Large-scale Matrix Research Articles

Weave elgamal encryption for secure outsourcing algebraic computations over p

This paper addresses the secure outsourcing problem for large-scale matrix computation to a public cloud. We propose a novel public-key weave ElGamal encryption (WEE) scheme for encrypting a matrix over the field Zp. The scheme has the echelon transformation property. We can apply a series of elementary row/column operations to transform an encrypted matrix under our WEE scheme into the row/column echelon form. The decrypted result matches the result of the corresponding operations performed on the original matrix. For security, our WEE scheme is shown to be entry irrecoverable for non-zero entries under the computational Diffie-Hellman assumption. By using our WEE scheme, we propose five secure outsourcing protocols of Gaussian elimination, Gaussian-Jordan elimination, matrix determinant, linear system solver, and matrix inversion. Each of these protocols preserves data privacy for clients (data owners). Furthermore, the linear system solver and matrix inversion protocols provide a cheating-resistant mechanism to verify correctness of computation results. Our experimental result shows that our protocols gain efficiency significantly for an outsourcer. Our outsourcing protocol solves a linear system of n = 1, 000 equations and m = 1, 000 unknown variables about 472 times faster than a non-outsourced version. The efficiency gain is more substantial when (n,m) gets larger. For example, when n = 10, 000 and m = 10, 000, the protocol can solve it about 56, 274 times faster. Our protocols can also be easily implemented in parallel computation architecture to get more efficiency improvement.

A POD projection method for large-scale algebraic Riccati equations

The solution of large-scale matrix algebraic Riccati equations is important for instance in control design and model reduction and remains an active area of research. We consider a class of matrix algebraic Riccati equations (AREs) arising from a linear system along with a weighted inner product. This problem class often arises from a spatial discretization of a partial differential equation system. We propose a projection method to obtain low rank solutions of AREs based on simulations of linear systems coupled with proper orthogonal decomposition. The method can take advantage of existing (black box) simulation code to generate the projection matrices. We also develop new weighted norm residual computations and error bounds. We present numerical results demonstrating that the proposed approach can produce highly accurate approximate solutions. We also briefly discuss making the proposed approach completely data-based so that one can use existing simulation codes without accessing system matrices.

Guaranteed Matrix Completion via Non-Convex Factorization

Matrix factorization is a popular approach for large-scale matrix completion. The optimization formulation based on matrix factorization, even with huge size, can be solved very efficiently through the standard optimization algorithms in practice. However, due to the non-convexity caused by the factorization model, there is a limited theoretical understanding of whether these algorithms will generate a good solution. In this paper, we establish a theoretical guarantee for the factorization-based formulation to correctly recover the underlying low-rank matrix. In particular, we show that under similar conditions to those in previous works, many standard optimization algorithms converge to the global optima of a factorization-based formulation and recover the true low-rank matrix. We study the local geometry of a properly regularized objective and prove that any stationary point in a certain local region is globally optimal. A major difference of this paper from the existing results is that we do not need resampling (i.e., using independent samples at each iteration) in either the algorithm or its analysis.

Innovated scalable efficient estimation in ultra-large Gaussian graphical models

Large-scale precision matrix estimation is of fundamental importance yet challenging in many contemporary applications for recovering Gaussian graphical models. In this paper, we suggest a new approach of innovated scalable efficient estimation (ISEE) for estimating large precision matrix. Motivated by the innovated transformation, we convert the original problem into that of large covariance matrix estimation. The suggested method combines the strengths of recent advances in high-dimensional sparse modeling and large covariance matrix estimation. Compared to existing approaches, our method is scalable and can deal with much larger precision matrices with simple tuning. Under mild regularity conditions, we establish that this procedure can recover the underlying graphical structure with significant probability and provide efficient estimation of link strengths. Both computational and theoretical advantages of the procedure are evidenced through simulation and real data examples.

