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Fix a finite ordinal \(n\geq 3\) and let \(\alpha\) be an arbitrary ordinal. Let \(\mathsf{CA}_n\) denote the class of cylindric algebras of dimension \(n\) and \(\sf RA\) denote the class of relation algebras. Let \(\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})\) stand for the class of polyadic (equality) algebras of dimension \(\alpha\). We reprove that the class \(\mathsf{CRCA}_n\) of completely representable \(\mathsf{CA}_n\)s, and the class \(\sf CRRA\) of completely representable \(\mathsf{RA}\)s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \(\sf V\) between polyadic algebras of dimension \(n\) and diagonal free \(\mathsf{CA}_n\)s. We show that that the class of completely and strongly representable algebras in \(\sf V\) is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class \(\sf CRRA\) is not closed under \(\equiv_{\infty,\omega}\). In contrast, we show that given \(\alpha\geq \omega\), and an atomic \(\mathfrak{A}\in \mathsf{PEA}_{\alpha}\), then for any \(n

Fix a finite ordinal n ≥ 3 and let α be an arbitrary ordinal. Let CAn denote the class of cylindric algebras of dimension n and RA denote the class of relation algebras. Let PAα(PEAα) stand for the class of polyadic (equality) algebras of dimension α. We reprove that the class CRCAn of completely representable CAns, and the class CRRA of completely representable RAs are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety V between polyadic algebras of dimension n and diagonal free CAns. We show that that the class of completely and strongly representable algebras in V is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class CRRA is not closed under =1,ω. In contrast, we show that given α ≥ ω, and an atomic A 2 PEAα, then for any n < !, NrnA is a completely representable PEAn. We show that for any α ≥ ω, the class of completely representable algebras in certain reducts of PAαs, that happen to be varieties, is elementary. We show that for α ≥ ω, the the class of polyadic-cylindric algebras dimension α, introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension n and PAαs for α an infinite ordinal, proving negative results for the first and positive ones for the second. [ABSTRACT FROM AUTHOR]

We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions. Finally, we introduce the sandwich higher polyadic regularity with generalized idempotents. [ABSTRACT FROM AUTHOR]

Canonical polyadic decomposition (CPD) of multi-subject complex-valued fMRI data can be used to provide spatially and temporally shared components among groups with both magnitude and phase information. However, the CPD model is not well formulated due to the large subject variability in the spatial and temporal modalities, as well as the high noise level in complex-valued fMRI data. Considering that the shift-invariant CPD can model temporal variability across subjects, we propose to further impose a phase sparsity constraint on the shared spatial maps to denoise the complex-valued components and to model the inter-subject spatial variability as well. More precisely, subject-specific time delays are first estimated for the complex-valued shared time courses in the framework of real-valued shift-invariant CPD. Source phase sparsity is then imposed on the complex-valued shared spatial maps. A smoothed l0 norm is specifically used to reduce voxels with large phase values after phase de-ambiguity based on the small phase characteristic of BOLD-related voxels. The results from both the simulated and experimental fMRI data demonstrate improvements of the proposed method over three complex-valued algorithms, namely, tensor-based spatial ICA, shift-invariant CPD and CPD without spatiotemporal constraints. When comparing with a real-valued algorithm combining shift-invariant CPD and ICA, the proposed method detects 178.7% more contiguous task-related activations. [ABSTRACT FROM AUTHOR]

All known vibration-rotation absorption lines of 13CH12CH accessing levels up to 6750 cm-1 were gathered from the literature. They were fitted simultaneously to J-dependent Hamiltonian matrices exploiting the well known vibrational polyad or cluster block diagonalization, in terms of the pseudo-quantum-numbers Ns=v1+v2+v3 and Nr=5v1+3v2+5v3+v4+v5, and accounting also for l parity and e/f symmetry properties. The anharmonic interaction coupling terms known to occur from a pure vibrational fit in this acetylene isotopologue [Robert et al., J. Chem. Phys. 123, 174302 (2005)] were included in the model. A total of 12 703 transitions accessing 158 different (v1v2v3v4v5,l4l5) vibrational states was fitted with a dimensionless standard deviation of 0.99, leading to the determination of 216 vibration-rotation parameters. The experimental data included very weak vibration-rotation transitions accessing 18 previously unreported states, some of them forming Q branches with very irregular patterns. [ABSTRACT FROM AUTHOR]

