Partitioning Of Matrices Research Articles

A method to shorten the time of high-dimensional matrix operations and convolution operations

Serverless computing, or Function-as-a-Service (FaaS), is a cloud computing execution model where the cloud provider dynamically manages the allocation and provisioning of servers which is a new way to solve the problem that the need of client.By serverless computing clients can easily solve their problems anytime,anywhere.However, the complex convolution operations and high-dimensional matrix operations can still consume a lot of resources such as computation time and memory occupancy in serverless computing. In this paper, firstly, the author selects two frequently used models in serverless computing, and then divide them into different numbers of partitions respectively to study the impact of partitioning methods on the computation time and memory occupancy. The author show that different partitioning methods can make an obviously impact on the computation time and memory occupancy,whitch shows that the proper partitioning of high-dimensional matrices and tensors can make a significantly reduction of calculation time and memory occupancy.

OPERATIONS ON BLOCK PREDICATE MATRICES

This article discusses the possibility of performing basic logical operations on matrices that are divided into rectangular blocks in an arbitrary manner. The results obtained are illustrated using the example of predicate logical matrices defined over a field of finite predicates of arbitrary arity. The possibility of extending the results obtained to Boolean matrices is also justified (they are considered to be defined over a field of finite predicates of zero arity). At the same time, using the apparatus of multivalued logic, starting from Boolean logical matrices, the results obtained can also be extended to the study of logical objects that are not Boolean. Block partitioning of matrices allows you to divide information specified in matrix form into parts, process these parts separately, and then combine them into a single whole, adhering to certain rules and restrictions described in this article. These methods can find their application in graph theory, in the theory of algorithms, programming and in other areas of theoretical and practical activity that are in one way or another related to mathematical logic.

Flexible Distributed Matrix Multiplication

The distributed matrix multiplication problem with an unknown number of stragglers is considered, where the goal is to efficiently and flexibly obtain the product of two massive matrices by distributing the computation across <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> servers. There are up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N - R$ </tex-math></inline-formula> stragglers but the exact number is not known a priori. Motivated by reducing the computation load of each server, a flexible solution is proposed to fully utilize the computation capability of available servers. The computing task for each server is separated into several subtasks, constructed based on Entangled Polynomial codes by Yu et al. The final results can be obtained from either a larger number of servers with a smaller amount of computation completed per server or a smaller number of servers with a larger amount of computation completed per server. The required finite field size of the proposed solution is less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2N$ </tex-math></inline-formula> . Moreover, the optimal design parameters such as the partitioning of the input matrices are discussed. Our constructions can also be generalized to other settings such as batch distributed matrix multiplication and secure distributed matrix multiplication.

Cayley--Hamilton theorem for $\alpha$-upper level partition of fuzzy square matrices

Cayley--Hamilton theorem for $\alpha$-upper level partition of fuzzy square matrices

On equitable partition of matrices and its applications

ABSTRACT A partition of a square matrix A is said to be equitable if all the blocks of the partitioned matrix have constant row sums and each of the diagonal blocks are of square order. A quotient matrix Q of a square matrix A corresponding to an equitable partition is a matrix whose entries are the constant row sums of the corresponding blocks of A. A quotient matrix is a useful tool to find some eigenvalues of the matrix A. In this paper we determine some matrices whose eigenvalues are those eigenvalues of A which are not the eigenvalues of a quotient matrix of A. Using this result we find eigenvalue localization theorems for matrices having an equitable partition. In particular, we find eigenvalue localization theorems for stochastic matrices and give a suitable example to compare with the existing results.

On the Capacity and Straggler-Robustness of Distributed Secure Matrix Multiplication

Computationally efficient matrix multiplication is a fundamental requirement in various fields, including and particularly in data analytics. To do so, the computation task of large-scale matrix multiplication is typically outsourced to multiple servers. However, due to data misusage at the servers, security is typical of concern. In this paper, we first study the two-sided secure matrix multiplication problem, where a user is interested in the matrix product AB of two finite field private matrices A and B from an information-theoretic perspective. In this problem, the user exploits the computational resources of N servers to compute the matrix product but simultaneously tries to conceal the private matrices from the servers. Our goal is twofold: (i) to maximize the downlink communication rate, and (ii) to minimize the effective number of server observations needed to determine AB, while preserving security, where we allow for up to l ≤ N servers to collude. To this end, we propose two schemes - an aligned secret sharing scheme (A3S) and a secure cross subspace alignment (SCSA) scheme. For A3S, we optimize the partitioning of matrices A and B in order to either optimize objective (i) or (ii) as a function of the system parameters (e.g., N and l). A proposed inductive approach gives us analytical, close-to-optimal solutions for both (i) and (ii). The SCSA, on the other hand, is shown to be (rate) capacity-optimal for the general J-sided distributed secure matrix multiplication problem Π j=1 J M j . We show this by developing a recursive information-theoretic upper bound (converse) on the downlink rate for the J-sided secure matrix multiplication problem. With respect to (i), both A3S and SCSA, significantly outperform the state-of-the-art in terms of (a) communication rate, (b) maximum tolerable number of colluding servers, and (c) computational complexity. Overall SCSA (A3S) is the preferred choice when the focus is on the downlink (uplink).

