Smaller Submatrices Research Articles

A High-Speed Floating Point Matrix Multiplier Implemented in Reconfigurable Architecture

Matrix multiplication is a fundamental operation in computational applications across various domains. This paper introduces a novel reconfigurable co-processor that enhances the efficiency of matrix multiplication by concurrently executing addition and multiplication operations upon matrix elements of different sizes. The proposed design aims to reduce computation time and improve efficiency for matrix multiplication equations. Experimental evaluations were conducted on matrices of different sizes to demonstrate the effectiveness of the processor. The results reveal substantial improvements in both time and efficiency when compared to traditional approaches. The reconfigurable transformation processor harnesses parallel processing capabilities, enabling the simultaneous execution of addition and multiplication operations by partitioning input matrices into smaller submatrices and performing parallel computations, thus the processor achieves faster results. Additionally, the design incorporates configurable arithmetic units that dynamically adapt to matrix characteristics, further optimizing performance. The experimental evaluations provide evidence of reduction in computation time and improvement in efficiency. present significant benefits over traditional sequential methods. This makes this co-processor ideally fit for domains that require intensive linear algebra computations such as computer vision, machine learning, and signal processing.

Multivariate curve resolution-soft independent modelling of class analogy (MCR-SIMCA)

BackgroundVarious classification, class modeling, and clustering techniques operate within abstract spaces, utilizing Principal Components (e.g., Linear Discriminant Analysis (LDA), Principal Component Analysis (PCA)) or latent variable spaces (e.g., Partial Least Squares Discriminant Analysis (PLS-DA)). It's important to note that PCA, despite being a mathematical tool, defines its Principal Components under certain mathematical constraints, it has a wide range of applications in the analysis of real-world systems. In this research, we assess the viability of employing the Multivariate Curve Resolution (MCR) subspace within class modeling techniques, as an alternative to the PC subspace. (92). ResultsThis study evaluates the use of the MCR subspace in class modeling methods, specifically in tandem with soft independent modeling of class analogy (SIMCA), to investigate the advantages of employing the meaningful physico-chemical subspace of MCR over the mathematical subspace of PCA. In the MCR-SIMCA strategy, the model is constructed by applying MCR to training samples from a target class. The MCR model effectively partitions the data into two smaller sub-matrices: the contribution matrix and the corresponding response matrix. In the next step, the contribution matrix resulting from the decomposition of the training set develops a distance plot (DP). First, the theory of the MCR-SIMCA model is discussed in detail. Next, two real experimental datasets were analyzed, and their performance was compared with the DD-SIMCA model. In most cases, the results were as good as or even more satisfactory than those obtained with the DD-SIMCA model. (146). SignificanceThe suggested class modeling method presents a promising avenue for the analysis of real-world natural systems. The study's results emphasize the practical utility of the MCR approach, underscoring the significance of the MCR subspace advantages over the PCA subspace. (39).

Multiplication of medium-density matrices using TensorFlow on multicore CPUs

Matrix multiplication is an essential part of many applications, such as linear algebra, image processing and machine learning. One platform used in such applications is TensorFlow, which is a machine learning library whose structure is based on dataflow programming paradigm. In this work, a method for multiplication of medium-density matrices on multicore CPUs using TensorFlow platform is proposed. This method, called tbt_matmul, utilizes TensorFlow built-in methods tf.matmul and tf.sparse_matmul. By partitioning each input matrix into four smaller sub-matrices, called tiles, and applying an appropriate multiplication method to each pair depending on their density, the proposed method outperforms the built-in methods for matrices of medium density and matrices of significantly uneven distribution of non-zeros.

An Improved Matrix Approximation for Recommender Systems Based on Context-Information and Transfer Learning

As we all known, transfer learning is an effective way to alleviate the sparsity problem in recommender systems by transferring the shared knowledge cross multiple related domains. However, additional related domain is not always available, and auxiliary data may be noisy and this leads to negative transfer. In this paper, we suppose that different parts of one domain also have the shared knowledge and put forward a novel in-domain collaborative filtering framework, which utilizes contextual information to divide an original user-item interaction matrix into some smaller sub-matrices and regards the selected sub-matrices as “multiple domains” to establish transfer learning. The proposed framework no longer needs additional domain information and has a better adaptive ability. Also, considering more actual situation that users may have multiple personalities and items may have diverse attributes, we resort to Rating-Matrix Generative Model (RMGM) to generate the shared cluster-level rating pattern. Experiments on dataset Douban with three different categories demonstrate that the proposed framework can improve the prediction accuracy as well as the top-N recommendation performance.

Big Feature Data Analytics: Split and Combine Linear Discriminant Analysis (SC-LDA) for Integration Towards Decision Making Analytics

This paper introduces a novel big feature data analytics scheme for integration toward data analytics with decision making. In this scheme, a split and combine approach for a linear discriminant analysis (LDA) algorithm termed SC-LDA is proposed. The SC-LDA replaces the full eigenvector decomposition of LDA with much cheaper eigenvector decompositions on smaller sub-matrices, and then recombines the intermediate results to obtain the exact reconstruction as for the original algorithm. The splitting or decomposition can be further applied recursively to obtain a multi-stage SC-LDA algorithm. The smaller sub-matrices can then be computed in parallel to reduce the time complexity for big data applications. The approach is discussed for an LDA algorithm variation (LDA/QR), which is suitable for the analytics of Big Feature data sets. The projected data vectors into the LDA subspace can then be integrated toward the decision-making process involving classification. Experiments are conducted on real-world data sets to confirm that our approach allows the LDA problem to be divided into the size-reduced sub-problems and can be solved in parallel while giving an exact reconstruction as for the original LDA/QR.

