Algorithm For Matrices Research Articles

Application of Paired Correlation Algorithms for the Distance Matrices Between DNA Chains

Authors:This paper is a continuation of some previous works by the authors. We consider various algorithms for calculating distances between genomes of similar species (we use primarily mitochondrial DNA, mtDNA) and various distance matrices between the same genomes obtained on the basis of these algorithms. We can say, just to simplify the situation a little, that all our publications on the subject of DNA analysis are associated with various applications of metrics set on such matrices. The paper also has a second subject, i.e., the study of the obtained distance matrices using special statistical characteristics. We consider two matrices obtained for the mtDNA of 32 species of monkeys; the species were selected so that they all belong to different genera. For them, we have obtained 2 matrices of distances between genomes corresponding to the Jaro –Winkler’s and Needleman –Wunsch’s algorithms. Next, we considered all the triangles obtained in these matrices, and for each of them we used a specially calculated badness. It is actually a measure of the deviation of the resulting triangle from some acuteangled isosceles one. For two sequences of such badness, we have considered variants of paired correlation. At the same time, in addition to the two standard pair correlation algorithms (Spearman’s and Kendall’s ones), we also considered a new algorithm proposed by us. The reason for considering this new algorithm is as follows. In the usual way of calculating the correlation, we consider only the set of pairwise values of two random variables, without taking into account the pairs themselves. Vice versa, in both of the mentioned pair correlation algorithms, despite their slight difference, we consider only the order of the elements in these pairs, not paying attention to the values themselves; we specifically note that this also applies to Spearman’s criterion, which is usually written about as being more suitable for measurements made on an ordinal scale. In our proposed algorithm, we tried to take into account both the value of both random variables and their order in pairs. The results obtained are of interest. Thus, the “pole” variants (i.e., the usual correlation formula and standard pair correlation algorithms) show some (though very small) correlation between two sequences of 4960 pairs of triangles: from 0.1 to 0.4, depending on the specific algorithm, on whether preliminary normalization was carried out, etc. And the “intermediate” variant (taking into account both the order of pairs and the values of random variables) showed a complete lack of connection: the absolute value of the correlation coefficient did not exceed 0.006. Even more interesting is another result obtained in the work, which can be called a small connection between two well-known algorithms for determining the distances between genomes, namely, algorithms of Jaro –Winkler and Needleman –Wunsch.

Hybrid-triggered H∞ control for Markov jump systems with quantizations and hybrid attacks

The hybrid-triggered quantized H∞ control problem is investigated for discrete-time Markov jump systems (MJSs) under hybrid cyber attacks. A novel hybrid-triggered mechanism obeying Bernoulli distribution between the time-triggered mechanism and the adaptive event-triggered mechanism is introduced. The triggered condition considers the average value between current measured output and latest triggered output to avoid the unnecessary triggered data released. Meanwhile, a quantizer is adopted to optimize the data transmission rate and an observer-based controller is designed to resist the impact of deception attacks and aperiodic DoS attacks on the system. Utilizing Lyapunov stability theory and iterative methods, sufficient conditions are obtained to ensure that the closed-loop MJSs are asymptotically mean-square stable with H∞ performance. Then, an algorithm for gain matrices and triggered matrices is given. Finally, the effectiveness and availability of the proposed method are verified by a numerical example and a DC motor model.

Counting vanishing matrix-vector products

Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces. Let v∈Qd be a rational vector, (T1,T2…,Tm) a list of d×d rational matrices, S∈Qh×d a rational matrix not necessarily square and k a parameter. The goal is to compute the number of ways one can choose k matrices Ti1,Ti2,…,Tik from the list such that STik⋯Ti1v=0∈Qh.In this paper, we show that this problem is #W[2]-hard for parameter k. As a consequence, computing the k-th homotopy group of a d-dimensional 1-connected topological space for d>3 is #W[2]-hard for parameter k. We also discuss a decision version of the problem and its several modifications for which we show W[1]/W[2]-hardness. This is in contrast to the parameterized k-sum problem, which is only W[1]-hard (Abboud-Lewi-Williams, ESA'14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized by the matrix dimensions and the order of the field.

Spectral Map for Slow Collective Variables, Markovian Dynamics, and Transition State Ensembles.