Atomic-pair theorem for universal matrix representatives of molecules and atomic clusters in non-relativistic Born-Oppenheimer approximation

The mathematical underpinnings of a universal atomic-pair based approach to ab initio calculations of the electronic wave functions and energies of molecules and other atomic clusters are reported in non-relativistic Born-Oppenheimer approximation. The general approach utilizes the closure of an orthonormal (Eisenschitz-London) outer product of atomic spectral eigenstates to support totally antisymmetric aggregate eigenstates in the absence of explicit term-by-term atomic-product basis antisymmetry. Meaningful definitions of atomic-pair interaction operators and of their corresponding universal matrix representatives on the atomic product basis are obtained in this way for calculations on physical and chemical aggregates. Symmetric-group-based permutation-symmetry methods for partitioning of aggregate Hamlitonian matrices, performed to date by unitary transformations to remove unphysical “non-Pauli” states after aggregate Hamiltonian matrix construction, are supplemented here with a new two-step procedure. The new approach, which combines explicit antisymmetrization of atomic pair matrices with unitary transformation of aggregate Hamiltonian matrices, is described here in complete mathematical detail. Conditions are established from the new development under which enforcement of pairwise-atomic antisymmetry alone can be sufficient for calculations of electronic energies, providing an efficient approach for practical applications, as well as long-absent theoretical support for related semi-empirical and ad hoc methods widely employed in computational molecular physics and quantum chemistry. An exact expression is obtained for total electronic energies from the solution of the Schrödinger equation in the universal form of sums over pairwise-atomic interaction energies uniquely defined within a molecule in the absence of subjective additional conditions. Wigner rotation matrices are employed to orient pairs of interacting atoms within an aggregate in the implementation of the approach, requiring calculations of the universal pair interaction matrices as functions of radial separations only. Such information can be obtained once and for all from molecular diatomic calculations and retained for subsequent use, avoiding repeated evaluations of the electronic integrals commonly required in conventional variational determinations of accurate potential energy surfaces. Diagonalizations of large Hamiltonian matrices are avoided in favor of optical potential, Krylov-Lanczos, and previously described Stieltjes-Techebycheff methods in determination of aggregate Schrödinger eigenspectra.

Partitioning Models for Scaling Parallel Sparse Matrix-Matrix Multiplication

We investigate outer-product--parallel, inner-product--parallel, and row-by-row-product--parallel formulations of sparse matrix-matrix multiplication (SpGEMM) on distributed memory architectures. For each of these three formulations, we propose a hypergraph model and a bipartite graph model for distributing SpGEMM computations based on one-dimensional (1D) partitioning of input matrices. We also propose a communication hypergraph model for each formulation for distributing communication operations. The computational graph and hypergraph models adopted in the first phase aim at minimizing the total message volume and balancing the computational loads of processors, whereas the communication hypergraph models adopted in the second phase aim at minimizing the total message count and balancing the message volume loads of processors. That is, the computational partitioning models reduce the bandwidth cost and the communication hypergraph models reduce the latency cost. Our extensive parallel experiments on up to 2048 processors for a wide range of realistic SpGEMM instances show that although the outer-product--parallel formulation scales better, the row-by-row-product--parallel formulation is more viable due to its significantly lower partitioning overhead and competitive scalability. For computational partitioning models, our experimental findings indicate that the proposed bipartite graph models are attractive alternatives to their hypergraph counterparts because of their lower partitioning overhead. Finally, we show that by reducing the latency cost besides the bandwidth cost through using the communication hypergraph models, the parallel SpGEMM time can be further improved up to 32%.

