Showing papers on "Matrix (mathematics) published in 2020"

Tensor Robust Principal Component Analysis with a New Tensor Nuclear Norm

Abstract: In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product) [14] . Induced by the t-product, we first rigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor average rank, and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm. These definitions, their relationships and properties are consistent with matrix cases. Equipped with the new tensor nuclear norm, we then solve the TRPCA problem by solving a convex program and provide the theoretical guarantee for the exact recovery. Our TRPCA model and recovery guarantee include matrix RPCA as a special case. Numerical experiments verify our results, and the applications to image recovery and background modeling problems demonstrate the effectiveness of our method.

GMC: Graph-Based Multi-View Clustering

Abstract: Multi-view graph-based clustering aims to provide clustering solutions to multi-view data. However, most existing methods do not give sufficient consideration to weights of different views and require an additional clustering step to produce the final clusters. They also usually optimize their objectives based on fixed graph similarity matrices of all views. In this paper, we propose a general G raph-based M ulti-view C lustering (GMC) to tackle these problems. GMC takes the data graph matrices of all views and fuses them to generate a unified graph matrix. The unified graph matrix in turn improves the data graph matrix of each view, and also gives the final clusters directly. The key novelty of GMC is its learning method, which can help the learning of each view graph matrix and the learning of the unified graph matrix in a mutual reinforcement manner. A novel multi-view fusion technique can automatically weight each data graph matrix to derive the unified graph matrix. A rank constraint without introducing a tuning parameter is also imposed on the graph Laplacian matrix of the unified matrix, which helps partition the data points naturally into the required number of clusters. An alternating iterative optimization algorithm is presented to optimize the objective function. Experimental results using both toy data and real-world data demonstrate that the proposed method outperforms state-of-the-art baselines markedly.

Formulas for Data-Driven Control: Stabilization, Optimality, and Robustness

Abstract: In a paper by Willems et al., it was shown that persistently exciting data can be used to represent the input-output behavior of a linear system. Based on this fundamental result, we derive a parametrization of linear feedback systems that paves the way to solve important control problems using data-dependent linear matrix inequalities only. The result is remarkable in that no explicit system's matrices identification is required. The examples of control problems we solve include the state and output feedback stabilization, and the linear quadratic regulation problem. We also discuss robustness to noise-corrupted measurements and show how the approach can be used to stabilize unstable equilibria of nonlinear systems.

Quantum Fisher information matrix and multiparameter estimation

Abstract: Quantum Fisher information matrix (QFIM) is a core concept in theoretical quantum metrology due to the significant importance of quantum Cram\'{e}r-Rao bound in quantum parameter estimation. However, studies in recent years have revealed wide connections between QFIM and other aspects of quantum mechanics, including quantum thermodynamics, quantum phase transition, entanglement witness, quantum speed limit and non-Markovianity. These connections indicate that QFIM is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. In this paper, we summarize the properties and existing calculation techniques of QFIM for various cases, and review the development of QFIM in some aspects of quantum mechanics apart from quantum metrology. On the other hand, as the main application of QFIM, the second part of this paper reviews the quantum multiparameter Cram\'{e}r-Rao bound, its attainability condition and the associated optimal measurements. Moreover, recent developments in a few typical scenarios of quantum multiparameter estimation and the quantum advantages are also thoroughly discussed in this part.

Image Compressed Sensing Using Convolutional Neural Network

Abstract: In the study of compressed sensing (CS), the two main challenges are the design of sampling matrix and the development of reconstruction method. On the one hand, the usually used random sampling matrices (e.g., GRM) are signal independent, which ignore the characteristics of the signal. On the other hand, the state-of-the-art image CS methods (e.g., GSR and MH) achieve quite good performance, however with much higher computational complexity. To deal with the two challenges, we propose an image CS framework using convolutional neural network (dubbed CSNet) that includes a sampling network and a reconstruction network, which are optimized jointly. The sampling network adaptively learns the sampling matrix from the training images, which makes the CS measurements retain more image structural information for better reconstruction. Specifically, three types of sampling matrices are learned, i.e., floating-point matrix, {0, 1}-binary matrix, and {−1, +1}-bipolar matrix. The last two matrices are specially designed for easy storage and hardware implementation. The reconstruction network, which contains a linear initial reconstruction network and a non-linear deep reconstruction network, learns an end-to-end mapping between the CS measurements and the reconstructed images. Experimental results demonstrate that CSNet offers state-of-the-art reconstruction quality, while achieving fast running speed. In addition, CSNet with {0, 1}-binary matrix, and {−1, +1}-bipolar matrix gets comparable performance with the existing deep learning-based CS methods, outperforms the traditional CS methods. Experimental results further suggest that the learned sampling matrices can improve the traditional image CS reconstruction methods significantly.