Understanding the behavior of complex molecular systems is a fundamental problem in physical chemistry. To describe the long-time dynamics of such systems, which is responsible for their most informative characteristics, we can identify a few slow collective variables (CVs) while treating the remaining fast variables as thermal noise. This enables us to simplify the dynamics and treat it as diffusion in a free-energy landscape spanned by slow CVs, effectively rendering the dynamics Markovian. Our recent statistical learning technique, spectral map [Rydzewski, J. J. Phys. Chem. Lett. 2023, 14(22), 5216-5220], explores this strategy to learn slow CVs by maximizing a spectral gap of a transition matrix. In this work, we introduce several advancements into our framework, using a high-dimensional reversible folding process of a protein as an example. We implement an algorithm for coarse-graining Markov transition matrices to partition the reduced space of slow CVs kinetically and use it to define a transition state ensemble. We show that slow CVs learned by spectral map closely approach the Markovian limit for an overdamped diffusion. We demonstrate that coordinate-dependent diffusion coefficients only slightly affect the constructed free-energy landscapes. Finally, we present how spectral maps can be used to quantify the importance of features and compare slow CVs with structural descriptors commonly used in protein folding. Overall, we demonstrate that a single slow CV learned by spectral map can be used as a physical reaction coordinate to capture essential characteristics of protein folding.

Algorithm 1043: Faster Randomized SVD with Dynamic Shifts

Aiming to provide a faster and convenient truncated SVD algorithm for large sparse matrices from real applications (i.e., for computing a few of the largest singular values and the corresponding singular vectors), a dynamically shifted power iteration technique is applied to improve the accuracy of the randomized SVD method. This results in a d yn a mic sh ifts-based randomized SVD (dashSVD) algorithm, which also collaborates with the skills for handling sparse matrices. An accuracy-control mechanism is included in the dashSVD algorithm to approximately monitor the per vector error bound of computed singular vectors with negligible overhead. Experiments on real-world data validate that the dashSVD algorithm largely improves the accuracy of a randomized SVD algorithm or attains the same accuracy with fewer passes over the matrix, and provides an efficient accuracy-control mechanism to the randomized SVD computation, while demonstrating the advantages on runtime and parallel efficiency. A bound of the approximation error of the randomized SVD with the shifted power iteration is also proved.

Improving compressed matrix multiplication using control variate method

The seminal work by Pagh [1] proposed a matrix multiplication algorithm for real-valued squared matrices called Compressed Matrix Multiplication (CMM) having a sparse matrix output product. The algorithm is based on a popular sketching technique called Count-Sketch [2] and Fast Fourier Transform (FFT). For input square matrices A and B of order n and the product matrix AB with Frobenius norm ||AB||F, the algorithm offers an unbiased estimate for each entry, i.e., (AB)i,j of the product matrix AB with a variance bounded by ||AB||F2/b, where b is the compressed bucket size. Thus, the variance will eventually become high for a small bucket size. In this work, we address the high variance problem of CMM with the help of a simple and practical technique based on classical variance reduction methods in statistics. Our techniques rely on the Control Variate (CV) method. We suggest rigorous theoretical analysis for variance reduction and complement it via supporting empirical evidence.

QR decomposition of dual matrices and its application

Dual number matrix decompositions have played an important role in fields such as kinematics and computer graphics in recent years. In this paper, we present a QR decomposition algorithm for dual number matrices. When dealing with large-scale problems, we present the thin QR decomposition of dual number matrices, along with its algorithm with column pivoting. In numerical experiments, we discuss the suitability of different QR algorithms when confronted with various large-scale dual matrices, providing their respective domains of applicability. Finally, we employ the QR decomposition of dual matrices to compute the DMPGI, attaining results of higher precision.

Computation of quasiseparable representations of Green matrices

The well-known Asplund theorem states that the inverse of a (possibly one-sided) band matrix A is a Green matrix. In accordance with quasiseparable theory, such a matrix admits a quasiseparable representation in its rank-structured part. Based on this idea, we derive algorithms that compute a quasiseparable representation of A−1 with linear complexity.Many inversion algorithms for band matrices exist in the literature. However, algorithms based on a computation of the rank structure performed theoretically via the Asplund theorem appear for the first time in this paper. Numerical experiments confirm complexity estimates and offer insight into stability properties.

More Numerically Accurate Algorithm for Stiff Matrix Exponential

In this paper, we propose a novel, highly accurate numerical algorithm for matrix exponentials (MEs). The algorithm is based on approximating Putzer’s algorithm by analytically solving the ordinary differential equation (ODE)-based coefficients and approximating them. We show that the algorithm outperforms other ME algorithms for stiff matrices for several matrix sizes while keeping the computation and memory consumption asymptotically similar to these algorithms. In addition, we propose a numerical-error- and complexity-optimized decision tree model for efficient ME computation based on machine learning and genetic programming methods. We show that, while there is not one ME algorithm that outperforms the others, one can find a good algorithm for any given matrix according to its properties.

