Independent Vectors Research Articles

On fluctuation-theoretic decompositions via Lindley-type recursions

Consider a Lévy process Y(t) over an exponentially distributed time Tβ with mean 1/β. We study the joint distribution of the running maximum Ȳ(Tβ) and the time epoch G(Tβ) at which this maximum last occurs. Our main result is a fluctuation-theoretic distributional equality: the vector (Ȳ(Tβ),G(Tβ)) can be written as a sum of two independent vectors, the first one being (Ȳ(Tβ+ω),G(Tβ+ω)) and the second one being the running maximum and corresponding time epoch under the restriction that the Lévy process is only observed at Poisson(ω) inspection epochs (until Tβ). We first provide an analytic proof for this remarkable decomposition, and then a more elementary proof that gives insight into the occurrence of the decomposition and into the fact that ω only appears in the right hand side of the decomposition. The proof technique underlying the more elementary derivation also leads to further generalizations of the decomposition, and to some fundamental insights into a generalization of the well known Lindley recursion.

Tensors with Constant Components in the Constitutive Equations of a Hemitropic Micropolar Solids

The present paper is devoted to elastic potentials and the constitutive equations of mechanics of anisotropic micropolar solids, the kinematics of which can be specified by two independent vector fields: a contravariant field of translational displacements and a contravariant pseudovector field of microrotations of weight +1. The quadratic stress potential is represented by three constitutive tensors of the fourth rank, two of which are pseudotensor in nature and can be assigned weights –2 and –1. Such a solid is completely specified by the 171st micropolar elastic modulus. The main attention is focused on the model of a hemitropic (half-isotropic, demitropic) micropolar elastic solid characterized by nine constitutive constants. The components of the constitutive pseudo-tensor of weight ‒1 turn out to be sensitive to mirror reflection transformations in three-dimensional space. A peculiar algebraic structure of the constitutive tensors of a hemitropic solid, more precisely, their absolute analogues obtained by multiplying by integer powers of a pseudoscalar unity, is studied. It is shown that these tensors can always be constructed from isomers (isomer) of a tensor with constant components (generally insensitive to any transformations of the coordinate system) and one additional fourth-rank tensor constructed, in turn, from the components of the metric tensor.

Accessibility of covariance information creates vulnerability in Federated Learning frameworks.

Federated Learning (FL) is gaining traction in various fields as it enables integrative data analysis without sharing sensitive data, such as in healthcare. However, the risk of data leakage caused by malicious attacks must be considered. In this study, we introduce a novel attack algorithm that relies on being able to compute sample means, sample covariances, and construct known linearly independent vectors on the data owner side. We show that these basic functionalities, which are available in several established FL frameworks, are sufficient to reconstruct privacy-protected data. Additionally, the attack algorithm is robust to defense strategies that involve adding random noise. We demonstrate the limitations of existing frameworks and propose potential defense strategies analyzing the implications of using differential privacy. The novel insights presented in this study will aid in the improvement of FL frameworks. The code examples are provided at GitHub (https://github.com/manuhuth/Data-Leakage-From-Covariances.git). The CNSIM1 dataset, which we used in the manuscript, is available within the DSData R package (https://github.com/datashield/DSData/tree/main/data).

Improvement of independent vector analysis for closely spaced sources

Independent vector analysis (IVA) is one of the most widely-used blind source separation methods. Theoretically, its performance is not affected by the location of the sound source, however, in practical applications with the presence of diffuse noise, the separation performance severely degrades when sources are closely spaced. In this technical note, the reason for this deterioration is explored and an effective modification of IVA cost function is proposed to compensate for the separation performance in closely-spaced source scenarios. Numerous experiments validate the effectiveness of the proposed method with its application to auxiliary function-based IVA.

Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations

Consider two random vectors x˜=Az+C11∕2x∈Rp and y˜=Bz+C21∕2y∈Rq, where x∈Rp, y∈Rq and z∈Rr are independent random vectors with i.i.d. entries of zero mean and unit variance, C1 and C2 are p×p and q×q deterministic population covariance matrices, and A and B are p×r and q×r deterministic factor loading matrices. With n independent observations of x˜ and y˜, we study the sample canonical correlations between them. Under the sharp fourth moment condition on the entries of x, y and z, we prove the BBP transition for the sample canonical correlation coefficients (CCCs). More precisely, if a population CCC is below a threshold, then the corresponding sample CCC converges to the right edge of the bulk eigenvalue spectrum of the sample canonical correlation matrix and satisfies the famous Tracy-Widom law; if a population CCC is above the threshold, then the corresponding sample CCC converges to an outlier that is detached from the bulk eigenvalue spectrum. We prove our results in full generality, in the sense that they also hold for near-degenerate population CCCs and population CCCs that are close to the threshold.

Mutually orthogonal frequency rectangles

A frequency rectangle of type FR(m,n;q) is an m×n matrix such that each symbol from a set of size q appears n/q times in each row and m/q times in each column. Two frequency rectangles of the same type are said to be orthogonal if, upon superimposition, each possible ordered pair of symbols appear the same number of times. A set of k frequency rectangles in which every pair is orthogonal is called a set of mutually orthogonal frequency rectangles, denoted by k–MOFR(m,n;q). We show that a k–MOFR(2,2n;2) and an orthogonal array OA(2n,k,2,2) are equivalent. We also show that an OA(mn,k,2,2) implies the existence of a k–MOFR(2m,2n;2). We construct (4a−2)–MOFR(4,2a;2) assuming the existence of a Hadamard matrix of order 4a.A k–MOFR(m,n;q) is said to be t–orthogonal, if each subset of size t, when superimposed, contains each of the qt possible ordered t-tuples of entries exactly mn/qt times. A set of vectors over a finite field Fq is said to be t-independent if each subset of size t is linearly independent. We describe a method to obtain a set of t–orthogonal k–MOFR(qM,qN,q) corresponding to a set of t–independent vectors in (Fq)M+N. We also discuss upper and lower bounds on the sizes of sets of t–independent vectors and give a table of values for binary vectors of length N⩽16.A frequency rectangle of type FR(n,n;q) is called a frequency square and a set of k mutually orthogonal frequency squares is denoted by k–MOFS(n;q) or k–MOFS(n) when there is no ambiguity about the symbol set. For p an odd prime, we show that there exists a set of (p−1) binary MOFS(2p), hence improving the lower bounds in (Britz et al. 2020) for p⩾19.

The effect of source sparsity on independent vector analysis for blind source separation

In this paper, the effect of source sparsity on the performance of the independent vector analysis (IVA) algorithm for blind source separation is investigated. The IVA algorithm was originally developed under the assumption of statistical independence between the sources and has made great advances in recent years. However, its performance under different sparsity conditions is rarely studied. This study begins by mathematically analyzing the performance of IVA in permutation alignment, which is proved to directly correlate with the degree of frame-level W-disjoint orthogonality (F-WDO) of the sources. We further prove that IVA can theoretically achieve the optimal separation in the cases where the sources are F-WDO. Experimental results show a strong positive correlation between a quantitative measure of F-WDO and the IVA algorithm’s performance under various conditions.

Efficient Overdetermined Independent Vector Analysis Based on Iterative Projection with Adjustment

In this paper, a computationally efficient optimization algorithm for independent vector analysis (IVA) is proposed to accelerate iterative convergence speed and enhance the overdetermined convolutive blind speech separation performance. An iterative projection with adjustment (IPA) is investigated to estimate the unmixing matrix for OverIVA. The IPA algorithm jointly executes the iterative projection (IP) algorithm and the iterative source steering (ISS) algorithm to jointly update one row and one column of the mixing matrix, which can perform computationally-efficient blind source separation. It is achieved by updating one demixing filter and jointly adjusting all the other sources along its current direction. Motivated by its technology superiorities, this paper proposes a modified algorithm for the OverIVA, fully exploiting the computational efficiency of IPA optimization scheme. Experimental results corroborate the proposed OverIVA-IPA algorithm converges faster and performs better than the existing state-of-the-arts algorithms.