All known vibration-rotation absorption lines of 13CH12CH accessing levels up to 6750 cm-1 were gathered from the literature. They were fitted simultaneously to J-dependent Hamiltonian matrices exploiting the well known vibrational polyad or cluster block diagonalization, in terms of the pseudo-quantum-numbers Ns=v1+v2+v3 and Nr=5v1+3v2+5v3+v4+v5, and accounting also for l parity and e/f symmetry properties. The anharmonic interaction coupling terms known to occur from a pure vibrational fit in this acetylene isotopologue [Robert et al., J. Chem. Phys. 123, 174302 (2005)] were included in the model. A total of 12 703 transitions accessing 158 different (v1v2v3v4v5,l4l5) vibrational states was fitted with a dimensionless standard deviation of 0.99, leading to the determination of 216 vibration-rotation parameters. The experimental data included very weak vibration-rotation transitions accessing 18 previously unreported states, some of them forming Q branches with very irregular patterns. [ABSTRACT FROM AUTHOR]

For β an ordinal, let PEAβ (SetPEAβ) denote the class of polyadic equality (set) algebras of dimension β. We show that for any infinite ordinal α, if A ∈ PEA α is atomic, then for any n < ω, the n-neat reduct of A , in symbols ℜ r n A → B , is a completely representable PEAn (regardless of the representability of A). That is to say, for all non-zero a ∈ ℜ r n A , there is a B a ∈ SetPEA n and a homomorphism f a : ℜ r n A → B such that fa(a) ≠ 0 and f a (∑ X) = ∪ x ∈ X f a (x) for any X ⊂ = A for which ∑ X exists. We give new proofs that various classes consisting solely of completely representable algebras of relations are not elementary; we further show that the class of completely representable relation algebras is not closed under ≡∞,ω. Various notions of representability (such as 'satisfying the Lyndon conditions', weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CAns satisfying the Lyndon conditions coincides with the class of atomic algebras in ElScNrnCAω, where El denotes 'elementary closure' and Sc is the operation of forming complete subalgebras. [ABSTRACT FROM AUTHOR]

We explore performance appraisal in project‐based organisations and provide novel insights into appraisal processes in this context. These include the central role of employees in orchestrating the appraisal process, the multiple actors that have input to appraisal including project managers, the distance between employees and their official line managers, and the weak coordinating role of human resource specialists in these systems. We draw attention to the drawbacks of current theorising on appraisal to predict and explain outcomes from appraisal systems that are not premised on stable line manager/employee dyads. Theorising based primarily on social exchange theories needs to be reconsidered in this context and new theories developed. We also question how human resource specialists can better support employees, and managers of all kinds, in their implementation roles in polyadic human resource management systems to ensure transparency, equity, and fairness of appraisal processes in a project‐based organisational context. [ABSTRACT FROM AUTHOR]

Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. Relations between operations arising from the structure definitions, however, lead to the restrictions which determine their possible arity shapes and lead us to formulate a partial arity freedom principle. Polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. Elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, operator norms, isometries and projections are introduced, as well as polyadic C*-algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators. It is shown that congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced (see Definition 7.17), and Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat's Last Theorem are formulated. For polyadic numbers neither of these statements holds. Polyadic versions of Frolov's theorem and the Tarry-Escott problem are presented. [ABSTRACT FROM AUTHOR]

Using a variational approach applied to generalized Rayleigh functionals, we extend the concepts of singular values and singular functions to trivariate functions defined on a rectangular parallelepiped. We also consider eigenvalues and eigenfunctions for trivariate functions whose domain is a cube. For a general finite-rank trivariate function, we describe an algorithm for computing the canonical polyadic (CP) decomposition, provided that the CP factors are linearly independent in two variables. All these notions are computed using Chebfun3; a part of Chebfun for numerical computing with 3D functions. Application in finding the best rank-1 approximation of trivariate functions is investigated. We also prove that if the function is analytic and two-way orthogonally decomposable (odeco), then the CP values decay geometrically, and optimal finite-rank approximants converge at the same rate. [ABSTRACT FROM AUTHOR]

