Matrix Perturbation Research Articles

Multiple-analyte mass detection in a coupled microresonator array

Resonant sensors using coupled micro-cantilever arrays have found wide applications in the ultrasensitive mass detection of biomolecules and chemical analytes. Experimental observations indicate that a target mass deposited on one of the cantilevers can be detected by measuring the change in resonant frequencies or in eigenmodes. Analytical works have studied eigenvalue and eigenmode sensitivities, but for a single analyte only. Since a resonator array consists of several cantilevers, it offers an opportunity for the simultaneous detection of multiple analytes. However, multiple-analyte mass detection has not been investigated so far. In this paper, an analytical foundation for the detection of multiple analytes, through the measurement of eigenvalue shifts, is developed using matrix perturbation theory. The formulation presents a system of over-determined linear equations in terms of unknown analyte masses. A novel approach based on solving the equations in a least square sense is proposed and it is shown that it gives far better estimation accuracy than using a subset of equations for direct solution. The approach is demonstrated through numerical simulation for a typical three-cantilever array for the detection of two analyte masses. Estimation errors are studied for a range of analyte masses and presented in the form of an error surface. The effect of interconnection stiffness and array size on estimation error is also investigated. The robustness of the method is further tested against manufacturing variations and it is shown that an envelope guideline of maximum estimation error can be constructed for the user.

Robust high dimensional factor models with applications to statistical machine learning.

Factor models are a class of powerful statistical models that have been widely used to deal with dependent measurements that arise frequently from various applications from genomics and neuroscience to economics and finance. As data are collected at an ever-growing scale, statistical machine learning faces some new challenges: high dimensionality, strong dependence among observed variables, heavy-tailed variables and heterogeneity. High-dimensional robust factor analysis serves as a powerful toolkit to conquer these challenges. This paper gives a selective overview on recent advance on high-dimensional factor models and their applications to statistics including Factor-Adjusted Robust Model selection (FarmSelect) and Factor-Adjusted Robust Multiple testing (FarmTest). We show that classical methods, especially principal component analysis (PCA), can be tailored to many new problems and provide powerful tools for statistical estimation and inference. We highlight PCA and its connections to matrix perturbation theory, robust statistics, random projection, false discovery rate, etc., and illustrate through several applications how insights from these fields yield solutions to modern challenges. We also present far-reaching connections between factor models and popular statistical learning problems, including network analysis and low-rank matrix recovery.

Understanding Persuasion Cascades in Online Product Rating Systems: Modeling, Analysis, and Inference

Online product rating systems have become an indispensable component for numerous web services such as Amazon, eBay, Google Play Store, and TripAdvisor. One functionality of such systems is to uncover the product quality via product ratings (or reviews) contributed by consumers. However, a well-known psychological phenomenon called “ message-based persuasion ” lead to “ biased ” product ratings in a cascading manner (we call this the persuasion cascade ). This article investigates: (1) How does the persuasion cascade influence the product quality estimation accuracy? (2) Given a real-world product rating dataset, how to infer the persuasion cascade and analyze it to draw practical insights? We first develop a mathematical model to capture key factors of a persuasion cascade. We formulate a high-order Markov chain to characterize the opinion dynamics of a persuasion cascade and prove the convergence of opinions. We further bound the product quality estimation error for a class of rating aggregation rules including the averaging scoring rule, via the matrix perturbation theory and the Chernoff bound. We also design a maximum likelihood algorithm to infer parameters of the persuasion cascade. We conduct experiments on both synthetic data and real-world data from Amazon and TripAdvisor. Experiment results show that our inference algorithm has a high accuracy. Furthermore, persuasion cascades notably exist, but the average scoring rule has a small product quality estimation error under practical scenarios.