A novel chaos based optical image encryption using fractional Fourier transform and DNA sequence operation

Abstract: In this paper, we propose a new method for optical image encryption using fractional Fourier transform, DNA sequence operation and chaos theory. Random phase masks are generated using iterative Lorenz map and the plain image is transformed to a DNA matrix. This matrix is combined with the random phase mask and then transformed three times using the fractional Fourier transform. An Optical implementation of the encryption algorithm is proposed in our work. According to the experiment results and security analysis, we find that our algorithm has good encryption effect, larger secret key space and high sensitivity to the secret key. It can resist to most known attacks, such as statistical analysis and exhaustive attacks. All these features show that our encryption algorithm is very suitable for digital image encryption.

New Insights and Perspectives on the Natural Gradient Method

Abstract: Natural gradient descent is an optimization method traditionally motivated from the perspective of information geometry, and works well for many applications as an alternative to stochastic gradient descent. In this paper we critically analyze this method and its properties, and show how it can be viewed as a type of 2nd-order optimization method, with the Fisher information matrix acting as a substitute for the Hessian. In many important cases, the Fisher information matrix is shown to be equivalent to the Generalized Gauss-Newton matrix, which both approximates the Hessian, but also has certain properties that favor its use over the Hessian. This perspective turns out to have significant implications for the design of a practical and robust natural gradient optimizer, as it motivates the use of techniques like trust regions and Tikhonov regularization. Additionally, we make a series of contributions to the understanding of natural gradient and 2nd-order methods, including: a thorough analysis of the convergence speed of stochastic natural gradient descent (and more general stochastic 2nd-order methods) as applied to convex quadratics, a critical examination of the oft-used "empirical" approximation of the Fisher matrix, and an analysis of the (approximate) parameterization invariance property possessed by natural gradient methods (which we show also holds for certain other curvature, but notably not the Hessian).

Randomized numerical linear algebra: Foundations and algorithms

Abstract: This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problems. The paper treats both the theoretical foundations of the subject and practical computational issues. Topics include norm estimation, matrix approximation by sampling, structured and unstructured random embeddings, linear regression problems, low-rank approximation, subspace iteration and Krylov methods, error estimation and adaptivity, interpolatory and CUR factorizations, Nystrom approximation of positive semidefinite matrices, single-view (‘streaming’) algorithms, full rank-revealing factorizations, solvers for linear systems, and approximation of kernel matrices that arise in machine learning and in scientific computing.

Inductive Matrix Completion Based on Graph Neural Networks

Abstract: We propose an inductive matrix completion model without using side information. By factorizing the (rating) matrix into the product of low-dimensional latent embeddings of rows (users) and columns (items), a majority of existing matrix completion methods are transductive, since the learned embeddings cannot generalize to unseen rows/columns or to new matrices. To make matrix completion inductive, content (side information), such as user's age or movie's genre, has to be used previously. However, high-quality content is not always available, and can be hard to extract. Under the extreme setting where not any side information is available other than the matrix to complete, can we still learn an inductive matrix completion model? In this paper, we investigate this seemingly impossible problem and propose an Inductive Graph-based Matrix Completion (IGMC) model without using any side information. It trains a graph neural network (GNN) based purely on local subgraphs around (user, item) pairs generated from the rating matrix and maps these subgraphs to their corresponding ratings. Our model achieves highly competitive performance with state-of-the-art transductive baselines. In addition, since our model is inductive, it can generalize to users/items unseen during the training (given that their ratings exist), and can even transfer to new tasks. Our transfer learning experiments show that a model trained out of the MovieLens dataset can be directly used to predict Douban movie ratings and works surprisingly well. Our work demonstrates that: 1) it is possible to train inductive matrix completion models without using any side information while achieving state-of-the-art performance; 2) local graph patterns around a (user, item) pair are effective predictors of the rating this user gives to the item; and 3) we can transfer models trained on existing recommendation tasks to new tasks without any retraining.

A Compressive Sensing-Based Approach to End-to-End Network Traffic Reconstruction

Abstract: Estimation of end-to-end network traffic plays an important role in traffic engineering and network planning. The direct measurement of a network's traffic matrix consumes large amounts of network resources and is thus impractical in most cases. How to accurately construct traffic matrix remains a great challenge. This paper studies end-to-end network traffic reconstruction in large-scale networks. Applying compressive sensing theory, we propose a novel reconstruction method for end-to-end traffic flows. First, the direct measurement of partial Origin-Destination (OD) flows is determined by random measurement matrix, providing partial measurements. Then, we use the K-SVD approach to obtain a sparse matrix. Combined with compressive sensing, this partially known OD flow matrix can be used to recover the entire end-to-end network traffic matrix. Simulation results show that the proposed method can reconstruct end-to-end network traffic with a high degree of accuracy. Moreover, in comparison with previous methods, our approach exhibits a significant performance improvement.