Deep computation models achieve super performance for big data feature learning. However, training a deep computation model poses a significant challenge since a deep computation model typically involves a large number of parameters. Specially, it needs a high-performance computing server with a large-scale memory and a powerful computing unit to train a deep computation model, making it difficult to increase the size of a deep computation model further for big data feature learning on low-end devices such as conventional desktops and portable CPUs. In this paper, we propose an improved deep computation model based on the canonical polyadic decomposition scheme to compress the parameters and to improve the training efficiency. Furthermore, we devise a learning algorithm based on the back-propagation strategy to train the parameters of the proposed model. The learning algorithm can be directly performed on the compressed parameters to improve the training efficiency. Finally, we carry on the experiments on three representative datasets, i.e., CUAVE, SNAE2, and STL-10, to evaluate the performance of the proposed model by comparing with the conventional deep computation model and other two improved deep computation models based on the Tucker decomposition and the tensor-train network. Results demonstrate that the proposed model can compress parameters greatly and improve the training efficiency significantly with a low classification accuracy drop. [ABSTRACT FROM AUTHOR]

A semigroup (R, ·) is said to be a UA-ring if there exists a unique binary operation + making (R, ·, +) into a ring. We study finitely presented Ẑ-modules with UA-rings of endomorphisms. [ABSTRACT FROM AUTHOR]

Joint blind source separation (J-BSS) is an emerging data-driven technique for multi-set data-fusion. In this paper, J-BSS is addressed from a tensorial perspective. We show how, by using second-order multi-set statistics in J-BSS, a specific double coupled canonical polyadic decomposition (DC-CPD) problem can be formulated. We propose an algebraic DC-CPD algorithm based on a coupled rank-1 detection mapping. This algorithm converts a possibly underdetermined DC-CPD to a set of overdetermined CPDs. The latter can be solved algebraically via a generalized eigenvalue decomposition based scheme. Therefore, this algorithm is deterministic and returns the exact solution in the noiseless case. In the noisy case, it can be used to effectively initialize optimization based DC-CPD algorithms. In addition, we obtain the deterministic and generic uniqueness conditions for DC-CPD, which are shown to be more relaxed than their CPD counterpart. We also introduce optimization based DC-CPD methods, including alternating least squares, and structured data fusion based methods. Experiment results are given to illustrate the superiority of DC-CPD over standard CPD based BSS methods and several existing J-BSS methods, with regards to uniqueness and accuracy. [ABSTRACT FROM AUTHOR]

To ensure interpretability of extracted sources in tensor decomposition, we introduce in this paper a dictionary-based tensor canonical polyadic decomposition, which enforces one factor to belong exactly to a known dictionary. A new formulation of sparse coding is proposed, which enables high-dimensional tensors dictionary-based canonical polyadic decomposition. The benefits of using a dictionary in tensor decomposition models are explored both in terms of parameter identifiability and estimation accuracy. Performances of the proposed algorithms are evaluated on the decomposition of simulated data and the unmixing of hyperspectral images. [ABSTRACT FROM AUTHOR]

We introduce periodic words that are connected with the Tribonacci-Lucas numbers and investigate their properties. [ABSTRACT FROM AUTHOR]

Gears are key components in rotation machinery and its fault vibration signals usually show strong nonlinear and non-stationary characteristics. It is not easy for classical time-frequency domain analysis methods to recognize different gear working conditions. Therefore, this paper presents a joint fault diagnosis scheme for gear fault classification via tensor nuclear norm canonical polyadic decomposition (TNNCPD) and multi-scale permutation entropy (MSPE). Firstly, the one-dimensional vibration data of different gear fault conditions is converted into a three-dimensional tensor data, and a new tensor canonical polyadic decomposition method based on nuclear norm and convex optimization called TNNCPD is proposed to extract the low rank component of the data, which represents the feature information of the measured signal. Then, the MSPE of the extracted feature information about different gear faults can be calculated as the feature vector in order to recognize fault conditions. Finally, this researched scheme is validated by practical gear vibration data of different fault conditions. The result demonstrates that the proposed scheme can effectively recognize different gear fault conditions. [ABSTRACT FROM AUTHOR]