FF-SKPCCA: Kernel probabilistic canonical correlation analysis

Several information fusion methods are developed for increasing the recognition accuracy in multimodal systems. Canonical correlation analysis (CCA), cross-modal factor analysis (CFA) and their kernel versions are known as successful fusion techniques but they cannot digest the data variability. Probabilistic CCA (PCCA) is suggested as a linear fusion method to capture input variability. A new kernel PCCA (KPCCA) is proposed here to capture both the nonlinear correlations of sources and input variability. The functionality of KPCCA decreases when the number of samples, which determines the size of kernel matrix increases. In the conventional fusion methods the latent variables of different modalities are concatenated; consequently, a large-scale covariance matrix with just limited number of samples must be estimated To overcome this drawback, a sparse KPCCA (SKPCCA) is introduced which scarifies the covariance matrix elements at the cost of decreasing its rank. In the final stage of the gradual evolution of KPCCA, a new feature fusion manner is proposed for SKPCCA (FF-SKPCCA) as a second stage fusion. This proposed method unifies the latent variables of two modalities into a feature vector with an acceptable size. Audio-visual databases like M2VTS (for speech recognition) eNTERFACE and RML (for emotion recognition) are applied to assess FF-SKPCCA compared to state-of-the-art fusion methods. The comparative results indicate the superiority of the proposed method in most cases.

Development of press-and-sinter Al2024-based nanocomposites reinforced with multiwalled carbon nanotubes

To fabricate large-scale multi-walled carbon nanotube-reinforced aluminum matrix composites, in this study, nanocomposites powder is successfully consolidated with simple processing methods. Ball milling is used to disperse multi-walled carbon nanotubes in the aluminum alloy 2024 matrix, and sequential cold press, hot press, and sintering techniques are selected for fabricating fully dense samples. As a result, an aluminum alloy 2024 composite with 4 vol.% multi-walled carbon nanotubes shows a yield strength of ∼700 MPa and an elongation to failure of 1.5% with Young’s modulus of 86.4 GPa. The large-scale composite can be used for automobile parts such as swash plates or piston heads.

Improved square-root cubature information filter

In this paper, a theoretical comparison between existing the sigma-point information filter (SPIF) framework and the unscented information filter (UIF) framework is presented. It is shown that the SPIF framework is identical to the sigma-point Kalman filter (SPKF). However, the UIF framework is not identical to the classical SPKF due to the neglect of one-step prediction errors of measurements in the calculation of state estimation error covariance matrix. Thus SPIF framework is more reasonable as compared with UIF framework. According to the theoretical comparison, an improved cubature information filter (CIF) is derived based on the superior SPIF framework. Square-root CIF (SRCIF) is also developed to improve the numerical accuracy and stability of the proposed CIF. The proposed SRCIF is applied to a target tracking problem with large sampling interval and high turn rate, and its performance is compared with the existing SRCIF. The results show that the proposed SRCIF is more reliable and stable as compared with the existing SRCIF. Note that it is impractical for information filters in large-scale applications due to the enormous computational complexity of large-scale matrix inversion, and advanced techniques need to be further considered.

Iterative-method performance evaluation for multiple vectors associated with a large-scale sparse matrix

ABSTRACT Ensemble computing, which is an instance of capacity computing, is an effective computing scenario for exascale parallel supercomputers. In ensemble computing, there are multiple linear systems associated with a common coefficient matrix. We improve the performance of iterative solvers for multiple vectors by solving them at the same time, that is, by solving for the product of the matrices. We implemented several iterative methods and compared their performance. The maximum performance on Sparc VIIIfx was 7.6 times higher than that of a naïve implementation. Finally, to deal with the different convergence processes of linear systems, we introduced a control method to eliminate the calculation of already converged vectors.

Iterative-method performance evaluation for multiple vectors associated with a large-scale sparse matrix

Ensemble computing, which is an instance of capacity computing, is an effective computing scenario for exascale parallel supercomputers. In ensemble computing, there are multiple linear systems ass...

Tangential interpolation-based eigensystem realization algorithm for MIMO systems

ABSTRACTThe eigensystem realization algorithm (ERA) is a commonly used data-driven method for system identification and reduced-order modelling of dynamical systems. The main computational difficulty in ERA arises when the system under consideration has a large number of inputs and outputs, requiring to compute a singular value decomposition (SVD) of a large-scale dense Hankel matrix. In this work, we present an algorithm that aims to resolve this computational bottleneck via tangential interpolation. This involves projecting the original impulse response sequence onto suitably chosen directions. The resulting data-driven reduced model preserves stability and is endowed with an a priori error bound. Numerical examples demonstrate that the modified ERA algorithm with tangentially interpolated data produces accurate reduced models while, at the same time, reducing the computational cost and memory requirements significantly compared to the standard ERA. We also give an example to demonstrate the limitations of the proposed method.

