Matrix Multiplication Research Articles

FPGA Accelerated Implementation of 3D Mesh Secret Sharing Based on Symmetric Similarity of Model

Secret sharing is particularly important in the field of information security, which allows for the reconstruction of secret information from secure shares. However, due to the large amount of data and non-integer data type of 3D (three-dimensional) models, it is less efficient to perform sharing compared to the 2D(two-dimensional) digital images. In order to enable efficient sharing of 3D models as well, this article designs different circuit modules to parallelize the process of sharing 3D models. The proposed structure exploits the vertex extension to perform a complete reconstruction of the 3D model. In the sharing phase, the symmetric similarity of 3D model is exploited to realize the sharing in parallel, which improves the efficiency of sharing. In the reconstruction phase, the Strassen algorithm is used to optimize matrix multiplication, which saves circuit resources. Simulation results show that the structure performs more than 100 times faster than software. The utilization of circuit resources and the size of the share are both optimized. The validity of the designed hardware architecture is confirmed after analyzing the data from experiments performed on several 3D models.

Improved Lower Bound, and Proof Barrier, for Constant Depth Algebraic Circuits

We show that any product-depth Δ algebraic circuit for the Iterated Matrix Multiplication polynomial IMM n, d (when d = O (log n /log log n ) must be of size at least \(n^{\Omega (d^{1/(\varphi ^2)^{\Delta }})}\) , where φ = 1.618 … is the golden ratio. This improves the recent breakthrough result of Limaye, Srinivasan, and Tavenas (FOCS’21), who showed a super polynomial lower bound of the form \(n^{\Omega (d^{1/4^{\Delta }})}\) for constant-depth circuits. One crucial idea of the (LST21) result was to use set-multilinear polynomials where each set in the variables’ underlying partition could be of different sizes. By picking the set sizes more carefully (depending on the depth we are working with), we first show that any product-depth \(\Delta\) set-multilinear circuit for \(\mathrm{IMM}_{n,d}\) (when \(d = O(\log n)\) ) needs size at least \(n^{\Omega (d^{1/\varphi ^{\Delta }})}\) . This improves the \(n^{\Omega (d^{1/2^{\Delta }})}\) lower bound of (LST21). We then use their Hardness Escalation technique to lift this to general circuits. We also show that these techniques cannot improve our lower bound significantly. For the specific two set sizes used in (LST21), they showed that their lower bound cannot be improved. We show that for any \(d^{o(1)}\) set sizes (out of maximum possible d ), the scope for improving our lower bound is minuscule. There exists a set-multilinear circuit that has product-depth \(\Delta\) and size almost matching our lower bound such that the value of the measure used to prove the lower bound is maximum for this circuit. This results in a barrier to further improvement using the same measure.

Semi-analytical solution of Brent equations

A parametrization of Brent equations is proposed which admit for a several times reduction of the number of unknowns and equations. The arising equations are solved numerically, and for the resulting fast matrix multiplication algorithms many known values of rank are reproduced and even improved, in particular, the designs (4,4,4;48) and (2,4,5;32) are found.

Unveiling the strategic impact of big data analytics capabilities in the Saudi Arabian banking sector: an explorative approach