In this letter, we study the deterministic sampling patterns for the completion of low-rank matrix, when corrupted with a sparse noise, also known as robust matrix completion. We extend the recent results on the deterministic sampling patterns in the absence of noise based on the geometric analysis on the Grassmannian manifold. A special case where each column has a certain number of noisy entries is considered, where our probabilistic analysis performs very efficiently. Furthermore, assuming that the rank of the original matrix is not given, we provide an analysis to determine if the rank of a valid completion is indeed the actual rank of the data corrupted with sparse noise by verifying some conditions. [ABSTRACT FROM AUTHOR]

Several program verification techniques assist in showing that software adheres to the required security policies. Such policies may be sensitive to the flow of execution and the verification may be supported by combinations of type systems and Hoare logics. However, this requires user assistance and to obtain full automation we shall explore the over-approximating nature of static analysis. We demonstrate that the use of atomistic Galois insertions constitutes a stable framework in which to obtain sound and fully automatic enforcement of flow sensitive integrity. The framework is illustrated on a concurrent language with local storage and polyadic synchronous communication. [ABSTRACT FROM AUTHOR]

Accurately determining dependency structure is critical to understanding a complex system's organization. We recently showed that the transfer entropy fails in a key aspect of this--measuring information flow--due to its conflation of dyadic and polyadic relationships. We extend this observation to demonstrate that Shannon information measures (entropy and mutual information, in their conditional and multivariate forms) can fail to accurately ascertain multivariate dependencies due to their conflation of qualitatively different relations among variables. This has broad implications, particularly when employing information to express the organization and mechanisms embedded in complex systems, including the burgeoning efforts to combine complex network theory with information theory. Here, we do not suggest that any aspect of information theory is wrong. Rather, the vast majority of its informational measures are simply inadequate for determining the meaningful relationships among variables within joint probability distributions. We close by demonstrating that such distributions exist across an arbitrary set of variables. [ABSTRACT FROM AUTHOR]

Tensor diagonalization means transforming a given tensor to an exactly or nearly diagonal form through multiplying the tensor by non-orthogonal invertible matrices along selected dimensions of the tensor. It has a link to an approximate joint diagonalization (AJD) of a set of matrices. In this paper, we derive (1) a new algorithm for a symmetric AJD, which is called two-sided symmetric diagonalization of an order-three tensor, (2) a similar algorithm for a non-symmetric AJD, also called a two-sided diagonalization of an order-three tensor, and (3) an algorithm for three-sided diagonalization of order-three or order-four tensors. The latter two algorithms may serve for canonical polyadic (CP) tensor decomposition, and in certain scenarios they can outperform traditional CP decomposition methods. Finally, we propose (4) similar algorithms for tensor block diagonalization, which is related to tensor block-term decomposition. The proposed algorithm can either outperform the existing block-term decomposition algorithms, or produce good initial points for their application. [ABSTRACT FROM AUTHOR]

Tensor decompositions such as the canonical polyadic decomposition (CPD) or the block term decomposition (BTD) are basic tools for blind signal separation. Most of the literature concerns instantaneous mixtures/memoryless channels. In this paper, we focus on convolutive extensions. More precisely, we present a connection between convolutive CPD/BTD models and coupled but instantaneous CPD/BTD. We derive a new identifiability condition dedicated to convolutive low-rank factorization problems. We explain that under this condition, the convolutive extension of CPD/BTD can be computed by means of an algebraic method, guaranteeing perfect source separation in the noiseless case. In the inexact case, the algorithm can be used as a cheap initialization for an optimization-based method. We explain that, in contrast to the memoryless case, convolutive signal separation is in certain cases possible despite only two-way diversities (e.g., space $\times$ time). [ABSTRACT FROM PUBLISHER]

We consider the problem of low canonical polyadic (CP) rank tensor completion. A completion is a tensor whose entries agree with the observed entries and its rank matches the given CP rank. We analyze the manifold structure corresponding to the tensors with the given rank and define a set of polynomials based on the sampling pattern and CP decomposition. Then, we show that finite completability of the sampled tensor is equivalent to having a certain number of algebraically independent polynomials among the defined polynomials. Our proposed approach results in characterizing the maximum number of algebraically independent polynomials in terms of a simple geometric structure of the sampling pattern, and therefore we obtain the deterministic necessary and sufficient condition on the sampling pattern for finite completability of the sampled tensor. Moreover, assuming that the entries of the tensor are sampled independently with probability p and using the mentioned deterministic analysis, we propose a combinatorial method to derive a lower bound on the sampling probability p, or equivalently, the number of sampled entries that guarantees finite completability with high probability. We also show that the existing result for the matrix completion problem can be used to obtain a loose lower bound on the sampling probability p. In addition, we obtain deterministic and probabilistic conditions for unique completability. It is seen that the number of samples required for finite or unique completability obtained by the proposed analysis on the CP manifold is orders-of-magnitude lower than that is obtained by the existing analysis on the Grassmannian manifold. [ABSTRACT FROM AUTHOR]