Efficient solution of the non-unitary time-dependent Schrodinger equation on a quantum computer with complex absorbing potential

We explore the possibility of adding complex absorbing potential at the boundaries when solving the one-dimensional real-time Schrödinger evolution on a grid using a quantum computer with a fully quantum algorithm described on a n qubit register. Due to the complex potential, the evolution mixes real- and imaginary-time propagation and the wave function can potentially be continuously absorbed during the time propagation. We use the dilation quantum algorithm to treat the imaginary-time evolution in parallel to the real-time propagation. This method has the advantage of using only one reservoir qubit at a time, that is measured with a certain success probability to implement the desired imaginary-time evolution. We propose a specific prescription for the dilation method where the success probability is directly linked to the physical norm of the continuously absorbed state evolving on the mesh. We expect that the proposed prescription will have the advantage of keeping a high probability of success in most physical situations. Applications of the method are made on one-dimensional wave functions evolving on a mesh. Results obtained on a quantum computer identify with those obtained on a classical computer. We finally give a detailed discussion on the complexity of implementing the dilation matrix. Due to the local nature of the potential, for n qubits, the dilation matrix only requires 2n CNOT and 2n unitary rotation for each time step, whereas it would require of the order of 4n+1 C-NOT gates to implement it using the best-known algorithm for general unitary matrices.

Machine Learning in Quasi-Newton Methods

In this article, we consider the correction of metric matrices in quasi-Newton methods (QNM) from the perspective of machine learning theory. Based on training information for estimating the matrix of the second derivatives of a function, we formulate a quality functional and minimize it by using gradient machine learning algorithms. We demonstrate that this approach leads us to the well-known ways of updating metric matrices used in QNM. The learning algorithm for finding metric matrices performs minimization along a system of directions, the orthogonality of which determines the convergence rate of the learning process. The degree of learning vectors’ orthogonality can be increased both by choosing a QNM and by using additional orthogonalization methods. It has been shown theoretically that the orthogonality degree of learning vectors in the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method is higher than in the Davidon–Fletcher–Powell (DFP) method, which determines the advantage of the BFGS method. In our paper, we discuss some orthogonalization techniques. One of them is to include iterations with orthogonalization or an exact one-dimensional descent. As a result, it is theoretically possible to detect the cumulative effect of reducing the optimization space on quadratic functions. Another way to increase the orthogonality degree of learning vectors at the initial stages of the QNM is a special choice of initial metric matrices. Our computational experiments on problems with a high degree of conditionality have confirmed the stated theoretical assumptions.

Predicting optimal sparse general matrix-matrix multiplication algorithm on GPUs

Sparse General Matrix-Matrix Multiplication (SpGEMM) has played an important role in a number of applications. So far, many efficient algorithms have been proposed to improve the performance of SpGEMM on GPUs. However, the performance of each algorithm for matrices of different structures varies a lot. There is no algorithm that can achieve the optimal performance of SpGEMM computation on all matrices. In this article, we design a machine learning based approach for predicting the optimal SpGEMM algorithm on input matrices. By extracting features from input matrices, we utilize LightGBM and XGBoost to train different lightweight models. The models are capable of predicting the best performing algorithm with low inference overhead and high accuracy for the given input matrices. We also investigate the impact of tree depth on model accuracy and inference overhead. Our evaluation shows that an increase in tree depth leads to a corresponding increase in prediction accuracy, reaching a maximum of approximately 85%, while resulting in increased inference overhead of approximately 10 µs. Compared with the state-of-the-art algorithms on three GPU platforms, our method achieves better overall performance.

Spectral clustering of Markov chain transition matrices with complex eigenvalues

The Robust Perron Cluster Analysis (PCCA+) has become a popular spectral clustering algorithm for coarse-graining transition matrices of nearly decomposable Markov chains with transition states. Originally developed for reversible Markov chains, the algorithm only worked for transition matrices with real eigenvalues. In this paper, we therefore extend the theoretical framework of PCCA+ to Markov chains with a complex eigen-decomposition. We show that by replacing a complex conjugate pair of eigenvectors by their real and imaginary components, a real representation of the same subspace is obtained, which is suitable for the cluster analysis. We show that our approach leads to the same results as the generalized PCCA+ (GPCCA), which replaces the complex eigen-decomposition by a conceptually more difficult real Schur decomposition. We apply the method on non-reversible Markov chains, including circular chains, and demonstrate its efficiency compared to GPCCA. The experiments are performed in the Matlab programming language and codes are provided.

On exact computational complexity of triangular factorization algorithms for general banded matrices

On exact computational complexity of triangular factorization algorithms for general banded matrices

Long-time adaptive coherent detection of small targets in sea clutter by fast inversion algorithm of block tridiagonal speckle covariance matrices

Low-velocity small target detection in maritime surveillance radars is always a challenging task. Low signal-to-clutter ratio requires long-time coherent integration to obtain enough gain of target returns. However, long-time coherent integration encounters insufficient secondary data due to the spatial inhomogeneity of sea clutter. In this paper, considering the decorrelation time of speckle component of sea clutter short up to a dozen of milliseconds, the spherical invariant random vector (SIRV) model with block tridiagonal speckle covariance matrix and the inverse Gamma distributed texture is proposed to model sea clutter sequences in several tenths of a second. In this model, a long-time adaptive generalized likelihood ratio test with linear threshold detector (GLRT-LTD) is constructed. Owing to the block tridiagonal structure of speckle covariance matrices, the adaptive detection requires much less reference cells for speckle covariance matrix estimation and much lower computational cost for its inversion. The proposed detector is verified by an X-band high-resolution island-based radar data with an unmanned aerial vehicle (UAV) as test target. The experimental result shows that it obtains competitive detection performance in comparison with the state-of-the-art detectors.