Analysis of regularized least-squares in reproducing kernel Kreĭn spaces

In this paper, we study the asymptotic properties of regularized least squares with indefinite kernels in reproducing kernel Krein spaces (RKKS). By introducing a bounded hyper-sphere constraint to such non-convex regularized risk minimization problem, we theoretically demonstrate that this problem has a globally optimal solution with a closed form on the sphere, which makes approximation analysis feasible in RKKS. Regarding to the original regularizer induced by the indefinite inner product, we modify traditional error decomposition techniques, prove convergence results for the introduced hypothesis error based on matrix perturbation theory, and derive learning rates of such regularized regression problem in RKKS. Under some conditions, the derived learning rates in RKKS are the same as that in reproducing kernel Hilbert spaces (RKHS), which is actually the first work on approximation analysis of regularized learning algorithms in RKKS.

Molecular and phenotypic analysis of rodent models reveals conserved and species-specific modulators of human sarcopenia

Sarcopenia, the age-related loss of skeletal muscle mass and function, affects 5–13% of individuals aged over 60 years. While rodents are widely-used model organisms, which aspects of sarcopenia are recapitulated in different animal models is unknown. Here we generated a time series of phenotypic measurements and RNA sequencing data in mouse gastrocnemius muscle and analyzed them alongside analogous data from rats and humans. We found that rodents recapitulate mitochondrial changes observed in human sarcopenia, while inflammatory responses are conserved at pathway but not gene level. Perturbations in the extracellular matrix are shared by rats, while mice recapitulate changes in RNA processing and autophagy. We inferred transcription regulators of early and late transcriptome changes, which could be targeted therapeutically. Our study demonstrates that phenotypic measurements, such as muscle mass, are better indicators of muscle health than chronological age and should be considered when analyzing aging-related molecular data.

A new iteration regularization method for dynamic load identification of stochastic structures

In this paper, we propose a novel fast convergence iterative regularization method on the basis of a new iterative regularization operator to solve the dynamic load identification of stochastic structures. Firstly, the validity and convergence of the proposed algorithm is proved by strict mathematical theory and numerical results of engineering example. Secondly, the dynamic load identification of uncertain structures can be transformed into a series of deterministic inverse problems by the matrix perturbation method. Finally, we also assess the statistical characteristics of identified loads using the statistical theory. Particularly, the proposed method successfully solves the dynamic load identification of stochastic structure. Numerical simulations are given to validate the feasibility of the presented method in this paper.

Approximate Graph Laplacians for Multimodal Data Clustering.

One of the important approaches of handling data heterogeneity in multimodal data clustering is modeling each modality using a separate similarity graph. Information from the multiple graphs is integrated by combining them into a unified graph. A major challenge here is how to preserve cluster information while removing noise from individual graphs. In this regard, a novel algorithm, termed as CoALa, is proposed that integrates noise-free approximations of multiple similarity graphs. The proposed method first approximates a graph using the most informative eigenpairs of its Laplacian which contain cluster information. The approximate Laplacians are then integrated for the construction of a low-rank subspace that best preserves overall cluster information of multiple graphs. However, this approximate subspace differs from the full-rank subspace which integrates information from all the eigenpairs of each Laplacian. Matrix perturbation theory is used to theoretically evaluate how far approximate subspace deviates from the full-rank one for a given value of approximation rank. Finally, spectral clustering is performed on the approximate subspace to identify the clusters. Experimental results on several real-life cancer and benchmark data sets demonstrate that the proposed algorithm significantly and consistently outperforms state-of-the-art integrative clustering approaches.

DEGENERATE BOUNDARY-VALUE PROBLEMS WITH A PERTURBING MATRIX FOR A DERIVATIVE

In the paper, there is considered degenerated Noether boundary value problem with a perturbing matrix for a derivative, in which the boundary condition is given by a linear vector functional. We have proposed an algorithm to consrtuct a set of linearly independent solutions of boundary value problems with a small parameter in the general case, when the number of boundary conditions given by a linear vector functional does not match with the number of unknowns in a degenerate differential system. There is used the technique of pseudoinverse Moore-Penrose matrices. Applying the Vishik-Lyusternik method, the solution of the boundary value problem is obtained as part of the Laurent series in powers of small parameter. We obtain conditions for the bifurcation of solutions of linear degenerated Noether boundary-value problems with a small parameter under the assumption that the unperturbed degenerated differential system can be reduced to central canonical form.