Purpose The purpose of this study is to identify a critical pathway of the effect of big data analytics capabilities (BDACs) on strategic vigilance based on hierarchical process and a capability approach. Design/methodology/approach The researcher adopted a qualitative approach using interviews and a quantitative approach based on the interpretative structural modeling (ISM) fuzzy cross-impact matrix multiplication applied to classification (MICMAC) approach. A primary theoretical approach was also conducted to identify BDACs previously cited in the literature. Findings Four main subdivisions of BDACs were identified: management capabilities, infrastructure flexibility, talent capability and technology. Management capabilities followed by big data technical knowledge and associated with talent capabilities generate a flexible infrastructure to enhance SV. A dynamic capability perspective of knowledge and information is also required for SV. Research limitations/implications Despite the opportunity of this research and the originality of results, some limitations have to be mentioned and can constitute further directives for future researchers, such as the problem of result generalization. First, this research was based in Saudi Arabia, and a comparative approach to defining BDAC on an international level can be more beneficial in providing an exhaustive list of these capabilities. Second, reliability issues, in this research can be addressed due to the use of qualitative data collection which is considered by many researchers as unspecified and can lack scientific rigor. Future studies can improve the number of interviews during the data collection process and data process using an advanced methodological approach. Third, the effect of BDAC in SV according to the hierarchical final modal is not quantified, future work can use this research model to appreciate each effect using a quantitative approach such as correlation and structural equation modeling while considering respondents with different profiles to take into account different point of view in this concern. Practical implications This research enriches the BDAC and MICMAC literature and contributes to this aspect in three main levels. First, by providing an additional empirical asset in this field, this study offers by the way a new case to the big data literature on the banking sector. Based on the limited knowledge as well as results collected from different databases and rigorously analyzed, this subject was not treated previously and the author could not find similar studies with the same approach dealing with the key BDACs in Saudi Arabia. Social implications This research presents three main implications for policymakers and researchers interested in big data analytics (BDA) through a capability and strategic perspective. First, to attain SV, they should prioritize the development of interactive interfaces and open platforms as the primary step before collecting information and deconstructing it to guarantee the generation of knowledge and make decisions effectively. Second, policymakers must introduce organizational technologies in terms of technology management, technical knowledge and technology for decision-making. This requires simultaneous sharing and communication according to relational management. Third, the research conclusions have many critical managerial ramifications for banks in Saudi Arabia while considering the adoption of BDAC. The importance of BDACs (especially technical aspects) in shaping the decision-making to be strategically vigilant emphasizes policymakers’ orientation by paying close attention to these aspects and specific training programs to facilitate the use of such technologies and guarantee strong security measures. Moreover, findings support a balance between technical and functional BDAC. Originality/value The adoption of a knowledge-based dynamic capabilities (KBDCs) view to analyze the interaction between different BDACs in banks in Saudi Arabia to be strategically vigilant using a mixed approach.

ApSpGEMM: Accelerating Large-scale SpGEMM with Heterogeneous Collaboration and Adaptive Panel

The Sparse General Matrix-Matrix multiplication (SpGEMM) is a fundamental component for many applications, such as algebraic multigrid methods (AMG), graphic processing, and deep learning. However, the unbearable latency of computing high-dimensional, large-scale sparse matrix multiplication on GPUs hinders the development of these applications. An effective approach is heterogeneous cores collaborative computing, but this method must address three aspects: (1) irregular non-zero elements lead to load imbalance and irregular memory access, (2) different core computing latency differences reduce computational parallelism, and (3) temporary data transfer between different cores introduces additional latency overhead. In this work, we propose an innovative framework for collaborative large-scale sparse matrix multiplication on CPU-GPU heterogeneous cores, named ApSpGEMM. ApSpGEMM is based on sparsity rules and proposes reordering and splitting algorithms to eliminate the impact of non-zero element distribution features on load and memory access. Then adaptive panels allocation with affinity constraints among cores improves computational parallelism. Finally, carefully arranged asynchronous data transmission and computation balance communication overhead. Compared with state-of-the-art SpGEMM methods, our approach provides excellent absolute performance on matrices with different sparse structures. On heterogeneous cores, the GFlops of large-scale sparse matrix multiplication is improved by 2.25 to 7.21 times.

Innovative Solutions for IK: PROA and Clonal Selection Algorithms Unveiled

Calculating joint angles for sequential manipulators consists of studying the correlation between Cartesian and joint variables. The problem-solving technique encounters two main hurdles described as direct and inverse kinematics. Matrix multiplications usually simplify the direct kinematic problem. However, inverse kinematic problems are harder as they require solving many nonlinear equations and eliminating variables a lot. In our work, we introduce two new methods of handling the complicated inverse kinematic problem for robotic manipulator arms; Poor and Rich Optimization Algorithm and Clonal Selection Algorithm (CSA). These advanced techniques enhance greatly the estimation of various joints in the arm which makes the solution more precise and efficient. To demonstrate the effectiveness, robustness, and potential benefits of these approaches for complicated kinematic problems we present extensive simulation results thereby enabling better performance of robots.

