M-dimensional Vector Research Articles

From positional representation of numbers to positional representation of vectors

To represent real m-dimensional vectors, a positional vector system given by a non-singular matrix M ∈ ℤm×m and a digit set Ɗ ⊂ ℤm is used. If m = 1, the system coincides with the well known numeration system used to represent real numbers. We study some properties of the vector systems which are transformable from the case m = 1 to higher dimensions. We focus on an algorithm for parallel addition and on systems allowing an eventually periodic representation of vectors with rational coordinates.

ПРЕДСТАВЛЕНИЕ ПАРНОГО ПОТЕНЦИАЛА ВЗАИМОДЕЙСТВИЯ В ВИДЕ МНОГОМЕРНЫХ РАЦИОНАЛЬНЫХ РЯДОВ В ПЕРЕМЕННЫХ ЯКОБИ ДЛЯ ЗАДАЧ МНОГИХ ТЕЛ

Scalar power functions of the form x1 + + xN -v Î are in some cases found in physical problems and applications, especially in many-body problems with paired interactions. There are known decompositions for two vectors in three-dimensional space. In this paper, we consider analogous decompositions with any number of N arbitrary M-dimensional vectors in Euclidean space as a product of a multidimensional rational series with respect to spatial variables and hyperspheric functions on the unit sphere SM-1. Such an advantage of expansion arises in three-body problems when solving the Faddeev equation, where it is known that the main problem is the approximate choice of approximation of interaction potentials, in which the t-matrix scattering elements acquired a separable form.

On positional representation of integer vectors

We show that any m×m matrix M with integer entries and det⁡M=Δ≠0 can be equipped by a finite digit set D⊂Zm such that any integer m-dimensional vector belongs to the setFinD(M)={∑k∈IMkdk:∅≠I finite subset of Z and dk∈D for each k∈I}⊂⋃k∈N1ΔkZm. We also characterize the matrices M for which the sets FinD(M) and ⋃k∈N1ΔkZm coincide.

Joint Poisson distribution of prime factors in sets

AbstarctGiven disjoint subsets T1, …, Tm of “not too large” primes up to x, we establish that for a random integer n drawn from [1, x], the m-dimensional vector enumerating the number of prime factors of n from T1, …, Tm converges to a vector of m independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when T1, …, Tm are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors.

Graph of graphs analysis for multiplexed data with application to imaging mass cytometry.

Imaging Mass Cytometry (IMC) combines laser ablation and mass spectrometry to quantitate metal-conjugated primary antibodies incubated in intact tumor tissue slides. This strategy allows spatially-resolved multiplexing of dozens of simultaneous protein targets with 1μm resolution. Each slide is a spatial assay consisting of high-dimensional multivariate observations (m-dimensional feature space) collected at different spatial positions and capturing data from a single biological sample or even representative spots from multiple samples when using tissue microarrays. Often, each of these spatial assays could be characterized by several regions of interest (ROIs). To extract meaningful information from the multi-dimensional observations recorded at different ROIs across different assays, we propose to analyze such datasets using a two-step graph-based approach. We first construct for each ROI a graph representing the interactions between the m covariates and compute an m dimensional vector characterizing the steady state distribution among features. We then use all these m-dimensional vectors to construct a graph between the ROIs from all assays. This second graph is subjected to a nonlinear dimension reduction analysis, retrieving the intrinsic geometric representation of the ROIs. Such a representation provides the foundation for efficient and accurate organization of the different ROIs that correlates with their phenotypes. Theoretically, we show that when the ROIs have a particular bi-modal distribution, the new representation gives rise to a better distinction between the two modalities compared to the maximum a posteriori (MAP) estimator. We applied our method to predict the sensitivity to PD-1 axis blockers treatment of lung cancer subjects based on IMC data, achieving 97.3% average accuracy on two IMC datasets. This serves as empirical evidence that the graph of graphs approach enables us to integrate multiple ROIs and the intra-relationships between the features at each ROI, giving rise to an informative representation that is strongly associated with the phenotypic state of the entire image.