On a characterization theorem in the space [formula omitted

By Heyde's theorem, the class of Gaussian distributions on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. We prove an analogue of this theorem for two independent random vectors taking values in the space Rn. The obtained class of distributions consists of convolutions of Gaussian distributions and a distribution supported in a subspace, which is determined by coefficients of the linear forms.

Estimating the number of principal components via Split-Half Eigenvector Matching (SHEM)

Estimating the number of principal components to retain for dimension reduction is a critical step in many applications of principal component analysis. Common methods may not be optimal, however. The current paper presents an alternative procedure that aims to recover the true number of principal components, in the sense of the number of independent vectors involved in the generation of the data.•Data are split into random halves repeatedly.•For each split, the eigenvectors in one half are compared to those in the other.•The split between high and low similarities is used to estimate the number of principal components.The method is a proof of principle that similarity over split-halves of the data may provide a useful approach to estimating the number of components in dimension reduction, or of similar dimensions in other models.

Global Asymptotic Stability of Submodule Capacitor Voltage Self-converging Modular Multilevel Converters Based on Characteristic Root Method

This study presents global asymptotic stability verification of submodule capacitor voltage self-converging modular multilevel converters (SC-MMCs) according to the characteristic root method, which constructs a submodule switching state matrix for voltage balancing control. The mathematical model of MMC is established in form of the state-space equation. It is proved that SC-MMCs have the characteristic of global asymptotic stability at a certain time from the perspective of switching state matrix characteristic roots, according to the theories of Lyapunov stability and continuous-time piecewise linear system stability. Then, the submodule capacitor voltage self-convergence over a period of time is deduced based on the capacitor voltage deviation vector. Under the constraints of linearly independent switching state matrix row vectors, submodule capacitor voltage self-convergence without real-time voltage calculation and feedback can be achieved. The global asymptotic stability of SC-MMCs and the efficacy of submodule capacitor voltage self-convergence based on switching state matrix are testified by simulations.

Using deep LSD to build operators in GANs latent space with meaning in real space.

Generative models rely on the idea that data can be represented in terms of latent variables which are uncorrelated by definition. Lack of correlation among the latent variable support is important because it suggests that the latent-space manifold is simpler to understand and manipulate than the real-space representation. Many types of generative model are used in deep learning, e.g., variational autoencoders (VAEs) and generative adversarial networks (GANs). Based on the idea that the latent space behaves like a vector space Radford et al. (2015), we ask whether we can expand the latent space representation of our data elements in terms of an orthonormal basis set. Here we propose a method to build a set of linearly independent vectors in the latent space of a trained GAN, which we call quasi-eigenvectors. These quasi-eigenvectors have two key properties: i) They span the latent space, ii) A set of these quasi-eigenvectors map to each of the labeled features one-to-one. We show that in the case of the MNIST image data set, while the number of dimensions in latent space is large by design, 98% of the data in real space map to a sub-domain of latent space of dimensionality equal to the number of labels. We then show how the quasi-eigenvectors can be used for Latent Spectral Decomposition (LSD). We apply LSD to denoise MNIST images. Finally, using the quasi-eigenvectors, we construct rotation matrices in latent space which map to feature transformations in real space. Overall, from quasi-eigenvectors we gain insight regarding the latent space topology.

Three new triangular thick plate/shell elements based on absolute nodal coordinate formulation

In this paper, three new triangular fully parameterized thick plate/shell elements are proposed based on the absolute nodal coordinate formulation. The volume parameters and corresponding independent volume gradient vectors are used to simplify the element construction. The node-wise structural parameter is adopted to circumvent the slope discontinuity issue associated with the volume gradient vectors. Two three-node elements inherit the interpolation strategy in the membrane direction of a well verified thin plate element. For the thickness direction, linear and quadratic interpolation are used in the three-node elements. Inspired by the quadrilateral plate elements with gradient vectors only along thickness direction, a six-node triangular element is presented with quadratic interpolation in both membrane and thickness direction. Three static analyses with large deformation and one dynamic analysis are conducted to evaluate the performance of the new elements. It concludes that the three-node element with linear interpolation in the thickness direction performs not well due to the incompatible/low interpolation in the thickness direction. The other two elements show good accuracy in all the numerical tests. Though the six-node element exhibits slightly higher accuracy than the quadratic three-node element, its DOFs are much more than the quadratic three-node element, which makes it less attractive.