Performance comparison of the Shack-Hartmann and pyramid wavefront sensors with a laser guide star for 40 m telescopes

Context. Upcoming giant segmented mirror telescopes will use laser guide stars (LGS) for their adaptive optics (AO) systems. Two options of wavefront sensors (WFSs) are the Shack-Hartmann wavefront sensor (SHWFS) and the pyramid wavefront sensor (PWFS). Aims. In this paper, we compare the noise performance of the PWFS and the SHWFS. We aim to identify which of the two is best to use in the context of a single or tomographic configuration. Methods. To compute the noise performance, we extended a noise model developed for the PWFS to be used with the SHWFS. To do this, we expressed the centroiding algorithm of the SHWFS as a matrix-vector multiplication, which allowed us to use the statistics of noise to compute its propagation through the AO loop. We validated the noise model with end-to-end simulations for telescopes of 8 and 16 m in diameter. Results. For an AO system with only one WFS, we found that given the same number of subapertures, the PWFS outperforms the SHWFS. For a 40 m telescope, the limiting magnitude of the PWFS is around one magnitude higher than the SHWFS. When using multiple WFS and a generalized least-squares estimator to combine the signal, our model predicts that in a tomographic system, the SHWFS performs better than the PWFS (with a limiting magnitude that is higher by a 0.3 magnitude. When using sub-electron RON detectors for the PWFS, the performance quality is almost identical for the two WFSs. Conclusions. We find that when using a single WFS with LGS, PWFS is a better alternative than the SH. For a tomographic system, both sensors would give roughly the same performance.

Analysing the radiation reliability, performance and energy consumption of low-power SoC through heterogeneous parallelism

This study focuses on the low-power Tegra X1 System-on-Chip (SoC) from the Jetson Nano Developer Kit, which is increasingly used in various environments and tasks. As these SoCs grow in prevalence, it becomes crucial to analyse their computational performance, energy consumption, and reliability, especially for safety-critical applications. A key factor examined in this paper is the SoC’s neutron radiation tolerance. This is explored by subjecting a parallel version of matrix multiplication, which has been offloaded to various hardware components via OpenMP, to neutron irradiation. Through this approach, this researcher establishes a correlation between the SoC’s reliability and its computational and energy performance. The analysis enables the identification of an optimal workload distribution strategy, considering factors such as execution time, energy efficiency, and system reliability. Experimental results reveal that, while the GPU executes matrix multiplication tasks more rapidly and efficiently than the CPU, using both components only marginally reduces execution time. Interestingly, GPU usage significantly increases the SoC’s critical section, leading to an escalated error rate for both Detected Unrecoverable Errors (DUE) and Silent Data Corruptions (SDC), with the CPU showing a higher average number of affected elements per SDC.

A block α-circulant based preconditioned MINRES method for wave equations

In this work, we propose an absolute value block α-circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Motivated by the absolute value block circulant preconditioner proposed in [E. McDonald, J. Pestana, and A. Wathen. SIAM J. Sci. Comput., 40(2):A1012-A1033, 2018], we propose an absolute value version of the block α-circulant preconditioner. Since the original block α-circulant preconditioner is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Our proposed preconditioner is the first Hermitian positive definite variant of the block α-circulant preconditioner for the concerned wave equations, which fills the gap between block α-circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for a properly chosen α the MINRES solver with the proposed preconditioner achieves a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance the concerned problem. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved.

16-channel photonic solver for optimization problems on a silicon chip

A programmable photonic solver for quadratic unconstrained binary optimization (QUBO) problems is demonstrated with a hybrid optoelectronic scheme, which consists of a photonic chip and an electronic driving board. The photonic chip is employed to perform the optical vector-matrix multiplication (OVMM) to calculate the cost function of the QUBO problem, while the electronic processor runs the heuristic algorithm to search for the optimal solution. Due to the parallel and low-latency propagation of lightwave, the calculation of the cost function can be accelerated. The photonic chip is fabricated on the silicon on insulator (SOI) substrate and integrates 16 high-speed electro-optic modulators, 88 thermo-optic phase shifters, and 16 balanced photodetectors. The computing speed of the photonic chip is 1.66 TFLOP/s. As a proof of principle, two randomly generated 16-dimensional QUBO problems are solved with high successful probabilities. These results present the potential of fast solving optimization problems with integrated photonic systems.