Normal approximation and confidence region of singular subspaces

This paper is on the normal approximation of singular subspaces when the noise matrix has i.i.d. entries. Our contributions are three-fold. First, we derive an explicit representation formula of the empirical spectral projectors. The formula is neat and holds for deterministic matrix perturbations. Second, we calculate the expected projection distance between the empirical singular subspaces and true singular subspaces. Our method allows obtaining arbitrary k-th order approximation of the expected projection distance. Third, we prove the non-asymptotical normal approximation of the projection distance with different levels of bias corrections. By the ⌈log(d1+d2)⌉-th order bias corrections, the asymptotical normality holds under optimal signal-to-noise ratio (SNR) condition where d1 and d2 denote the matrix sizes. In addition, it shows that higher order approximations are unnecessary when |d1−d2|=O((d1+d2)1∕2). Finally, we provide comprehensive simulation results to merit our theoretic discoveries. Unlike the existing results, our approach is non-asymptotical and the convergence rates are established. Our method allows the rank r to diverge as fast as o((d1+d2)1∕3). Moreover, our method requires no eigen-gap condition (except the SNR) and no constraints between d1 and d2.

Convergence of Newton-MR under Inexact Hessian Information

Recently, there has been a surge of interest in designing variants of the classical Newton-CG in which the Hessian of a (strongly) convex function is replaced by suitable approximations. This is mainly motivated by large-scale finite-sum minimization problems that arise in many machine learning applications. Going beyond convexity, inexact Hessian information has also been recently considered in the context of algorithms such as trust-region or (adaptive) cubic regularization for general nonconvex problems. Here, we do that for Newton-MR, which extends the application range of the classical Newton-CG beyond convexity to invex problems. Unlike the convergence analysis of Newton-CG, which relies on spectrum preserving Hessian approximations in the sense of Löwner partial order, our work here draws from matrix perturbation theory to estimate the distance between the range spaces underlying the exact and approximate Hessian matrices. Numerical experiments demonstrate a great degree of resilience to such Hessian approximations, amounting to a highly efficient algorithm in large-scale problems.

Bearing Damage Detection of a Bridge under the Uncertain Conditions Based on the Bayesian Framework and Matrix Perturbation Method

The bearing of a bridge, as a critical component, is important in the force transformation of the superstructure; however, due to the service condition and repeated impact load, the bearing is prone to be damaged but difficult to detect the damage; the present research has few studies that focused on the damage detection of the structural bearing. Meanwhile, practical engineering is always surrounded by variational environmental conditions, and sometimes, the element and bearing damage both exist in the structure. Thus, these uncertain conditions all cause inaccurate damage identification results using the vibration‐based damage detection method. In order to detect the damage of the structural bearing and improve the precision, firstly, the structural dynamic characteristic equation considering uncertain conditions has been deduced; then, a damage detection framework constructed by the Bayesian theory and perturbation method has been developed in this article; a numerical example of an 8‐span concrete continuous beam and a practical example of I‐40 steel‐concrete composite bridge are utilized to validate the feasibility of the proposed method, and single type and two types of damage cases are studied. The outcomes demonstrate that the damage of structural elements and bearings can be detected with high accuracy. The proposed method is of great applicability and good potential.