Spherical fuzzy Dombi aggregation operators and their application in group decision making problems

Abstract: Spherical fuzzy sets (SFSs), recently proposed by Ashraf, is one of the most important concept to describe the fuzzy information in the process of decision making. In SFSs the sum of the squares of memberships grades lies in close unit interval and hence accommodate more uncertainties. Thus, this set outperforms over the existing structures of fuzzy sets. In real decision making problems, there is often a treat regarding a neutral character towards the membership and non-membership degrees expressed by the decision-makers. To get a fair decision during the process, in this paper, we define some new operational laws by Dombi t-norm and t-conorm. In the present study, we propose Spherical fuzzy Dombi weighted averaging (SFDWA), Spherical fuzzy Dombi ordered weighted averaging (SFDOWA), Spherical fuzzy Dombi hybrid weighted averaging (SFDHWA), Spherical fuzzy Dombi weighted geometric (SFDWG), Spherical fuzzy Dombi ordered weighted geometric (SFDOWG) and Spherical fuzzy Dombi hybrid weighted geometric (SFDHWG) aggregation operators and discuss several properties of these aggregation operators. These aforesaid operators are enormously used to help a successful solution of the decision problems. Then an algorithm by using spherical fuzzy set information in decision-making matrix is developed and applied the algorithm to decision-making problem to illustrate its applicability and effectiveness. Through this algorithm, we proved that our proposed approach is practical and provides decision makers a more mathematical insight before making decisions on their options. Besides this, a systematic comparison analysis with other existent methods is conducted to reveal the advantages of our method. Results indicate that the proposed method is suitable and effective for decision process to evaluate their best alternative.

The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral

Abstract: We propose that a class of new topologies, for which there is no classical solution, should be included in the path integral of three-dimensional pure gravity, and that their inclusion solves pathological negativities in the spectrum, replacing them with a nonperturbative shift of the BTZ extremality bound. We argue that a two-dimensional calculation using a dimensionally reduced theory captures the leading effects in the near extremal limit. To make this argument, we study a closely related two-dimensional theory of Jackiw-Teitelboim gravity with dynamical defects. We show that this theory is equivalent to a matrix integral.

Design of highly efficient deep-blue organic afterglow through guest sensitization and matrices rigidification

Abstract: Blue/deep-blue emission is crucial for organic optoelectronics but remains a formidable challenge in organic afterglow due to the difficulties in populating and stabilizing the high-energy triplet excited states. Here, a facile strategy to realize the efficient deep-blue organic afterglow is proposed via host molecules to sensitize the triplet exciton population of guest and water implement to suppress the non-radiative decays by matrices rigidification. A series of highly luminescent deep-blue (405–428 nm) organic afterglow materials with lifetimes up to 1.67 s and quantum yields of 46.1% are developed. With these high-performance water-responsive materials, lifetime-encrypted rewritable paper has been constructed for water-jet printing of high-resolution anti-counterfeiting patterns that can retain for a long time (>1 month) and be erased by dimethyl sulfoxide vapor in 15 min with high reversibility for many write/erase cycles. These results provide a foundation for the design of high-efficient blue/deep-blue organic afterglow and stimuli-responsive materials with remarkable applications. Though realizing organic materials with deep blue emission is attractive for next-generation display technologies, achieving this emission in afterglow molecules remains a challenge. Here, the authors report blue organic afterglow via a strategy involving guest sensitization and matrix rigidification.

MatRaptor: A Sparse-Sparse Matrix Multiplication Accelerator Based on Row-Wise Product

Abstract: Sparse-sparse matrix multiplication (SpGEMM) is a computation kernel widely used in numerous application domains such as data analytics, graph processing, and scientific computing. In this work we propose MatRaptor, a novel SpGEMM accelerator that is high performance and highly resource efficient. Unlike conventional methods using inner or outer product as the meta operation for matrix multiplication, our approach is based on row-wise product, which offers a better tradeoff in terms of data reuse and on-chip memory requirements, and achieves higher performance for large sparse matrices. We further propose a new hardware-friendly sparse storage format, which allows parallel compute engines to access the sparse data in a vectorized and streaming fashion, leading to high utilization of memory bandwidth. We prototype and simulate our accelerator architecture using gem5 on a diverse set of matrices. Our experiments show that MatRaptor achieves 129.2× speedup over single-threaded CPU, 8.8× speedup over GPU and 1.8× speedup over the state-of-the-art SpGEMM accelerator (OuterSPACE). MatRaptor also has 7.2× lower power consumption and 31.3× smaller area compared to OuterSPACE.