Wavelet multilevel augmentation method for linear boundary value problems

This work presents a new approach to numerically solve the general linear two-point boundary value problems with Dirichlet boundary conditions. Multilevel bases from the anti-derivatives of the Daubechies wavelets are constructed in conjunction with the augmentation method. The accuracy of numerical solutions can be improved by increasing the number of basis levels, but the computational cost also increases drastically. The multilevel augmentation method can be applied to reduce the computational time by splitting the coefficient matrix into smaller submatrices. Then the unknown coefficients in the higher level can be solved separately. The convergent rate of this method is $2^{s}$ , where $1 \leq s \leq p+1$ , when the anti-derivatives of the Daubechies wavelets order p are applied. Some numerical examples are also presented to confirm our theoretical results.

Neurodegenerative disease diagnosis using incomplete multi-modality data via matrix shrinkage and completion

In this work, we are interested in predicting the diagnostic statuses of potentially neurodegenerated patients using feature values derived from multi-modality neuroimaging data and biological data, which might be incomplete. Collecting the feature values into a matrix, with each row containing a feature vector of a sample, we propose a framework to predict the corresponding associated multiple target outputs (e.g., diagnosis label and clinical scores) from this feature matrix by performing matrix shrinkage following matrix completion. Specifically, we first combine the feature and target output matrices into a large matrix and then partition this large incomplete matrix into smaller submatrices, each consisting of samples with complete feature values (corresponding to a certain combination of modalities) and target outputs. Treating each target output as the outcome of a prediction task, we apply a 2-step multi-task learning algorithm to select the most discriminative features and samples in each submatrix. Features and samples that are not selected in any of the submatrices are discarded, resulting in a shrunk version of the original large matrix. The missing feature values and unknown target outputs of the shrunk matrix is then completed simultaneously. Experimental results using the ADNI dataset indicate that our proposed framework achieves higher classification accuracy at a greater speed when compared with conventional imputation-based classification methods and also yields competitive performance when compared with the state-of-the-art methods.

평균이동 분할계산기법을 사용한 역 컬러라이제이션 기반의 컬러영상압축

In this letter, we propose a method for colorization based coding, which divides the colorization matrix into smaller sub-matrices using the meanshift segmentation. Using the proposed method the computation speed becomes more than 30 times faster. Furthermore, the smearing artifact, which appears in conventional colorization based compression method, is greatly reduced.

An Integer Programming Approach for Analyzing the Measurement Redundancy in Structured Linear Systems

A linear system whose model matrix is of size n×p is considered structured if some p row vectors in the model matrix are linearly dependent. Computing the degree of redundancy for structured linear systems is proven NP-hard. Previous computation strategy is divide-and-conquer, materialized in a bound-and-decompose algorithm, which, when the required conditions are satisfied, can compute the degree of redundancy on a set of much smaller submatrices instead of directly on the original model matrix. The limitation of this algorithm is that the current decomposition conditions are still restrictive and not always satisfied for many applications. We present a mixed integer programming (MIP) formulation of the redundancy degree problem and solve it using an existing MIP solver. Our numerical studies indicate that our approach outperforms the existing methods for many applications, especially when the decomposition conditions are not satisfied. The main contribution of the paper is that we tackle this challenging problem from a different angle and test a promising new approach. The resulting approach points to a path that can potentially solve the problem in its entirety.

An Algebraic Substructuring Method for Large-Scale Eigenvalue Calculation

This paper is concerned with solving large-scale eigenvalue problems by algebraic substructuring. Algebraic substructuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. Through an algebraic analysis, we identify critical conditions under which a simple version of algebraic substructuring works well. This particular version of substructuring is identical to the component mode synthesis (CMS) method (see [R. R. Craig and M. C. C. Bampton, {\em Coupling of substructures for dynamic analysis}, AIAA J., 6 (1968), pp. 1313--1319] and [W. C. Hurty, {\em Vibrations of structure systems by component-mode synthesis}, J. Engrg. Mech., 86 (1960), pp. 51--69]) when the matrix reordering is based on a geometric partitioning of the computational domain. We observe an interesting connection between the accuracy of an approximate eigenpair obtained through substructuring and the distribution of the components of eigenvectors of a canonical matrix pencil congruent to the original problem. A priori error bounds for the smallest eigenpair approximation are developed. This development leads to a simple heuristic for choosing spectral components (modes) from each substructure. The effectiveness of such a heuristic is demonstrated with numerical examples. We show that algebraic substructuring can be effectively used to solve a generalized eigenvalue problem arising from the finite element analysis of an accelerator structure.

Partitioning a matrix with non-guillotine cuts to minimize the maximum cost

We consider the problem of partitioning a matrix of m rows and n columns of non-negative integers into M smaller submatrices. With each submatrix is associated a cost equal to the sum of its elements. The objective is to minimize the cost of the submatrix of maximum cost. We present a (0–1)-integer programming formulation of the problem and three different lower bounds. A heuristic procedure for finding a valid upper bound to the optimal solution cost is also described. Problem reduction tests derived from both the original problem and the lower bounds are given. Lower bounds and reduction tests are used in a tree search algorithm in order to find the optimal solution to the problem. Computational results on a number of randomly generated test problems are presented.

A network theoretic approach to inversion of large matrices

A network theoretic approach for inverting large order matrices has been given. Firstly the case of matrices partitioned in nine smaller submatrices has been considered. The case of sub-division into four submatrices is derived as a special case of the first. An experimental inversion procedure for simple matrices is also included.