This paper investigates a two-dimensional angle of arrival (2D AOA) estimation algorithm for the electromagnetic vector sensor (EMVS) array based on Type-2 block component decomposition (BCD) tensor modeling. Such a tensor decomposition method can take full advantage of the multidimensional structural information of electromagnetic signals to accomplish blind estimation for array parameters with higher resolution. However, existing tensor decomposition methods encounter many restrictions in applications of the EMVS array, such as the strict requirement for uniqueness conditions of decomposition, the inability to handle partially-polarized signals, etc. To solve these problems, this paper investigates tensor modeling for partially-polarized signals of an L-shaped EMVS array. The 2D AOA estimation algorithm based on rank-(L1, L2, .) BCD is developed, and the uniqueness condition of decomposition is analyzed. By means of the estimated steering matrix, the proposed algorithm can automatically achieve angle pair-matching. Numerical experiments demonstrate that the present algorithm has the advantages of both accuracy and robustness of parameter estimation. Even under the conditions of lower SNR, small angular separation and limited snapshots, the proposed algorithm still possesses better performance than subspace methods and the canonical polyadic decomposition (CPD) method. [ABSTRACT FROM AUTHOR]

In this paper, a way of generalizing the tensor renormalization group (TRG) is proposed. Mathematically, the connection between patterns of tensor renormalization group and the concept of truncation sequence in polytope geometry is discovered. A theoretical contraction framework is therefore proposed. Furthermore, the canonical polyadic decomposition is introduced to tensor network theory. A numerical verification of this method on the 3-D Ising model is carried out. [ABSTRACT FROM AUTHOR]

A selection of tensor decomposition techniques is presented for the detection of weak signals in electron energy loss spectroscopy (EELS) data. The focus of the analysis lies on the correct representation of the simulated spatial structure. An analysis scheme for EEL spectra combining two-dimensional and n-way decomposition methods is proposed. In particular, the performance of robust principal component analysis (ROBPCA), Tucker Decompositions using orthogonality constraints (Multilinear Singular Value Decomposition (MLSVD)) and Tucker decomposition without imposed constraints, canonical polyadic decomposition (CPD) and block term decompositions (BTD) on synthetic as well as experimental data is examined. [ABSTRACT FROM AUTHOR]

Systems of equations and their solution sets are studied in polyadic groups. We prove that a polyadic group (G,f) = der𝜃,b(G,⋅) is equational noetherian, if and only if the ordinary group (G,⋅) is equational noetherian. The structure of coordinate polyadic group of algebraic sets in equational noetherian polyadic groups is also determined. [ABSTRACT FROM PUBLISHER]

Tensor canonical polyadic decomposition (CPD), which recovers the latent factor matrices from multidimensional data, is an important tool in signal processing. In many applications, some of the factor matrices are known to have orthogonality structure, and this information can be exploited to improve the accuracy of latent factors recovery. However, existing methods for CPD with orthogonal factors all require the knowledge of tensor rank, which is difficult to acquire, and have no mechanism to handle outliers in measurements. To overcome these disadvantages, in this paper, a novel tensor CPD algorithm based on the probabilistic inference framework is devised. In particular, the problem of tensor CPD with orthogonal factors is interpreted using a probabilistic model, based on which an inference algorithm is proposed that alternatively estimates the factor matrices, recovers the tensor rank, and mitigates the outliers. Simulation results using synthetic data and real-world applications are presented to illustrate the excellent performance of the proposed algorithm in terms of accuracy and robustness. [ABSTRACT FROM AUTHOR]