Search for CP violation using textrm{t}overline{textrm{t}} events in the lepton+jets channel in pp collisions at sqrt{s} = 13 TeV

Results are presented on a search for CP violation in the production and decay of top quark-antiquark pairs in the lepton+jets channel. The search is based on data from proton-proton collisions at sqrt{s} = 13 TeV, collected with the CMS detector, corresponding to an integrated luminosity of 138 fb−1. Possible CP violation effects are evaluated by measuring asymmetries in observables constructed from linearly independent four-momentum vectors of the final-state particles. The dimensionless chromoelectric dipole moment of the top quark obtained from the observed asymmetries is measured to be 0.04 ± 0.10 (stat) ± 0.07(syst), and the asymmetries exhibit no evidence for CP-violating effects, consistent with expectations from the standard model.

Student Online Learning Behavior Characteristics Based on Multidimensional Cognitive Model

Analysis of student learning behavior characteristics is an important means for educators to better understand students and improve the quality and effectiveness of teaching in the field of education. It is necessary to refer to students' cognitive levels for analysis of student learning behavior characteristics. However, existing algorithms only focus on the overall performance and grades of students, ignoring the individual differences in learning cognitive levels among students, which affects the accuracy of the analysis results. Therefore, this paper conducts research on student online learning behavior characteristics based on a multidimensional cognitive model. Firstly, a multidimensional and multilevel model for evaluating students' cognitive levels is constructed, and the process of evaluating students' cognitive levels is sustainable and can be adjusted in real-time as students' cognitive levels change. By considering the differences in evaluation levels and students' cognitive levels, targeted observation and extraction of students' online learning behavior characteristics can be achieved. A new model based on variational autoencoder neural network is proposed to perform decoupled representation of students' implicit preferences. By using a regularization term based on maximum mean difference, the model can learn independent hidden vectors sensitive to dynamic and static factors from students' online learning behavior history data and multidimensional cognitive evaluation history data. The experimental results verify the effectiveness of the constructed model.

Simple polarization-insensitive coherent RoF link with increased capacity.

A simple polarization-insensitive coherent radio-over-fiber (RoF) link with increased spectrum efficiency and transmission capacity is proposed and demonstrated. Instead of using two polarization splitters (PBSs), two 90° hybrids, and four pairs of balanced photodetectors (PDs) in a conventional polarization-diversity coherent receiver (PDCR), a simplified PDCR with only one PBS, one optical coupler (OC), and two PDs is employed in the coherent RoF link. At the simplified receiver, a novel, to the best of our knowledge, digital signal processing (DSP) algorithm is proposed to achieve polarization-insensitive detection and demultiplexing of two spectrally overlapping microwave vector signals as well as the elimination of the joint phase noise originating from the transmitter and the local oscillator (LO) laser sources. An experiment is performed. The transmission and detection of two independent 16QAM microwave vector signals at identical microwave carrier frequencies of 3 GHz with a symbol rate of 0.5 GSym/s over a 25-km single-mode fiber (SMF) is demonstrated. Thanks to the spectrum superposition of the two microwave vector signals, the spectral efficiency as well as the data transmission capacity is increased.

A Scalable Approach to Independent Vector Analysis by Shared Subspace Separation for Multi-Subject fMRI Analysis.

Joint blind source separation (JBSS) has wide applications in modeling latent structures across multiple related datasets. However, JBSS is computationally prohibitive with high-dimensional data, limiting the number of datasets that can be included in a tractable analysis. Furthermore, JBSS may not be effective if the data's true latent dimensionality is not adequately modeled, where severe overparameterization may lead to poor separation and time performance. In this paper, we propose a scalable JBSS method by modeling and separating the "shared" subspace from the data. The shared subspace is defined as the subset of latent sources that exists across all datasets, represented by groups of sources that collectively form a low-rank structure. Our method first provides the efficient initialization of the independent vector analysis (IVA) with a multivariate Gaussian source prior (IVA-G) specifically designed to estimate the shared sources. Estimated sources are then evaluated regarding whether they are shared, upon which further JBSS is applied separately to the shared and non-shared sources. This provides an effective means to reduce the dimensionality of the problem, improving analyses with larger numbers of datasets. We apply our method to resting-state fMRI datasets, demonstrating that our method can achieve an excellent estimation performance with significantly reduced computational costs.