Combined high order compact schemes for non-self-adjoint nonlinear Schrödinger equations

Some combined high order compact (CHOC) schemes are proposed for non-self-adjoint and nonlinear Schrödinger equation (NSANLSE). There are first order and second order spatial derivatives ux‾, uxx in the NSANLSE. If one uses classical high order compact schemes to approximate uxx and ux‾ separately, it will widen the bandwidth in practical coding due to matrix multiplication. This will partly counteract the advantages of high order compact. To overcome the deficiency, one solves the spatial derivatives simultaneously by combining them. In other words, it solves uxjn and uxxjn simultaneously in terms of uj. The idea is applied to discretize NSANLSE in space. Two efficient numerical schemes are proposed for NSANLSE. The stability and convergence of the new schemes are analyzed theoretically. Numerical experiments are reported to verify the new schemes.

Algorithms and Data Structures for Numerical Computations with Automatic Precision Estimation

We introduce data structures and algorithms to count numerical inaccuracies arising from usage of floating numbers described in IEEE 754. Here we describe how to estimate precision for some collection of functions most commonly used for array manipulations and training of neural networks. For highly optimized functions like matrix multiplication, we provide a fast estimation of precision and some hint how the estimation can be strengthened.

Global-power-split-tree architecture for large-scale coherent optical matrix multiplication

Global-power-split-tree architecture for large-scale coherent optical matrix multiplication

Optical control of multiple resistance levels in graphene for memristic applications

Neuromorphic computing has emphasized the need for memristors with non-volatile, multiple conductance levels. This paper demonstrates the potential of hexagonal boron nitride (hBN)/graphene heterostructures to act as memristors with multiple resistance states that can be optically tuned using visible light. The number of resistance levels in graphene can be controlled by modulating doping levels, achieved by varying the electric field strength or adjusting the duration of optical illumination. Our measurements show that this photodoping of graphene results from the optical excitation of charge carriers from the nitrogen-vacancy levels of hBN to its conduction band, with these carriers then being transferred to graphene by the gate-induced electric field. We develop a qualitative model to describe our observations. Additionally, utilizing our device architecture, we propose a memristive crossbar array for vector-matrix multiplications.

Enhancing the Efficiency of Distributed Storage Systems through XOR-Optimized Cauchy Reed-Solomon Codes

In today‘s society, we have entered the era of big data and the Internet of Things. The rise of services such as artificial intelligence, cloud storage, and the metaverse has led to increased demands for data transmission and storage. People now require not only greater capacity but also higher efficiency, more powerful hardware, and advanced algorithms. storageThe Reed-Solomon (RS) code, as a highly effective error correction code, is officially applied in fields such as data storage and data transmission. The Cauchy Reed-Solomon (CRS) Code, which is based on the RS code, utilizes a Cauchy matrix instead of the original Vandermonde matrix and replaces the original matrix multiplication with XOR operations. This paper primarily explores how the XOR-optimized Cauchy Reed-Solomon Code improves the efficiency of distributed storage systems, It was discovered that using lookup tables to store frequently used data and operations can further enhance the efficiency of storage systems and its future applications.

Extension of accurate numerical algorithms for matrix multiplication based on error-free transformation

Extension of accurate numerical algorithms for matrix multiplication based on error-free transformation

Variational quantum algorithm for the Poisson equation based on the banded Toeplitz systems