Convex combination of data matrices: PCA perturbation bounds for multi-objective optimal design of mechanical metafilters

<p style='text-indent:20px;'>In the present study, matrix perturbation bounds on the eigenvalues and on the invariant subspaces found by principal component analysis is investigated, for the case in which the data matrix on which principal component analysis is performed is a convex combination of two data matrices. The application of the theoretical analysis to multi-objective optimization problems – e.g., those arising in the design of mechanical metamaterial filters – is also discussed, together with possible extensions.</p>

Placing Grid-Forming Converters to Enhance Small Signal Stability of PLL-Integrated Power Systems

The modern power grid features the high penetration of power converters, which widely employ a phase-locked loop (PLL) for grid synchronization. However, it has been pointed out that PLL can give rise to small-signal instabilities under weak grid conditions. This problem can be potentially resolved by operating the converters in grid-forming mode, namely, without using a PLL. Nonetheless, it has not been theoretically revealed how the placement of grid-forming converters enhances the small-signal stability of power systems integrated with large-scale PLL-based converters. This paper aims at filling this gap. Based on matrix perturbation theory, we explicitly demonstrate that the placement of grid-forming converters is equivalent to increasing the power grid strength and thus improving the small-signal stability of PLL-based converters. Furthermore, we investigate the optimal locations to place grid-forming converters by increasing the smallest eigenvalue of the weighted and Kron-reduced Laplacian matrix of the power network. The analysis in this paper is validated through high-fidelity simulation studies on a modified two-area test system and a modified 39-bus test system. This paper potentially lays the foundation for understanding the interaction between PLL-based (i.e., grid-following) converters and grid-forming converters, and coordinating their placements in future converter-dominated power systems.

Verified Error Bounds for Real Eigenvalues of Real Symmetric and Persymmetric Matrices

This paper mainly investigates the verification of real eigenvalues of the real symmetric and persymmetric matrices. For a real symmetric or persymmetric matrix, we use eig code in Matlab to obtain its real eigenvalues on the basis of numerical computation and provide an algorithm to compute verified error bound such that there exists a perturbation matrix of the same type within the computed error bound whose exact real eigenvalues are the computed real eigenvalues.

Spectral dissection of finite rank perturbations of normal operators

Finite rank perturbations T=N+K of a bounded normal operator N acting on a separable Hilbert space are studied thanks to a natural functional model of T; in its turn the functional model solely relies on a perturbation matrix/characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix encode in a closed-form the spectral behavior of T. Under mild geometric conditions on the spectral measure of N and some smoothness constraints on K we show that the operator T admits invariant subspaces, or even it is decomposable.

Azimuthal amplitude variation with offset parameterization and inversion for fracture weaknesses in tilted transversely isotropic media

The characterization of fracture-induced tilted transverse isotropy (TTI) seems to be more suitable to actual scenarios of geophysical exploration for fractured reservoirs. Fracture weaknesses enable us to describe fracture-induced anisotropy. With the incident and reflected PP-wave in TTI media, we have adopted a robust method of azimuthal amplitude variation with offset (AVO) parameterization and inversion for fracture weaknesses in a fracture-induced reservoir with TTI symmetry. Combining the linear-slip model with the Bond transformation, we have derived the stiffness matrix of a dipping-fracture-induced TTI medium characterized by normal and tangential fracture weaknesses and a tilt angle. Integrating the first-order perturbations in the stiffness matrix of a TTI medium and scattering theory, we adopt a method of azimuthal AVO parameterization for PP-wave reflection coefficient for the case of a weak-contrast interface separating two homogeneous weakly anisotropic TTI layers. We then adopt an iterative inversion method by using the partially incidence-angle-stacked seismic data with different azimuths to estimate the fracture weaknesses of a TTI medium when the tilt angle is estimated based on the image well logs prior to the seismic inversion. Synthetic examples confirm that the fracture weaknesses of a TTI medium are stably estimated from the azimuthal seismic reflected amplitudes for the case of moderate noise. A field data example demonstrates that geologically reasonable results of fracture weaknesses can be determined when the tilt angle is treated as the prior information. We determine that the azimuthal AVO inversion approach provides an available tool for fracture prediction in a dipping-fracture-induced TTI reservoir.