Performance improvement for machine learning‐based cooperative spectrum sensing by feature vector selection

To explore the potential of machine learning-based cooperative spectrum sensing (CSS) in training time, classification speed and classification performance, this study mainly focuses on studying the problem of the feature vectors selecting for machine learning-based CSS. First, a new machine learning-based CSS framework is presented, in which, energy vector forming module, feature vector conversion module, training module, classification module and training sample database are included. Second, a new two-dimensional distance vector is developed, and it is converted by an m-dimensional energy vector according to the distance measurement between vectors. Furthermore, six combination modes are obtained by combining three feature vectors (energy, probability and distance vectors) with two supervised machine learning methods, which are support vector machine (SVM) and weighted K-nearest-neighbour, respectively. From the proposed experimental simulations, the authors can find that the distance vector is obviously superior to the probability vector in computation time. Moreover, the probability vector and distance vector are superior to the energy vector in training time except for the case of poor signal and fewer users, and obviously superior to the energy vector in classification speed. At last, the probability vector and distance vector with SVM classifier show the best classification performance in six combination modes.

Quadratic and symmetric bilinear forms over finite fields and their association schemes

Let 𝒬(m,q) and 𝒮(m,q) be the sets of quadratic forms and symmetric bilinear forms on an m-dimensional vector space over 𝔽 q , respectively. The orbits of 𝒬(m,q) and 𝒮(m,q) under a natural group action induce two translation association schemes, which are known to be dual to each other. We give explicit expressions for the eigenvalues of these association schemes in terms of linear combinations of generalised Krawtchouk polynomials, generalising earlier results for odd q to the more difficult case when q is even. We then study d-codes in these schemes, namely subsets X of 𝒬(m,q) or 𝒮(m,q) with the property that, for all distinct A,B∈X, the rank of A-B is at least d. We prove tight bounds on the size of d-codes and show that, when these bounds hold with equality, the inner distributions of the subsets are often uniquely determined by their parameters. We also discuss connections to classical error-correcting codes and show how the Hamming distance distribution of large classes of codes over 𝔽 q can be determined from the results of this paper.

On polynomial-time solvable linear Diophantine problems

We obtain a polynomial-time algorithm that, given input (A, b), where A=(B|N) is an integer mxn matrix, m<n, with nonsingular mxm submatrix B and b is an m-dimensional integer vector, finds a nonnegative integer solution to the system Ax=b or determines that no such solution exists, provided that b is located sufficiently "deep" in the cone generated by the columns of B. This result improves on some of the previously known conditions that guarantee polynomial-time solvability of linear Diophantine problems.

Critical Behavior and Universality Classes for an Algorithmic Phase Transition in Sparse Reconstruction

Recovery of an N-dimensional, K-sparse solution $${\mathbf {x}}$$ from an M-dimensional vector of measurements $${\mathbf {y}}$$ for multivariate linear regression can be accomplished by minimizing a suitably penalized least-mean-square cost $$||{\mathbf {y}}-{\mathbf {H}} {\mathbf {x}}||_2^2+\lambda V({\mathbf {x}})$$ . Here $${\mathbf {H}}$$ is a known matrix and $$V({\mathbf {x}})$$ is an algorithm-dependent sparsity-inducing penalty. For ‘random’ $${\mathbf {H}}$$ , in the limit $$\lambda \rightarrow 0$$ and $$M,N,K\rightarrow \infty $$ , keeping $$\rho =K/N$$ and $$\alpha =M/N$$ fixed, exact recovery is possible for $$\alpha $$ past a critical value $$\alpha _c = \alpha (\rho )$$ . Assuming $${\mathbf {x}}$$ has iid entries, the critical curve exhibits some universality, in that its shape does not depend on the distribution of $${\mathbf {x}}$$ . However, the algorithmic phase transition occurring at $$\alpha =\alpha _c$$ and associated universality classes remain ill-understood from a statistical physics perspective, i.e. in terms of scaling exponents near the critical curve. In this article, we analyze the mean-field equations for two algorithms, Basis Pursuit ( $$V({\mathbf {x}})=||{\mathbf {x}}||_{1} $$ ) and Elastic Net ( $$V({\mathbf {x}})= ||{\mathbf {x}}||_{1} + \tfrac{g}{2} ||{\mathbf {x}}||_{2}^2$$ ) and show that they belong to different universality classes in the sense of scaling exponents, with mean squared error (MSE) of the recovered vector scaling as $$\lambda ^\frac{4}{3}$$ and $$\lambda $$ respectively, for small $$\lambda $$ on the critical line. In the presence of additive noise, we find that, when $$\alpha >\alpha _c$$ , MSE is minimized at a non-zero value for $$\lambda $$ , whereas at $$\alpha =\alpha _c$$ , MSE always increases with $$\lambda $$ .