In Part I of this paper, we have presented a link between multidimensional harmonic retrieval (MHR) and the recently proposed coupled canonical polyadic decomposition (CPD), which implies new uniqueness conditions for MHR that are more relaxed than the existing results based on a Vandermonde constrained CPD. In Part II, we explain that the coupled CPD also provides a computational framework for MHR. In particular, we present an algebraic method for MHR based on simultaneous matrix diagonalization that is guaranteed to find the exact solution in the noiseless case, under conditions discussed in Part I. Since the simultaneous matrix diagonalization method reduces the MHR problem into an eigenvalue problem, the proposed algorithm can be interpreted as an MHR generalization of the classical ESPRIT method for one-dimensional harmonic retrieval. We also demonstrate that the presented coupled CPD framework for MHR can algebraically support multirate sampling. We develop an efficient implementation which has about the same computational complexity for single-rate and multirate sampling. Numerical experiments demonstrate that by simultaneously exploiting the harmonic structure in all dimensions and making use of multirate sampling, the coupled CPD framework for MHR can lead to an improved performance compared to the conventional Vandermonde constrained CPD approaches. [ABSTRACT FROM PUBLISHER]

Canonical Polyadic Decomposition (CPD) of a third-order tensor is a minimal decomposition into a sum of rank-1 tensors. We find new mild deterministic conditions for the uniqueness of individual rank-1 tensors in CPD and present an algorithm to recover them. We call the algorithm “algebraic” because it relies only on standard linear algebra. It does not involve more advanced procedures than the computation of the null space of a matrix and eigen/singular value decomposition. Simulations indicate that the new conditions for uniqueness and the working assumptions for the algorithm hold for a randomly generated I × J × K tensor of rank R ≥ K ≥ J ≥ I ≥ 2 if R is bounded as R ≤ ( I + J + K − 2 ) / 2 + ( K − ( I − J ) 2 + 4 K ) / 2 at least for the dimensions that we have tested. This improves upon the famous Kruskal bound for uniqueness R ≤ ( I + J + K − 2 ) / 2 as soon as I ≥ 3 . In the particular case R = K , the new bound above is equivalent to the bound R ≤ ( I − 1 ) ( J − 1 ) which is known to be necessary and sufficient for the generic uniqueness of the CPD. An existing algebraic algorithm (based on simultaneous diagonalization of a set of matrices) computes the CPD under the more restrictive constraint R ( R − 1 ) ≤ I ( I − 1 ) J ( J − 1 ) / 2 (implying that R < ( J − 1 2 ) ( I − 1 2 ) / 2 + 1 ). We give an example of a low-dimensional but high-rank CPD that cannot be found by optimization-based algorithms in a reasonable amount of time while our approach takes less than a second. We demonstrate that, at least for R ≤ 24 , our algorithm can recover the rank-1 tensors in the CPD up to R ≤ ( I − 1 ) ( J − 1 ) . [ABSTRACT FROM AUTHOR]

Physiological signals are often organized in the form of multiple dimensions (e.g., channel, time, task, and 3-D voxel), so it is better to preserve original organization structure when processing. Unlike vector-based methods that destroy data structure, canonical polyadic decomposition (CPD) aims to process physiological signals in the form of multiway array, which considers relationships between dimensions and preserves structure information contained by the physiological signal. Nowadays, CPD is utilized as an unsupervised method for feature extraction in a classification problem. After that, a classifier, such as support vector machine, is required to classify those features. In this manner, classification task is achieved in two isolated steps. We proposed supervised CPD by directly incorporating auxiliary label information during decomposition, by which a classification task can be achieved without an extra step of classifier training. The proposed method merges the decomposition and classifier learning together, so it reduces procedure of classification task compared with that of respective decomposition and classification. In order to evaluate the performance of the proposed method, three different kinds of signals, synthetic signal, EEG signal, and MEG signal, were used. The results based on evaluations of synthetic and real signals demonstrated that the proposed method is effective and efficient. [ABSTRACT FROM AUTHOR]

This paper concerns the algebraic logic associated to probabilistic models of intuitionistic predicate calculus. We consider Bosbach's style states on polyadic Heyting algebras, define the polyadic states, the weak polyadic states, the state models and the weak state models of the states. The main result of this paper asserts that any state defined on a Heyting subalgebra of a polyadic Heyting algebra has a weak state model. This theorem has as consequence that any intuitionistic probability has a weak probabilistic model. [ABSTRACT FROM AUTHOR]

