in bipartite systems

Various inequalities (Boole inequality, Chung–Erdös inequality, Frechet inequality) for Kolmogorov (classical) probabilities are considered. Quantum counterparts of these inequalities are introduced, which have an extra ‘quantum correction’ term, and which hold for all quantum states. When certain sufficient conditions are satisfied, the quantum correction term is zero, and the classical version of these inequalities holds for all states. But in general, the classical version of these inequalities is violated by some of the quantum states. For example in bipartite systems, classical Boole inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. A logical approach to CHSH inequalities (which are related to the Frechet inequalities), is studied in this context. It is shown that CHSH inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. The reduction of the rank of a pure state by a quantum measurement with both orthogonal and coherent projectors, is studied. Bounds for the average rank reduction are given.

A New Approach for Developing Segmentation Algorithms for Strongly Imbalanced Data

During the past two decades, the problem of how to develop efficient segmentation algorithms for dealing with strongly imbalanced data has been drawing much attention of researchers and practitioners in the field of data mining. A typical approach for this difficult problem is represented by a random under-sampling approach, where the cardinality of the majority set is reduced to that of the minority set through random sampling, thereby enabling one to utilize standard classifiers such as Logistic Regression, Support Vector Machine (SVM) and Random Forest. When the resulting segmentation algorithm is applied to a set of testing data with the original imbalanced-ness, however, its performance could be rather limited. So as to improve the performance, a bagged under-sampling (BUS) approach has been introduced where a random under-sampling is repeated M times, though the effect of BUS turns out to be still not quite satisfactory. The first purpose of this paper is to enhance the performance of BUS by developing a novel way where BUS is employed in a repetitive manner. While the performance improvement of this approach (R-BUS) over BUS is recognizable, it is still not sufficient enough from a practical point of view, especially when the dimension of underlying binary profile vectors is quite large. The second purpose of this paper is to establish a rank reduction (RR) approach for reducing this large dimension. The combined use of R-BUS with RR provides an excellent performance, as we will see through a real-world application of large magnitude.

Tuning the Parameters for Precision Matrix Estimation Using Regression Analysis

Precision matrix, i.e., inverse covariance matrix, is widely used in signal processing, and often estimated from training samples. Regularization techniques, such as banding and rank reduction, can be applied to the covariance matrix or precision matrix estimation for improving the estimation accuracy when the training samples are limited. In this paper, exploiting regression interpretations of the precision matrix, we introduce two data-driven, distribution-free methods to tune the parameter for regularized precision matrix estimation. The numerical examples are provided to demonstrate the effectiveness of the proposed methods and example applications in the design of minimum mean squared error (MMSE) channel estimators for large-scale multiple-input multiple-output (MIMO) communication systems are demonstrated.

Real-Valued Near-Field Localization of Partially Polarized Noncircular Sources With a Cross-Dipole Array

In this paper, a novel near-field localization algorithm is proposed for partially polarized noncircular sources. The algorithm is composed of two stages. In the first stage, based on the cross-dipole, the array observations are transformed into the real-valued domain. Then, an extended covariance matrix is constructed and the structure of the real-valued steering vector is utilized to factorize the direction of arrival (DOA) parameter from other parameters. Based on the rank reduction (RARE) principle, the DOA parameters can be obtained through the one-dimensional (1-D) spectral search. In the second stage, with the DOA estimates, the range parameters are generated via another RARE estimator, which also only requires a 1-D spectral search. Compared with the conventional second-order statistics (SOS)-based methods, the proposed algorithm is computationally more efficient since it only requires 1-D spectral search and real-valued computations. Moreover, it can resolve more sources and promote the DOA estimation accuracy. Numerical simulation results corroborate the effectiveness and efficiency of the proposed method.

3-D prestack data preconditioning for seismic amplitude inversion and the estimation of geomechanical properties: A case study from the structurally complex Moncton sub-basin, Eastern Canada

3-D land seismic data collected in the Moncton sub-basin (Eastern Canada) are plagued by random and coherent nose attributed to difficult acquisition conditions and complexities in the subsurface. Prior to amplitude inversion or the estimation of geomechanical properties, data must be preconditioned to provide reliable results. The preconditioning approach includes AVO-friendly rank reduction filtering followed by frequency-wavenumber (f-k) trace regularization. First, a synthetic model is used to asses the ability of the rank reduction approach to preserve the AVO behavior under various noise conditions before it is applied to the field data. The rank reduction scheme more than doubles the signal-to-noise ratio (s/n) of near-offset and mid-offset traces, increases the s/n of far-offset traces by 1.5 times and reduced amplitude distortions by removing the coherent noise from the data. F-k trace regularization helped to fill gaps in the angle coverage induced by irregular and wide-spaced offsets. However, the modeling experiment showed that the performance of rank reduction decreases as the s/n deteriorates. For low s/n, the model suggests that noise significantly alters the AVO behavior. This makes it very challenging for the filtering method, as it not only has to preserve the AVO trend but somehow restore it. Although rank reduction is able to retrieve the general AVO character, it fails to fully restore the trend on the far offset. This is not only observed on the experimental results but also on the field data for which the far offset is frequently the nosiest receiver. This questions the validity of the AVO response on far offsets when land data are collected in noisy terrains. In such settings, the inversion of the density term from large angles reflections can be biased, adding uncertainty to the estimation of Young's Modulus and the brittleness index that involves the density term in their calculation.

Low-Complexity Reduced-Dimension Space–Time Adaptive Processing for Navigation Receivers

In this paper, we propose a low-complexity adaptive reduced-rank interference suppression scheme based on alternating low-rank decomposition techniques for navigation receivers. The proposed scheme, which employs a generalized sidelobe canceler structure, makes use of a projection matrix for rank reduction followed by a reduced-dimension receive filter. An alternating optimization strategy based on recursive least squares is devised to compute the basis vectors of the projection matrix and the receive filter adaptively. Simulation results show that the proposed algorithm achieves a better performance than existing methods with a reduced computational complexity.

RARE-based localization for mixed near-field and far-field rectilinear sources

In this paper, a novel localization method for mixed near-field (NF) and far-field (FF) rectilinear or strictly noncircular sources is proposed using the noncircular information for a symmetric uniform linear array (ULA). For FF case, we adopt the NC-MUSIC method to achieve the DOA parameter, for NF case, by exploiting the center symmetrical characteristic of the ULA, we decouple the array steering vectors into two new vectors: one related only to the DOA parameter, and the other dependent on both DOA and range parameters. Based on the principle of rank reduction (RARE), three MUSIC-like estimators are formed to estimate the direction of arrival (DOA) and the range of mixed NF and FF rectilinear sources successively. Meanwhile, distinguishing the types of sources is also solved. The deterministic Cramer–Rao bound (CRB) of the mixed rectilinear signals is derived by the Slepian–Bangs formulation. Simulation results are provided, showing that the proposed method yields a performance better than existing ones.