The lattice of subspaces of a vector space over a finite field

For finite m and q we study the lattice mathbf {L}(mathbf {V})=(L(mathbf {V}),+,cap ,{vec {0}},V) of subspaces of an m-dimensional vector space mathbf {V} over a field mathbf {K} of cardinality q. We present formulas for the number of d-dimensional subspaces of mathbf {V}, for the number of complements of a subspace and for the number of e-dimensional subspaces including a given d-dimensional subspace. It was shown in Eckmann and Zabey (Helv Phys Acta 42:420–424, 1969) that mathbf {L}(mathbf {V}) possesses an orthocomplementation only in case m=2 and {{,mathrm{char},}}mathbf {K}ne 2. Hence, only in this case mathbf {L}(mathbf {V}) can be considered as an orthomodular lattice. On the contrary, we show that a complementation ' on mathbf {L}(mathbf {V}) can be chosen in such a way that (L(mathbf {V}),+,cap ,{}') is both weakly orthomodular and dually weakly orthomodular. Moreover, we show that (L(mathbf {V}),+,cap ,{}^perp ,{vec {0}},V) is paraorthomodular in the sense of Giuntini et al. (Stud Log 104:1145–1177, 2016).

Circuit Theory in Projective Space and Homogeneous Circuit Models

This paper presents a general framework for linear circuit analysis based on elementary aspects of projective geometry. We use a flexible approach in which no a priori assignment of an electrical nature to the circuit branches is necessary. Such an assignment is eventually done just by setting certain model parameters, in a way which avoids the need for a distinction between voltage and current sources and, in addition, makes it possible to get rid of voltageor current-control assumptions on the impedances. This paves the way for a completely general m-dimensional reduction of any circuit defined by m twoterminal, uncoupled linear elements, contrary to most classical methods which at one step or another impose certain restrictions on the allowed devices. The reduction has the form (316) u = ( A P) s. Here, A and B capture the graph topology, whereas P, Q, and s comprise homogeneous descriptions of all the circuit elements; the unknown u is an m-dimensional vector of (say) “seed” variables from which currents and voltages are obtained as i = Pu - Qs and v = Qu + Ps. Computational implementations are straightforward. These models allow for a general characterization of non-degenerate configurations in terms of the multihomogeneous Kirchhoff polynomial, and in this direction, we present some results of independent interest involving the matrix-tree theorem. Our approach can be easily combined with classical methods by using homogeneous descriptions only for certain branches, yielding partially homogeneous models. We also indicate how to accommodate controlled sources and coupled devices in the homogeneous framework. Several examples illustrate the results.

An Efficient System for Color Image Retrieval Representing Semantic Information to Enhance Performance by Optimizing Feature Extraction

Image retrieval is a well discussed field in digital image processing. Images can be sort out from a big collection and database of images on the basis of text, color and shape of objects in images. In CBIR systems, using combined features get many most similar and relevant images. In a typical CBIR system, the ocular content of the images from the large collection is extracted and displayed by a storehouse like m-dimensional feature vectors framed out from images. This vector of the images in the collection of images in database is named as a database of features. Most sought out systems represent images with color feature most related to user and shape feature to relate image to exact object from collection and test dataset. In this paper we present introduction, research work on color scheme with HSV colour space and its combination with shape using coiflet wavelet methods. States of art design of overall system is delineated with results evaluated for mean average precision and mean average recall for overall system.