Regularized Spectral Clustering With Entropy Perturbation

Spectral clustering is a popular clustering method because it gives a natural way to reduce the dimensionality of data using eigenvectors. It is well known that the performance of spectral clustering could be improved via regularization. Nevertheless, it is hard to cope with the different cases by only one constant regularization parameter. To solve such a problem, a novel regularized spectral clustering method is proposed. Specifically, two modules are integrated in the proposed method. First, under matrix perturbation analysis, we prove that the entropy can be used as a rank score function to reveal the informative eigenvector, and the eigenvector corresponding to the minimal entropy will be the regularization to regularize the data matrix instead of a constant regularization parameter. Second, in order to ensure the perturbation on eigenspace is within the effective range, a perturbation boundary on eigenvectors is given. Numerical results showed that our proposal has superior performance than spectral clustering and k-means algorithm.

A probabilistic algorithm for aggregating vastly undersampled large Markov chains

Model reduction of large Markov chains is an essential step in a wide array of techniques for understanding complex systems and for efficiently learning structures from high-dimensional data. We present a novel aggregation algorithm for compressing such chains that exploits a specific low-rank structure in the transition matrix which, e.g., is present in metastable systems, among others. It enables the recovery of the aggregates from a vastly undersampled transition matrix which in practical applications may gain a speedup of several orders of magnitude over methods that require the full transition matrix. Moreover, we show that the new technique is robust under perturbation of the transition matrix. The practical applicability of the new method is demonstrated by identifying a reduced model for the large-scale traffic flow patterns from real-world taxi trip data.

Decentralized Optimization Over Time-Varying Directed Graphs With Row and Column-Stochastic Matrices

In this article, we provide a distributed optimization algorithm, termed as TV- $\mathcal {AB}$ , that minimizes a sum of convex functions over time-varying, random directed graphs. Contrary to the existing work, the algorithm we propose does not require eigenvector estimation to estimate the (non- $\mathbf {1}$ ) Perron eigenvector of a stochastic matrix. Instead, the proposed approach relies on a novel information mixing approach that exploits both row- and column-stochastic weights to achieve agreement toward the optimal solution when the underlying graph is directed. We show that TV- $\mathcal {AB}$ converges linearly to the optimal solution when the global objective is smooth and strongly convex, and the underlying time-varying graphs exhibit bounded connectivity, i.e., a union of every $C$ consecutive graphs is strongly connected. We derive the convergence results based on the stability analysis of a linear system of inequalities along with a matrix perturbation argument. Simulations confirm the findings in this article.

Second-Order Statistics-Based Semi-Blind Techniques for Channel Estimation in Millimeter-Wave MIMO Analog and Hybrid Beamforming

Semi-blind (SB) channel estimation is conceived for millimeter wave (mmWave) analog-beamforming (AB) and hybrid-beamforming (HB)-based multiple-input multiple-output (MIMO) systems, which also exploits the data symbols for improving the estimation accuracy. A novel aspect of the proposed framework is that it directly estimates the analog beamformer/ combiner weights without necessitating the estimation of the entire mmWave MIMO channel matrix. By involving powerful matrix perturbation theoretic techniques, a closed-form expression is derived for the mean-squared-error (MSE) of the mmWave-AB-SB algorithm. As a further novelty, our mmWave-HB-SB technique relies on the decomposition of the channel matrix as the product of a decorrelating and a unitary matrix. Subsequently, the former is estimated purely relying on the unknown data symbols, whereas the latter is estimated exclusively from the training vectors. A lower bound on the MSE of the proposed mmWave-HB-SB technique is derived using the constrained Cramer-Rao lower bound (CRLB) framework. Furthermore, the performance gain of our mmWave-HB-SB technique over the conventional purely training-based scheme is also quantified analytically. Our simulation results demonstrate the superiority of the techniques advocated over the existing solutions and also verify the accuracy of our analytical findings.