Low-rank tensor completion addresses the task of filling in missing entries in multi-dimensional data. It has proven its versatility in numerous applications, including context-aware recommender systems and multivariate function learning. To handle large-scale datasets and applications that feature high dimensions, the development of distributed algorithms is central. In this work, we propose novel, highly scalable algorithms based on a combination of the canonical polyadic (CP) tensor format with block coordinate descent methods. Although similar algorithms have been proposed for the matrix case, the case of higher dimensions gives rise to a number of new challenges and requires a different paradigm for data distribution. The convergence of our algorithms is analyzed and numerical experiments illustrate their performance on distributed-memory architectures for tensors from a range of different applications. [ABSTRACT FROM AUTHOR]

The dynamic potentials of highly excited vibrational states of deuterated phosphaethyne (DCP) in the D-C and C-P stretching coordinates with anharmonicity and Fermi coupling are studied in this article and the results show that the D-C-P bending vibration mode has weak effects on D-C and C-P stretching modes under different Polyad numbers (P number). Furthermore, the dynamic potentials and the corresponding phase space trajectories of DCP are given, as an example, in the case of P = 30. In the end, a comparative study between deuterated phosphaethyne (DCP) and phosphaethyne (HCP) with dynamic potential is done, and it is elucidated that the uncoupled mode makes the original horizontal reversed symmetry breaking between the dynamic potential of HCP (q3) and DCP (q1), but has little effect on the vertical reversed symmetry, between the dynamic potential of HCP (q2) and DCP (q3). [ABSTRACT FROM AUTHOR]

Canonical polyadic decomposition (CPD), also known as parallel factor analysis, is a representation of a given tensor as a sum of rank-one components. Traditional method for accomplishing CPD is the alternating least squares (ALS) algorithm. Convergence of ALS is known to be slow, especially when some factor matrices of the tensor contain nearly collinear columns. We propose a novel variant of this technique, in which the factor matrices are partitioned into blocks, and each iteration jointly updates blocks of different factor matrices. Each partial optimization is quadratic and can be done in closed form. The algorithm alternates between different random partitionings of the matrices. As a result, a faster convergence is achieved. Another improvement can be obtained when the method is combined with the enhanced line search of Rajih et al. Complexity per iteration is between those of the ALS and the Levenberg–Marquardt (damped Gauss–Newton) method. It is important, however, that the idea of alternating quadratic optimization with partitioned factor matrices is general and can be applied to other variants of the tensor decomposition problems, e.g., when non-Gaussian additive noise is considered. [ABSTRACT FROM PUBLISHER]

The FTIR spectra of CH 2 F 2 have been investigated in a region of atmospheric interest (1000–1300 cm −1 ) where four fundamentals ν 3 , ν 5 , ν 7 and ν 9 occur. These vibrations perturb each other by different Coriolis interactions and the forbidden ν 5 borrows intensity from the neighboring levels. Furthermore, the v 4 =2 state has been found to interact with the v 3 =1 and v 9 =1 states by anharmonic and c -type Coriolis resonances, respectively. The spectral analysis resulted in the assignment of about 7500 rovibrational transitions which have been simultaneously fitted, together with microwave data available in literature (Hirota E. J Mol Spectrosc 1978; 69: 409–420) [15] using the Watson׳s A-reduction Hamiltonian in the I r representation and the relevant perturbation operators. The model employed includes eight types of resonances within the pentad ν 3 /ν 5 /ν 7 /ν 9 /2ν 4 . A set of spectroscopic constants for the four fundamentals as well as parameters for the v 4 =2 state and eighteen coupling terms have been determined. The simulations performed in different spectral regions well reproduce the experimental data. [ABSTRACT FROM AUTHOR]

The global modeling of 15 N 2 16 O line positions in the 4–12,516 cm −1 region has been performed using the polyad model of effective Hamiltonian. The effective Hamiltonian parameters were fitted to the line positions collected from an exhaustive review of the literature. The dimensionless weighted standard deviation of the fit is 1.31. The fitted set of 109 parameters allowed reproducing more than 18,000 measured line positions with an RMS value of 0.001 cm −1 . A line list was calculated for a reference temperature 296 K, natural abundance (1.32×10 −5 ), and an intensity cutoff 10 −30 cm/molecule. The line list is based on the fitted set of the effective Hamiltonian parameters for 15 N 2 16 O obtained in this work and the effective dipole moment parameters of the 15 N 2 16 O and 14 N 2 16 O isotopologues. Accurate values of the 15 N 2 16 O total partition function are also given. [ABSTRACT FROM AUTHOR]