Bases of software for computer simulation and multivariante prediction of economic even at uncertainty conditions on the base of n-component piecewise-linear economic-mathematical models in m-dimensional vector space

Bases of software for computer simulation and multivariante prediction of economic even at uncertainty conditions on the base of n-component piecewise-linear economic-mathematical models in m-dimensional vector space Azad Gabil oglu Aliyev Abstract For the last 15 years in periodic literature there has appeared a series of scientific publications that has laid the foundation of a new scientific direction on creation of piecewise-linear economic-mathematical models at uncertainty conditions in finite dimensional vector space. Representation of economic processes in finite-dimensional vector space, in particular in Euclidean space, at uncertainty conditions in the form of mathematical models in connected with complexity of complete account of such important issues as: spatial in homogeneity of occurring economic processes, incomplete macro, micro and social-political information; time changeability of multifactor economic indices, their duration and their change rate. The above-listed one in mathematical plan reduces the solution of the given problem to creation of very complicated economic-mathematical models of nonlinear type. In this connection, it was established in these works that all possible economic processes considered with regard to uncertainty factor in finite-dimensional vector space should be explicitly determined in spatial-time aspect. Owing only to the stated principle of spatial-time certainty of economic process at uncertainty conditions in finite dimensional vector space it is possible to reveal systematically the dynamics and structure of the occurring process. In addition, imposing a series of softened additional conditions on the occurring economic process, it is possible to classify it in finite-dimensional vector space and also to suggest a new science-based method of multivariant prediction of economic process and its control in finite-dimensional vector space at uncertainty conditions, in particular, with regard to unaccounted factors influence. Full Text: PDF DOI: 10.15640/arms.v7n1a3

Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments

We show that a Bayesian game where the type space of each agent is a bounded set of m-dimensional vectors with non-negative components and the utility of each agent depends linearly on its own type only is equivalent to a simultaneous competition in m basic games which is called a uniform multigame. The type space of each agent can be normalised to be given by the ( m − 1 ) -dimensional simplex. This class of m-dimensional Bayesian games, via their equivalence with uniform multigames, can model decision making in multi-environments in a variety of circumstances, including decision making in multi-markets and decision making when there are both material and social utilities for agents as in the Prisoner’s Dilemma and the Trust Game. We show that, if a uniform multigame in which the action set of each agent consists of one Nash equilibrium inducing action per basic game has a pure ex post Nash equilibrium on the boundary of its type profile space, then it has a pure ex post Nash equilibrium on the whole type profile space. We then develop an algorithm, linear in the number of types of the agents in such a multigame, which tests if a pure ex post Nash equilibrium on the vertices of the type profile space can be extended to a pure ex post Nash equilibrium on the boundary of its type profile space in which case we obtain a pure ex post Nash equilibrium for the multigame.

Accelerated orthogonal least-squares for large-scale sparse reconstruction

We study the problem of inferring a sparse vector from random linear combinations of its components. We propose the Accelerated Orthogonal Least-Squares (AOLS) algorithm that improves performance of the well-known Orthogonal Least-Squares (OLS) algorithm while requiring significantly lower computational costs. While OLS greedily selects columns of the coefficient matrix that correspond to non-zero components of the sparse vector, AOLS employs a novel computationally efficient procedure that speeds up the search by anticipating future selections via choosing L columns in each step, where L is an adjustable hyper-parameter. We analyze the performance of AOLS and establish lower bounds on the probability of exact recovery for both noiseless and noisy random linear measurements. In the noiseless scenario, it is shown that when the coefficients are samples from a Gaussian distribution, AOLS with high probability recovers a k-sparse m-dimensional sparse vector using O(klog⁡mk+L−1) measurements. Similar result is established for the bounded-noise scenario where an additional condition on the smallest nonzero element of the unknown vector is required. The asymptotic sampling complexity of AOLS is lower than the asymptotic sampling complexity of the existing sparse reconstruction algorithms. In simulations, AOLS is compared to state-of-the-art sparse recovery techniques and shown to provide better performance in terms of accuracy, running time, or both. Finally, we consider an application of AOLS to clustering high-dimensional data lying on the union of low-dimensional subspaces and demonstrate its superiority over existing methods.

Energy Efficient GNSS Signal Acquisition Using Singular Value Decomposition (SVD).