Variation partitioning of benthic diatom community matrices: Effects of multiple variables on benthic diatom communities in an Austral temperate river system

This study explores diatom community dynamics in a highly modified semi-arid temperate region river system characterised by inconsistent river flow. Various water and sediment environmental variables were assessed using a multi-faceted analysis approach to determine the spatio-temporal drivers of benthic diatom communities in the river system. Overall, the diatom community was generally dominated by pollution tolerant species, reflecting the anthropogenic intensity and activities on the river system. Diatom community composition was found to be largely determined by water column chemistry variables particularly nutrient concentrations in comparison to sediment chemistry and physical variables. Strong seasonal diatom species composition was also observed and this was driven by strong seasonal variations in nutrient loads and metal concentrations, a result of the variable water flow across the two seasons. However, the greater temporal variation in communities was observed in the smaller systems with the mainstream river system being more homogenous over time. In addition, diatom community composition and environmental variables were found to be different and more pronounced between streams and mainstream sites, than between canals and streams. The study highlights the complex interaction between water column, sediment and physical variables in determining the diatom species composition in small river systems. It also highlights the importance of river flow inconsistency as an indirect variable that alters primary drivers such as nutrient concentrations in the water column and heavy metal levels in the sediment.

Variation partitioning of diatom species data matrices: Understanding the influence of multiple factors on benthic diatom communities in tropical streams

Elucidating the confounding influence of multiple environmental factors on benthic diatom communities is important in developing water quality predictive models for better guidance of stream management efforts. The objective of this study was to explore the relative impact of metal pollution and hydromorphological alterations in, addition to nutrient enrichment and organic pollution, on diatom taxonomic composition with the view to improve stream diatom-based water quality inference models. Samples were collected twice at 20 sampling stations in the tropical Manyame Catchment, Zimbabwe. Diatom, macroinvertebrate communities and environmental factors were sampled and analysed. The variations in diatom community composition explained by different categories of environmental factors were analysed using canonical correspondence analysis using variance partitioning (partial CCA). The following variations were explained by the different predictor matrices: nutrient levels and organic pollution - 10.4%, metal pollution - 8.3% and hydromorphological factors - 7.9%. Thus, factors other than nutrient levels and organic pollution explain additional significant variation in these diatom communities. Development of diatom-based stream water quality inference models that incorporate metal pollution and hydromorphological alterations, where these are key issues, is thus deemed necessary.

Phylogenetic relationships in Asarum: Effect of data partitioning and a revised classification.

Generic boundaries and infrageneric relationships among the charismatic temperate magnoliid Asarum sensu lato (Aristolochiaceae) have long been uncertain. Previous molecular phylogenetic analyses used either plastid or nuclear loci alone and varied greatly in their taxonomic implications for the genus. We analyzed additional molecular markers from the nuclear and plastid genomes, reevaluated the possibility of a derived loss of autonomous self-pollination, and investigated the topological effects of matrix-partitioning-scheme choice. We sequenced seven plastid regions and the nuclear ITS1-ITS2 region of 58 individuals representing all previously recognized Asarum s.l. segregate genera and the monotypic genus Saruma. Matrices were partitioned using common a priori partitioning schemes and PartitionFinder. Topologies that were recovered using a priori partitioning of matrices differed from those recovered using a PartitionFinder-selected scheme, and by analysis method. We recovered six monophyletic groups that we circumscribed into three subgenera and six sections. Putative fungal mimic characters served as synapomorphies only for subgenus Heterotropa. Subgenus Geotaenium, a new subgenus, was recovered as sister to the remainder of Asarum by ML analyses of highly partitioned datasets. Section Longistylis, also newly named, is sister to section Hexastylis. Our analyses do not unambiguously support a single origin for all fungal-mimicry characters. Topologies recovered through the analysis of PartitionFinder-optimized matrices can differ drastically from those inferred from a priori partitioned matrices, and by analytical method. We recommend that investigators evaluate the topological effects of matrix partitioning using multiple methods of phylogenetic reconstruction.

Statistical modeling of Southern Ocean marine diatom proxy and winter sea ice data: Model comparison and developments

We compare the performance of the modern analog technique (MAT), the Imbrie and Kipp transfer function (IKTF), the generalized additive model (GAM) and weighted averaging partial least squares (WA PLS) on a southern hemisphere diatom relative abundance and winter sea ice concentration training data set. All relevant model assumptions are tested with a random 10-fold cross-validation, whilst a hold out cross-validation tested the explanatory power of each model on spatially independent validation data. We used auto correlograms on model residuals, variance partitioning, and principal coordinates analysis of neighbor matrices (PCNM) to investigate the importance of the spatial structure of our training database. A set of hierarchical logistic regression models (or Huisman–Olff–Fresco models) are used to infer the response of each diatom species along the winter sea ice gradient. Our analyses suggest that IKTF is an inappropriate sea ice estimation approach as its underlying statistical assumptions do not hold and the fit of IKTF to our data under cross-validation was poor. We conclude that MAT may be biased by spatial autocorrelation, and together with IKTF fails to provide unbiased estimates of winter sea ice. We find GAM and WA PLS are more appropriate than IKTF and MAT for the estimation of paleo winter sea ice cover throughout the Southern Ocean. However, as WA PLS is based on a unimodal species response, which is rarely exhibited by diatoms along the winter sea ice gradient, we ultimately advocate the application of GAM. GAM only uses diatoms with a statistically significant association, and ecologically based link, with sea ice. GAM outperformed all other models under cross-validation in terms of performance statistics, the fit of GAM to the training dataset and diagnostic tests for model assumptions. We also demonstrate that GAM provides a more detailed and potentially more accurate (based on a comparison with New Zealand and southeast Australian paleo climatic records) paleo winter sea ice record for the southwestern Pacific Ocean in comparison with IKTF, MAT and WA PLS.