Feature Selective Projection with Low-Rank Embedding and Dual Laplacian Regularization

Abstract: Feature extraction and feature selection have been regarded as two independent dimensionality reduction methods in most of the existing literature. In this paper, we propose to integrate both approaches into a unified framework and design an unsupervised linear feature selective projection (FSP) for feature extraction with low-rank embedding and dual Laplacian regularization, with the aim to exploit the intrinsic relationship among data and suppress the impact of noise. Specifically, a projection matrix with an $l_{2,1}$ l 2 , 1 -norm regularization is introduced to project original high dimensional data points into a new subspace with lower dimension, where the $l_{2,1}$ l 2 , 1 -norm regularization can endow the projection with good interpretability. We deploy a coefficient matrix with low rank constraint to reconstruct the data points and the $l_{2,1}$ l 2 , 1 -norm is imposed to regularize the data reconstruction errors in the low-dimensional subspace and make FSP robust to noise. Furthermore, a dual graph Laplacian regularization term is imposed on the low dimensional data and data reconstruction matrix for preserving the local manifold geometrical structure of data. Finally, an alternatively iterative algorithm is carefully designed for solving the proposed optimization model. Theoretical convergence and computational complexity analysis of the algorithm are also provided. Comprehensive experiments on various benchmark datasets have been carried out to evaluate the performance of the proposed FSP. As indicated, our algorithm significantly outperforms other state-of-the-art methods for feature extraction.

A new color image encryption scheme based on DNA encoding and spatiotemporal chaotic system

Abstract: In this paper, a new color image encryption scheme based on DNA operations and spatiotemporal chaotic system is presented. Firstly, to hide the distribution information of the plain image, we convert the plain image into three DNA matrices based on the DNA random encoding rules. Then, the DNA matrices are combined into a new matrix and is permutated by a scramble matrix generated by mixed linear-nonlinear coupled map lattices (MLNCML) system. In which, the key streams are associated with the secret keys and plain image, which can ensure our cryptosystem plain-image-dependent and improve the ability to resist known-plaintext or chosen-plaintext attacks. Thereafter, to resist statistical attacks, the scrambled matrix is decomposed into three matrices and diffused by DNA deletion-insertion operations. Finally, the three matrices are decoded based on DNA random decoding rules and recombined to three channels of the cipher image. Simulation results demonstrate that the proposed image cryptosystem has good security and can resist various potential attacks.

Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion

Abstract: The main aim of this paper is to develop a framelet representation of the tensor nuclear norm for third-order tensor recovery. In the literature, the tensor nuclear norm can be computed by using tensor singular value decomposition based on the discrete Fourier transform matrix, and tensor completion can be performed by the minimization of the tensor nuclear norm which is the relaxation of the sum of matrix ranks from all Fourier transformed matrix frontal slices. These Fourier transformed matrix frontal slices are obtained by applying the discrete Fourier transform on the tubes of the original tensor. In this paper, we propose to employ the framelet representation of each tube so that a framelet transformed tensor can be constructed. Because of framelet basis redundancy, the representation of each tube is sparsely represented. When the matrix slices of the original tensor are highly correlated, we expect the corresponding sum of matrix ranks from all framelet transformed matrix frontal slices would be small, and the resulting tensor completion can be performed much better. The proposed minimization model is convex and global minimizers can be obtained. Numerical results on several types of multi-dimensional data (videos, multispectral images, and magnetic resonance imaging data) have tested and shown that the proposed method outperformed the other testing methods.