A significant challenge in global navigation satellite system (GNSS) signal processing is a requirement for a very high sampling rate. The recently-emerging compressed sensing (CS) theory makes processing GNSS signals at a low sampling rate possible if the signal has a sparse representation in a certain space. Based on CS and SVD theories, an algorithm for sampling GNSS signals at a rate much lower than the Nyquist rate and reconstructing the compressed signal is proposed in this research, which is validated after the output from that process still performs signal detection using the standard fast Fourier transform (FFT) parallel frequency space search acquisition. The sparse representation of the GNSS signal is the most important precondition for CS, by constructing a rectangular Toeplitz matrix (TZ) of the transmitted signal, calculating the left singular vectors using SVD from the TZ, to achieve sparse signal representation. Next, obtaining the M-dimensional observation vectors based on the left singular vectors of the SVD, which are equivalent to the sampler operator in standard compressive sensing theory, the signal can be sampled below the Nyquist rate, and can still be reconstructed via minimization with accuracy using convex optimization. As an added value, there is a GNSS signal acquisition enhancement effect by retaining the useful signal and filtering out noise by projecting the signal into the most significant proper orthogonal modes (PODs) which are the optimal distributions of signal power. The algorithm is validated with real recorded signals, and the results show that the proposed method is effective for sampling, reconstructing intermediate frequency (IF) GNSS signals in the time discrete domain.

Ranking by rating

Each item in a given collection is characterized by a set of possible performances. A (ranking) method is a function that assigns an ordering of the items to every performance profile. Ranking by Rating consists in evaluating each item's performance by using an exogenous rating function, and ranking items according to their performance ratings. Any such method is separable: the ordering of two items does not depend on the performances of the remaining items. We prove that every separable method must be of the ranking-by-rating type if (i) the set of possible performances is the same for all items and the method is anonymous, or (ii) the set of performances of each item is ordered and the method is monotonic. When performances are m-dimensional vectors, a separable, continuous, anonymous, monotonic, and invariant method must rank items according to a weighted geometric mean of their performances along the m dimensions.

Particle swarm optimization-based automatic parameter selection for deep neural networks and its applications in large-scale and high-dimensional data.

In this paper, we propose a new automatic hyperparameter selection approach for determining the optimal network configuration (network structure and hyperparameters) for deep neural networks using particle swarm optimization (PSO) in combination with a steepest gradient descent algorithm. In the proposed approach, network configurations were coded as a set of real-number m-dimensional vectors as the individuals of the PSO algorithm in the search procedure. During the search procedure, the PSO algorithm is employed to search for optimal network configurations via the particles moving in a finite search space, and the steepest gradient descent algorithm is used to train the DNN classifier with a few training epochs (to find a local optimal solution) during the population evaluation of PSO. After the optimization scheme, the steepest gradient descent algorithm is performed with more epochs and the final solutions (pbest and gbest) of the PSO algorithm to train a final ensemble model and individual DNN classifiers, respectively. The local search ability of the steepest gradient descent algorithm and the global search capabilities of the PSO algorithm are exploited to determine an optimal solution that is close to the global optimum. We constructed several experiments on hand-written characters and biological activity prediction datasets to show that the DNN classifiers trained by the network configurations expressed by the final solutions of the PSO algorithm, employed to construct an ensemble model and individual classifier, outperform the random approach in terms of the generalization performance. Therefore, the proposed approach can be regarded an alternative tool for automatic network structure and parameter selection for deep neural networks.

Precise large deviations for sums of random vectors with dependent components of consistently varying tails

Let {X i = (X 1,i ,...,X m,i )⊤, i ≥ 1} be a sequence of independent and identically distributed nonnegative m-dimensional random vectors. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Here, the components of X 1 are allowed to be generally dependent. Moreover, let N(·) be a nonnegative integer-valued process, independent of the sequence {X i , i ≥ 1}. Under several mild assumptions, precise large deviations for S n = Σ i=1 n X i and S N(t) = Σ i=1 N(t) X i are investigated. Meanwhile, some simulation examples are also given to illustrate the results.

Alternative predictors in chaotic time series

In the scheme of reconstruction, non-polynomial predictors improve the forecast from chaotic time series. The algebraic manipulation in the Maple environment is the basis for obtaining of accurate predictors. Beyond the different times of prediction, the optional arguments of the computational routines optimize the running and the analysis of global mappings.