Simultaneous Input and Output Matrix Partitioning for Outer-Product--Parallel Sparse Matrix-Matrix Multiplication

For outer-product--parallel sparse matrix-matrix multiplication (SpGEMM) of the form $C\!=\!A\!\times\!B$, we propose three hypergraph models that achieve simultaneous partitioning of input and output matrices without any replication of input data. All three hypergraph models perform conformable one-dimensional (1D) columnwise and 1D rowwise partitioning of the input matrices $A$ and $B$, respectively. The first hypergraph model performs two-dimensional (2D) nonzero-based partitioning of the output matrix, whereas the second and third models perform 1D rowwise and 1D columnwise partitioning of the output matrix, respectively. This partitioning scheme induces a two-phase parallel SpGEMM algorithm, where communication-free local SpGEMM computations constitute the first phase and the multiple single-node-accumulation operations on the local SpGEMM results constitute the second phase. In these models, the two partitioning constraints defined on weights of vertices encode balancing computational loads of processors during the two separate phases of the parallel SpGEMM algorithm. The partitioning objective of minimizing the cutsize defined over the cut nets encodes minimizing the total volume of communication that will occur during the second phase of the parallel SpGEMM algorithm. An MPI-based parallel SpGEMM library is developed to verify the validity of our models in practice. Parallel runs of the library for a wide range of realistic SpGEMM instances on two large-scale parallel systems JUQUEEN (an IBM BlueGene/Q system) and SuperMUC (an Intel-based cluster) show that the proposed hypergraph models attain high speedup values.

Determinants of spider species richness in coastal dunes along a gradient of mediterraneity

Abstract. 1. The Iberian Peninsula is one of the most relevant areas in terms of species richness, rarity and endemism in the Mediterranean Basin. Using spiders as a model, we studied the relative importance of environmental and non‐environmental spatial variation along a gradient of mediterraneity on the western coast of the Iberian Peninsula.2. We performed a spatially explicit analysis to evaluate the contribution of pure environmental and pure spatial effects and their shared influence on spider species richness by variation partitioning and principal coordinates of neighbour matrices (PCNM). We dissect the spatial variation of species richness into additive scale‐specific models through PCNM analysis and estimate the relative importance of environmental variables for each model.3. Variation partitioning revealed that 72.8% of species richness variation could be explained by the environmental (climate and dune stability) and spatial variables. Most of this variation (51.3%) corresponds to the environmental spatially structured component.4. The influence of environmental variables was scale‐dependent. At the broadest scales, the latitudinal‐climatic gradient was the dominant factor that influenced spider species richness variation. At the finest scale, stability of the dune system was the most important component.5. In conclusion, spider species richness patterns were determined mostly by spatially structured environmental variation. More importantly, this study shows that different environmental factors act at multiple scales. Hence, our results reinforce the importance of the incorporation of both local and broad‐scale factors when examining species richness patterns.

Determinants of beta diversity of spiders in coastal dunes along a gradient of mediterraneity