Nuclear shell model in natural basis

Shell model calculation is a key tool in investigating the nuclear structure. Traditional shell model calculations deal with the many-body correlations through the configurations interaction, in which one performs large-scale matrix diagonalization to study the excitation spectra for system in many basis states. The crucial starting point in shell model calculations is the derivative of effective interaction, which be either a microscopic approach based on realistic nuclear forces or a phenomenological determined interaction. Conventional shell model calculation always build the many-body wave functions in harmonic oscillator basis, which is always bound and localized. However, nuclei far from stability have strong coupling with the scattering states, low particle emission threshold and the wave functions have large spatial dimensions, which may lead the harmonic oscillator basis fail in describing the weakly bound nuclei.

2.5D forward modeling and inversion of frequency-domain airborne electromagnetic data

Frequency-domain airborne electromagnetics is a proven geophysical exploration method. Presently, the interpretation is mainly based on resistivity—depth imaging and one-dimensional layered inversion; nevertheless, it is difficult to obtain satisfactory results for two- or three-dimensional complex earth structures using 1D methods. 3D forward modeling and inversion can be used but are hampered by computational limitations because of the large number of data. Thus, we developed a 2.5D frequency-domain airborne electromagnetic forward modeling and inversion algorithm. To eliminate the source singularities in the numerical simulations, we split the fields into primary and secondary fields. The primary fields are calculated using homogeneous or layered models with analytical solutions, and the secondary (scattered) fields are solved by the finite-element method. The linear system of equations is solved by using the large-scale sparse matrix parallel direct solver, which greatly improves the computational efficiency. The inversion algorithm was based on damping least-squares and singular value decomposition and combined the pseudo forward modeling and reciprocity principle to compute the Jacobian matrix. Synthetic and field data were used to test the effectiveness of the proposed method.

Compact InGaAsP/InP 3 × 3 multimode-interference coupler-based electro-optic switch.

Optical switches are key components in data links for optical communication networks requiring low crosstalk and insertion loss, high switching speed, and power efficiency. A multimode-interference (MMI)-coupler-based switch with multiple inputs and outputs serving as a switching unit is desired toward forming a large-scale switch matrix. Here we demonstrate a compact 3×3 MMI coupler electro-optic switch based on the carrier injection effect on InGaAsP/InP substrates. This switch device is 2780 μm long by 18 μm wide. The switching states can be controlled by two index-modulation regions through applied bias voltages. Our simulation results show that the device exhibits low crosstalk of <-22 dB, high extinction ratio of ≳23 dB, low electrical switching energy of ∼2.0 pJ/bit, maximum operational frequency of ∼1.0 GHz, and optical bandwidth of ∼20 nm in the C band. We experimentally validated one of the switching states on a fabricated device with maximum current injections of ∼25 mA under combined bias voltages of ∼2.5 and ∼3.0 V. Such monolithic integration schemes make them an ideal candidate for future on-chip photonic integrated circuits.

High Precision Low Complexity Matrix Inversion Based on Newton Iteration for Data Detection in the Massive MIMO

Currently, massive multiple-input multiple-output (MIMO) is one of the most promising wireless transmission technologies for 5G. Massive MIMO requires handling with large-scale matrix computation, especially for matrix inversion. In this letter, we find that matrix inversion based on Newton iteration (NI) is suitable for data detection in massive MIMO system. In contrast with recently proposed polynomial expansion (PE) method for matrix inversion, we analyze both the algorithm complexity and precision in detail, and propose a diagonal band Newton iteration (DBNI) method, which is an approximate method for NI. Compared with PE method, DBNI can obtain higher precision and approximately equal complexity, and we give an explanation of how to select the bandwidth of DBNI.

Fast SVD Computations for Synchrophasor Algorithms

Many singular value decomposition (SVD) problems in power system computations require only a few largest singular values of a large-scale matrix for the analysis. This letter introduces two fast SVD approaches recently developed in other domains to power systems for speeding up phasor measurement unit (PMU) based online applications. The first method is a randomized SVD algorithm that accelerates computation by introducing a low-rank approximation of a given matrix through randomness. The second method is the augmented Lanczos bidiagonalization, an iterative Krylov subspace technique that computes sequences of projections of a given matrix onto low-dimensional subspaces. Both approaches are illustrated on SVD evaluation within an ambient oscillation monitoring algorithm, namely stochastic subspace identification (SSI).