Abstract For solving the Poisson equation it is usually possible to discretize it into solving the corresponding linear system $Ax=b$. Variational quantum algorithms (VQAs) for \textcolor{red}{the discreted} Poisson equation have been studied before. We give a VQA based on the banded Toeplitz systems for solving the Poisson equation with respect to the structural features of matrix $A$. In detail, we decompose the matrix $A$ and $A^2$ into a linear combination of the corresponding banded Toeplitz matrix and sparse matrices with only a few non-zero elements. For the one-dimensional Poisson equation with different boundary conditions and the $d$-dimensional Poisson equation with Dirichlet boundary conditions, \textcolor{red}{the number of decomposition terms is less than the work in [Phys. Rev. A108, 032418 (2023)]}. Based on the decomposition of the matrix, we design quantum circuits that \textcolor{red}{evaluate efficiently} the cost function. \textcolor{red}{Additionally, numerical simulation verifies the feasibility of the proposed algorithm. In the end,} the VQAs for linear systems of equations and matrix–vector multiplications with $K$-banded Teoplitz matrix $T_n^K$ \textcolor{red}{are} given, where $T_n^K\in R^{n\times n}$ and $K\in O(ploy\log n)$.

Barriers to Developing a Citizen-Centric Smart City in China: From the Perspective of Citizens’ Sense of Gain

ABSTRACT Identifying barriers aligned with the development objectives and their interrelationships in Citizen-Centric Smart City (CCSC) is pivotal for devising tailored developmental strategies. Hence, this article aims to conduct a comprehensive exploration of these elements, with a specific focus on citizens’ sense of gain (CSG), a crucial component of citizen-centricity. The study conceptualizes the CCSC development objectives and identifies 20 barriers that align with them by employing a literature review and the Delphi method. Besides, the barriers are analyzed by adopting the Decision Making Trial and Evaluation Laboratory (DEMATEL)-Interpretative Structural Modeling (ISM)-Cross-impact matrix multiplication applied to classification (MICMAC) method. The study reveals that CCSC in China should focus on making CSG more comprehensive and sustainable in five dimensions decomposed from the objectives, “Lack of departments to interface with citizens’ feedback” is identified as the barrier with the highest causality and centrality, and “Lack of guiding framework and policy system for CCSC” is a fundamental barrier to the China’s CCSC development. By identifying the critical barriers and suggesting practical solutions, this study can help policymakers and practitioners worldwide to prioritize their efforts towards creating more sustainable and CCSCs.

A resource-aware workload scheduling method for unbalanced GEMMs on GPUs

Abstract GEMM (General Matrix Multiplication) serves as a fundamental operator for deep learning computations. Especially in attention-based deep learning models, such as Bert, GPT, and SAM, the sizes of matrices involved in GEMMs exhibit an unbalanced distribution due to the variable input, resulting in the low utilization of hardware resources. To address the issue, this paper proposes inserting a novel GEMM processing layer into the deep learning inference stack and using an adaptive load balancing method to partition and schedule GEMM computation tasks. The method is implemented with hardware runtime resource information, such as the occupancy of computing units, etc. Experiment results show the remarkable performance of our method in unbalanced input GEMM scenarios, achieving an average performance improvement of 2.3x. The method also performs well in attention-based models (GPT-2 and SAM), achieving an average inference speed improvement of 1.1x. These findings highlight the effectiveness of resource-aware algorithm optimization, especially for computation task scheduling.

Dimensions of Job Demands Among New-Generation Employees Based on Online Reviews by Employees

Based on employees’ online reviews, this article analyzes the dimensions of job demand for new-generation employees, using a combination model of Latent Dirichlet Allocation, decision-making trials and evaluation laboratory, interpretative structural modeling, and cross-impact matrix multiplication (LDA—DEMATEL—ISM—MICMAC). The results show that job demand is composed of 10 dimensions, and there is significant interdependence between the dimensions. Changing one dimension will quickly affect the other dimensions. The dimension with the greatest influence degree is leadership and team atmosphere (s1), while the dimension with the highest affected degree and centrality degree is welfare and promotion (s3). Leadership and team atmosphere (s1), company culture and industry (s4), overall environment and platform (s5), and platform and technology (s7) were identified as the key factors. They play a causal role in job demand and have a significant impact on other dimensions. Dimensions such as working relationship and intensity (s2), welfare and promotion (s3), opportunity and resources (s6), business and industry development (s8), corporate prospects and personal development (s9), and work stress and position (s10) are affected by other dimensions and require special attention. The underlying need is present for the dimension of leadership and team atmosphere (s1), which has a direct or indirect impact on other dimensions in different ways.