SpArch: Efficient Architecture for Sparse Matrix Multiplication

Abstract: Generalized Sparse Matrix-Matrix Multiplication (SpGEMM) is a ubiquitous task in various engineering and scientific applications. However, inner product based SpGEMM introduces redundant input fetches for mismatched nonzero operands, while outer product based approach suffers from poor output locality due to numerous partial product matrices. Inefficiency in the reuse of either inputs or outputs data leads to extensive and expensive DRAM access. To address this problem, this paper proposes an efficient sparse matrix multiplication accelerator architecture, SpArch, which jointly optimizes the data locality for both input and output matrices. We first design a highly parallelized streaming-based merger to pipeline the multiply and merge stage of partial matrices so that partial matrices are merged on chip immediately after produced. We then propose a condensed matrix representation that reduces the number of partial matrices by three orders of magnitude and thus reduces DRAM access by 5.4x. We further develop a Huffman tree scheduler to improve the scalability of the merger for larger sparse matrices, which reduces the DRAM access by another 1.8x. We also resolve the increased input matrix read induced by the new representation using a row prefetcher with near-optimal buffer replacement policy, further reducing the DRAM access by 1.5x. Evaluated on 20 benchmarks, SpArch reduces the total DRAM access by 2.8x over previous state-of-the-art. On average, SpArch achieves 4x, 19x, 18x, 17x, 1285x speedup and 6x, 164x, 435x, 307x, 62x energy savings over OuterSpace, MKL, cuSPARSE, CUSP, and ARM Armadillo, respectively.

Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing Quantum machine learning

Abstract: We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyen et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffices to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: l2-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

Variational Quantum Fidelity Estimation

Abstract: Computing quantum state fidelity will be important to verify and characterize states prepared on a quantum computer. In this work, we propose novel lower and upper bounds for the fidelity F(ρ,σ) based on the “truncated fidelity'” F(ρ_m,σ) which is evaluated for a state ρ_m obtained by projecting ρ onto its mm-largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically with mm. To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that involves three steps: (1) variationally diagonalize ρ, (2) compute matrix elements of σ in the eigenbasis of ρ, and (3) combine these matrix elements to compute our bounds. Our algorithm is aimed at the case where σ is arbitrary and ρ is low rank, which we call low-rank fidelity estimation, and we prove that no classical algorithm can efficiently solve this problem under reasonable assumptions. Finally, we demonstrate that our bounds can detect quantum phase transitions and are often tighter than previously known computable bounds for realistic situations.

Eigenstate Thermalization in a Locally Perturbed Integrable System.

Abstract: Eigenstate thermalization is widely accepted as the mechanism behind thermalization in generic isolated quantum systems. Using the example of a single magnetic defect embedded in the integrable spin-$1/2$ $XXZ$ chain, we show that locally perturbing an integrable system can give rise to eigenstate thermalization. Unique to such setups is the fact that thermodynamic and transport properties of the unperturbed integrable chain emerge in properties of the eigenstates of the perturbed (nonintegrable) one. Specifically, we show that the diagonal matrix elements of observables in the perturbed eigenstates follow the microcanonical predictions for the integrable model, and that the ballistic character of spin transport in the integrable model is manifest in the behavior of the off-diagonal matrix elements of the current operator in the perturbed eigenstates.

Stability and Stabilization of T–S Fuzzy Systems With Time-Varying Delays via Delay-Product-Type Functional Method

Abstract: This paper is concerned with the stability and stabilization problems of T–S fuzzy systems with time-varying delays. The purpose is to develop a new state-feedback controller design method with less conservatism. First, a novel Lyapunov–Krasovskii functional is constructed by combining delay-product-type functional method together with the state vector augmentation. By utilizing Wirtinger-based integral inequality and an extended reciprocally convex matrix inequality, a less conservative delay-dependent stability condition is developed. Then, the corresponding controller design method for the closed-loop delayed fuzzy system is derived based on parallel distributed compensation scheme. Finally, two classic numerical examples are given to show the effectiveness and merits of the proposed approaches.

A Multireference Quantum Krylov Algorithm for Strongly Correlated Electrons.

Abstract: We introduce a multireference selected quantum Krylov (MRSQK) algorithm suitable for quantum simulation of many-body problems. MRSQK is a low-cost alternative to the quantum phase estimation algorithm that generates a target state as a linear combination of non-orthogonal Krylov basis states. This basis is constructed from a set of reference states via real-time evolution; thus, avoiding the numerical optimization of parameters. An efficient algorithm for the evaluation of the off-diagonal matrix elements of the overlap and Hamiltonian matrices is discussed and a selection procedure is introduced to identify a basis of orthogonal references that ameliorates the linear dependency problem. Preliminary benchmarks on linear H6, H8, and BeH2 indicate that MRSQK can predict the energy of these systems accurately using very compact Krylov bases.

GASP Codes for Secure Distributed Matrix Multiplication

Abstract: We consider the problem of secure distributed matrix multiplication (SDMM) in which a user wishes to compute the product of two matrices with the assistance of honest but curious servers. We construct polynomial codes for SDMM by studying a combinatorial problem on a special type of addition table, which we call the degree table. The codes are based on arithmetic progressions, and are thus named GASP (Gap Additive Secure Polynomial) Codes. GASP Codes are shown to outperform all previously known polynomial codes for secure distributed matrix multiplication in terms of download rate.