Rank‐structured approximation of some Cauchy matrices with sublinear complexity

AbstractIn this article, we consider the rank‐structured approximation of one important type of Cauchy matrix. This approximation plays a key role in some structured matrix methods such as stable and efficient direct solvers and other algorithms for Toeplitz matrices and certain kernel matrices. Previous rank‐structured approximations (specifically hierarchically semiseparable, or HSS, approximations) for such a matrix of size cost at least complexity. Here, we show how to construct an HSS approximation with sublinear (specifically, ) complexity. The main ideas include extensive computation reuse and an analytical far‐field compression strategy. Low‐rank compression at each hierarchical level is restricted to just a single off‐diagonal block row, and a resulting basis matrix is then reused for other off‐diagonal block rows as well as off‐diagonal block columns. The relationships among the off‐diagonal blocks are rigorously analyzed. The far‐field compression uses an analytical proxy point method where we optimize the choice of some parameters so as to obtain accurate low‐rank approximations. Both the basis reuse ideas and the resulting analytical hierarchical compression scheme can be generalized to some other kernel matrices and are useful for accelerating relevant rank‐structured approximations (though not subsequent operations like matrix‐vector multiplications).

On singular value decomposition for split quaternion matrices and applications in split quaternionic mechanics

In the theoretical studies and numerical computations of split quaternionic quantum mechanics, the singular value decomposition theorem of split quaternion matrices plays an essential role. In general the problem of singular value decomposition over split quaternion algebra has hitherto remained tangential for split quaternion matrices. In this paper, by means of a real representation matrix of a split quaternion matrix, the singular value decomposition of split quaternion matrices is studied, the singular value decomposition theorem and the corresponding algorithm for split quaternion matrices are given. Finally, three applications for solving the split quaternion least squares problem and the wave separation problem are given based on the singular value decomposition over split quaternion algebra.

Finding community structure using the ordered random graph model.

Visualization of the adjacency matrix enables us to capture macroscopic features of a network when the matrix elements are aligned properly. Community structure, a network consisting of several densely connected components, is a particularly important feature and the structure can be identified through the adjacency matrix when it is close to a block-diagonal form. However, classical ordering algorithms for matrices fail to align matrix elements such that the community structure is visible. In this study, we propose an ordering algorithm based on the maximum-likelihood estimate of the ordered random graph model. We show that the proposed method allows us to more clearly identify community structures than the existing ordering algorithms.

Hierarchical structure optimal setting approach for production indexes of the rolling heating furnace temperature field

In order to improve the optimal setting temperature problem to achieve the global optimum of product performance, costs and benefits. In this article, a hierarchical structure optimal setting approach of production indexes for the rolling heating furnace temperature field (RHFTF) is proposed. It is composed of three layers with different functions to obtain the temperature control setting model of the RHFTF. In the first layer, the bi-feature Gaussian mixture model clustering (BFGMMC) algorithm of loading plan is proposed to optimize the setting of a limited number of slabs. In the second layer, the type-2 fuzzy rule interpolation (T2FRI) setting method is developed to obtain the optimal setting curve. Meanwhile, an improved KH (Kóczy-Hirota) α-cut distance (IKHCD) algorithm is proposed to get the miss information between any two adjacent interpolation points. In the third layer, knowledge feedforward compensation of rule matrices (KFCRM) algorithm is presented to improve the anti-interference ability of the setting model. The results of the study can demonstrate that the proposed method improves the accuracy of the model and optimizes the control strategy. Furthermore, the experimental results show that the proposed method meets the process technical requirements.

Anti-diagonalization theory and algorithm of matrices—from skew-symmetric matrices to arbitrary matrices

In this paper, a novel algorithm for anti-diagonalization of skew symmetric matrices via using orthogonal similarity transformations has been introduced. The theory and algorithm about the anti-triangular factorization of skew-symmetric matrices are proved. In the case of skew-symmetric matrices, we prove that the anti-diagonal form is always obtained, resulting in developing a new factorization scheme. Moreover, a theoretical algorithm is given based on the theory of double eigenvector system, which provides all the information for the factorization of arbitrary matrices. Finally, the proposed algorithm is verified effective and efficient through the numerical experiments of anti-diagonalization of matrices over a general number field.