AbstractAim The Mediterranean Basin is recognized for its high levels of species richness, rarity and endemicity. Our main aim was to evaluate the relative effects of environmental and spatial variables and their scale‐specific importance on beta diversity patterns along a gradient of mediterraneity, using spiders as a model group.Location This study was carried out in 18 coastal dune sites along the Portuguese Atlantic coast. This area encompasses 445 km and comprises two distinct biogeographic regions, Eurosiberian (northern coast) and Mediterranean (centre and south).Methods A forward selection procedure was carried out to select environmental and spatial variables responsible for determining beta diversity patterns. Variation partitioning and principal coordinates of neighbour matrices (PCNM) were used to estimate the contribution of pure environmental and pure spatial effects and their shared influence on beta diversity patterns and to estimate the relative importance of environmental structured variation and pure spatial variation at multiple spatial scales.Results Climate, ground vegetation dune cover and area were selected by a forward selection procedure. The same procedure identified three PCNM variables, all corresponding to large and medium spatial scales. Variation partitioning revealed that 46.1% of the variation of beta diversity patterns was explained by a combination of environmental and PCNM variables. Most of this variation (42.5%) corresponded to spatial variation (environmental spatially structured and pure spatial). Climate and vegetation structure influences were predominant at the PCNM1 and PCNM3 scales, while area was more important at the intermediate PCNM2 scale.Main conclusions Our study revealed that beta diversity of spiders was primarily controlled by a broad‐scale gradient of mediterraneity. The relative importance of environmental variables on the spider assemblage composition varied with spatial scale. This study highlights the need of considering the scale‐specific influence of niche and neutral processes on beta diversity patterns.

On Two-Dimensional Sparse Matrix Partitioning: Models, Methods, and a Recipe

We consider two-dimensional partitioning of general sparse matrices for parallel sparse matrix-vector multiply operation. We present three hypergraph-partitioning-based methods, each having unique advantages. The first one treats the nonzeros of the matrix individually and hence produces fine-grain partitions. The other two produce coarser partitions, where one of them imposes a limit on the number of messages sent and received by a single processor, and the other trades that limit for a lower communication volume. We also present a thorough experimental evaluation of the proposed two-dimensional partitioning methods together with the hypergraph-based one-dimensional partitioning methods, using an extensive set of public domain matrices. Furthermore, for the users of these partitioning methods, we present a partitioning recipe that chooses one of the partitioning methods according to some matrix characteristics.

Complex-Valued State Matrices for Simple Representation of Large Autonomous Microgrids Supplied by $PQ$ and $Vf$ Generation

This paper focuses on representing the state space model of a microgrid in which power regulated (PQ ) and voltage/frequency regulated (Vf) generation units share a distribution system. The generation units considered in this paper are inverter interfaced. This introduces some interesting modeling problems which are treated in the paper, such as the decoupled cascaded control schemes or the non-negligible grid dynamics. A modeling approach is proposed based on four defined complex vectors. These vectors allow for complex-valued system matrices to be formed in a quite automated way. Moreover, a convenient partition of the system matrices is proposed, which in turn allows fast and easy modifications. Additionally, a multivariable methodology is proposed to simultaneously find the control system gains in an optimal sense. A 69-bus radial system, supplied by 20 generation units, is used to demonstrate how the proposal is of easy implementation to conduct small-signal stability analyses.

First-order exchange energy of intermolecular interactions from coupled cluster density matrices and their cumulants

A new method for the calculation of the first-order intermolecular exchange energy is proposed. It is based on the partition of two-particle density matrices of monomers into the antisymmetrized product of one-particle density matrices and the remaining cumulant part. This partition is used to modify the formula for the first-order exchange energy developed by Moszynski et al. [J. Chem. Phys. 100, 5080 (1994)]. The new expression has been applied for the case of monomer density matrices derived from the expectation value expression for the coupled cluster singles and doubles wave function. In this way an accurate method of calculation of the first-order exchange energy for many-electron systems has been obtained, where both monomers are described on the coupled cluster level. Numerical results are presented for several benchmark van der Waals systems to illustrate the performance of the new approach.

Second-order exchange-induction energy of intermolecular interactions from coupled cluster density matrices and their cumulants

A new formulation of the second-order exchange-induction energy of symmetry-adapted perturbation theory is presented. In the proposed formalism the exchange-induction energy is expressed through one- and two-particle reduced density matrices of monomers, which are of zeroth and first order with respect to the effective electrostatic potential of another monomer. The resulting expression is further modified by using the partition of two-particle density matrices into the antisymmetrized product of one-particle density matrices and the remaining cumulant part. The proposed formalism has been applied to the case of closed-shell monomers and for density matrices obtained from the expectation-value expression with coupled cluster singles and doubles wave functions. The performance of the new approach has been demonstrated on several benchmark van der Waals systems, including dimers of argon, water, and ethyne.

An algorithm to check the nonnegativity of singular systems

In the literature, an important class of generalized inverse matrices corresponds to the group inverse, that is, matrices of index 1. Recently, the nonnegativity of a singular system has been applied to different fields. In this paper, an algorithm to check the nonnegativity of a singular linear control system of index 1 is presented. To this purpose, the nonnegativity of this kind of systems is characterized using a block partition of the original matrices. In this way, we can work with matrices having smaller sizes and keeping the original information. Finally, numerical examples illustrating the obtained results are shown.