Nearly optimal central limit theorem and bootstrap approximations in high dimensions

In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of n independent high-dimensional centered random vectors X1,…,Xn over the class of rectangles in the case when the covariance matrix of the scaled average is nondegenerate. In the case of bounded Xi’s, the implied bound for the Kolmogorov distance between the distribution of the scaled average and the Gaussian vector takes the form C(Bn2log3d/n)1/2logn, where d is the dimension of the vectors and Bn is a uniform envelope constant on components of Xi’s. This bound is sharp in terms of d and Bn, and is nearly (up to logn) sharp in terms of the sample size n. In addition, we show that similar bounds hold for the multiplier and empirical bootstrap approximations. Moreover, we establish bounds that allow for unbounded Xi’s, formulated solely in terms of moments of Xi’s. Finally, we demonstrate that the bounds can be further improved in some special smooth and moment-constrained cases.

A Novel Approach for Brain Tumor Classification Using an Ensemble of Deep and Hand-Crafted Features.

One of the most severe types of cancer caused by the uncontrollable proliferation of brain cells inside the skull is brain tumors. Hence, a fast and accurate tumor detection method is critical for the patient's health. Many automated artificial intelligence (AI) methods have recently been developed to diagnose tumors. These approaches, however, result in poor performance; hence, there is a need for an efficient technique to perform precise diagnoses. This paper suggests a novel approach for brain tumor detection via an ensemble of deep and hand-crafted feature vectors (FV). The novel FV is an ensemble of hand-crafted features based on the GLCM (gray level co-occurrence matrix) and in-depth features based on VGG16. The novel FV contains robust features compared to independent vectors, which improve the suggested method's discriminating capabilities. The proposed FV is then classified using SVM or support vector machines and the k-nearest neighbor classifier (KNN). The framework achieved the highest accuracy of 99% on the ensemble FV. The results indicate the reliability and efficacy of the proposed methodology; hence, radiologists can use it to detect brain tumors through MRI (magnetic resonance imaging). The results show the robustness of the proposed method and can be deployed in the real environment to detect brain tumors from MRI images accurately. In addition, the performance of our model was validated via cross-tabulated data.

ICA and IVA bounded multivariate generalized Gaussian mixture based hidden Markov models

Machine learning (ML), a branch of artificial intelligence (AI), is an area of computational science that is concerned with the analysis and interpretation of patterns and structures in data to enable learning and decision-making without the participation of a human. Hidden Markov models (HMMs), which have been acknowledged for decades but have recently made a significant revival in machine learning, are one of the most impressively powerful probabilistic models. HMMs are frequently employed in machine learning to model heterogeneous time series data. In this paper, we integrate independent component analysis (ICA) and ICA with a bounded multivariate generalized Gaussian mixture model (ICA-BMGGMM) into the HMM approach. One limitation of ICA is that it assumes the sources to be independent from each other. This assumption can be relaxed by combining independent vectors analysis (IVA) and IVA with the BMGGMM (IVA-BMGGMM) into the HMM approach to improve their modeling capability. We validate our proposed models using a variety of applications, such as human action recognition, speech recognition, and energy disaggregation. The results presented in the paper demonstrate the effectiveness of the proposed approaches for modeling different types of data. These data include KTH and Weizmann datasets for human action recognition, TIMIT and SDR for speech recognition, REDD dataset for energy disaggregation and EEG dataset for elliptic seizure classification. For all conducted experiments, our proposed models outperform other comparing models for all performance metrics such as accuracy, sensitivity, and precision. The best detection results were found using the IVABMGGMM-HMM for the reported experiments.