Spark-Based Large-Scale Matrix Inversion for Big Data Processing

Matrix inversion is a fundamental operation for solving linear equations for many computational applications, especially for various emerging big data applications. However, it is a challenging task to invert large-scale matrices of extremely high order (several thousands or millions), which are common in most Web-scale systems, such as social networks and recommendation systems. In this paper, we present an lower upper decomposition-based block-recursive algorithm for large-scale matrix inversion. We present its well-designed implementation with optimized data structure, reduction of space complexity, and effective matrix multiplication on the Spark parallel computing platform. The experimental evaluation results show that the proposed algorithm is efficient to invert large-scale matrices on a cluster composed of commodity servers and is scalable for inverting even larger matrices. The proposed algorithm and implementation will become a solid foundation for building a high-performance linear algebra library on Spark for big data processing and applications.

Low-Rank Solvers for Fractional Differential Equations

Many problems in science and technology can be cast using differential equations with both fractional time and spatial derivatives. To accurately simulate natural phenomena using this technology fine spatial and temporal discretizations are required, leading to large-scale linear systems or matrix equations, especially whenever more than one space dimension is considered. The discretization of fractional differential equations typically involves dense matrices with a Toeplitz or in the variable coefficient case Toeplitz-like structure . We combine the fast evaluation of Toeplitz matrices and their circulant preconditioners with state-of-the-art linear matrix equation methods to efficiently solve these problems, both in terms of CPU time and memory requirements. Additionally, we illustrate how these techniqes can be adapted when variable coefficients are present. Numerical experiments on typical differential problems with fractional derivatives in both space and time showing the effectiveness of the approaches are reported.

Induced Dimension Reduction Method for Solving Linear Matrix Equations

This paper discusses the solution of large-scale linear matrix equations using the Induced Dimension reduction method (IDR(s)). IDR(s) was originally presented to solve system of linear equations, and is based on the IDR(s) theorem. We generalize the IDR(s) theorem to solve linear problems in any finite-dimensional space. This generalization allows us to develop IDR(s) algorithms to approximate the solution of linear matrix equations. The IDR(s) method presented here has two main advantages; firstly, it does not require the computation of inverses of any matrix, and secondly, it allows incorporation of preconditioners. Additionally, we present a simple preconditioner to solve the Sylvester equation based on a fixed point iteration. Several numerical examples illustrate the performance of IDR(s) for solving linear matrix equations. We also present the software implementation.

2.5D Inversion Algorithm of Frequency-Domain Airborne Electromagnetics with Topography

We presented a 2.5D inversion algorithm with topography for frequency-domain airborne electromagnetic data. The forward modeling is based on edge finite element method and uses the irregular hexahedron to adapt the topography. The electric and magnetic fields are split into primary (background) and secondary (scattered) field to eliminate the source singularity. For the multisources of frequency-domain airborne electromagnetic method, we use the large-scale sparse matrix parallel shared memory direct solver PARDISO to solve the linear system of equations efficiently. The inversion algorithm is based on Gauss-Newton method, which has the efficient convergence rate. The Jacobian matrix is calculated by “adjoint forward modelling” efficiently. The synthetic inversion examples indicated that our proposed method is correct and effective. Furthermore, ignoring the topography effect can lead to incorrect results and interpretations.

Visual scenes are categorized by function.

How do we know that a kitchen is a kitchen by looking? Traditional models posit that scene categorization is achieved through recognizing necessary and sufficient features and objects, yet there is little consensus about what these may be. However, scene categories should reflect how we use visual information. Therefore, we test the hypothesis that scene categories reflect functions, or the possibilities for actions within a scene. Our approach is to compare human categorization patterns with predictions made by both functions and alternative models. We collected a large-scale scene category distance matrix (5 million trials) by asking observers to simply decide whether 2 images were from the same or different categories. Using the actions from the American Time Use Survey, we mapped actions onto each scene (1.4 million trials). We found a strong relationship between ranked category distance and functional distance (r = .50, or 66% of the maximum possible correlation). The function model outperformed alternative models of object-based distance (r = .33), visual features from a convolutional neural network (r = .39), lexical distance (r = .27), and models of visual features. Using hierarchical linear regression, we found that functions captured 85.5% of overall explained variance, with nearly half of the explained variance captured only by functions, implying that the predictive power of alternative models was because of their shared variance with the function-based model. These results challenge the dominant school of thought that visual features and objects are sufficient for scene categorization, suggesting instead that a scene's category may be determined by the scene's function.