We investigate the notion of a signature in Polyadic Inductive Logic and study the probability functions satisfying the Principle of Signature Exchangeability . We prove a representation theorem for such functions on binary languages and show that they satisfy a binary version of the Principle of Instantial Relevance. We discuss polyadic versions of the Principle of Instantial Relevance and Johnson's Sufficientness Postulate. [ABSTRACT FROM AUTHOR]

This article investigates the role of arity of second-order quantifiers in existential second-order logic, also known as Σ 1 1 . We identify fragments L of Σ 1 1 where second-order quantification of relations of arity k > 1 is (nontrivially) vacuous in the sense that each formula of L can be translated to a formula of (a fragment of) monadic Σ 1 1 . Let polyadic Boolean modal logic with identity ( PBML = ) be the logic obtained by extending standard polyadic multimodal logic with built-in identity modalities and with constructors that allow for the Boolean combination of accessibility relations. Let Σ 1 1 ( PBML = ) be the extension of PBML = with existential prenex quantification of accessibility relations and proposition symbols. The principal result of the article is that Σ 1 1 ( PBML = ) translates into monadic Σ 1 1 . As a corollary, we obtain a variety of decidability results for multimodal logic. The translation can also be seen as a step towards establishing whether every property of finite directed graphs expressible in Σ 1 1 ( FO 2 ) is also expressible in monadic Σ 1 1 . This question was left open in the 1999 paper of Grädel and Rosen in the 14th Annual IEEE Symposium on Logic in Computer Science. [ABSTRACT FROM AUTHOR]

Assuming the usual finite axiom schema of polyadic equality algebras, axiom (P10) is changed to a stronger version. It is proved that infinite dimensional, polyadic equality algebras satisfying the resulting system of axioms are representable. The foregoing stronger axiom is not given with a first order schema. The latter is to be expected knowing the negative results for the Halmos schema axiomatizability of the representable, infinite dimensional, polyadic equality algebras. Furthermore, Halmos' well-known classical theorem that 'locally finite polyadic equality algebras of infinite dimension α are representable' is generalized for locally- $${\mathfrak{m}}$$ polyadic equality algebras, where $${\mathfrak{m}}$$ is an arbitrary infinite cardinal and $${\mathfrak{m}}$$ < α. Also, a neat embedding theorem is proved for the foregoing classes of polyadic-like equality algebras (a neat embedding theorem does not exists for polyadic equality algebras). [ABSTRACT FROM AUTHOR]

We show that the usual axiom system of quasi polyadic equality algebras is strongly redundant. Then, so called non-commutative quasi-polyadic equality algebras are introduced (), in which, among others, the commutativity of cylindrifications is dropped. As is known, quasi-polyadic equality algebras are not representable in the classical sense, but we prove that algebras in are representable by quasi-polyadic relativized set algebras, or more exactly by algebras in . [ABSTRACT FROM AUTHOR]

Background Canonical polyadic decomposition (CPD) may face a local optimal problem when analyzing multi-subject fMRI data with inter-subject variability. Beckmann and Smith proposed a tensor PICA approach that incorporated an independence constraint to the spatial modality by combining CPD with ICA, and alleviated the problem of inter-subject spatial map (SM) variability. New method This study extends tensor PICA to incorporate additional inter-subject time course (TC) variability and to connect CPD and ICA in a new way. Assuming multiple subjects share common TCs but with different time delays, we accommodate subject-dependent TC delays into the CP model based on the idea of shift-invariant CP (SCP). We use ICA as an initialization step to provide the aggregating mixing matrix for shift-invariant CPD to estimate shared TCs with subject-dependent delays and intensities. We then estimate shared SMs using a least-squares fit post shift-invariant CPD. Results Using simulated fMRI data as well as actual fMRI data we demonstrate that the proposed approach improves the estimates of the shared SMs and TCs, and the subject-dependent TC delays and intensities. The default mode component illustrates larger TC delays than the task-related component. Comparison with existing method(s) The proposed approach shows improvements over tensor PICA in particular when TC delays are large, and also outperforms SCP with SM orthogonality constraint and SCP with ICA-based SM initialization. Conclusions TCs with subject-dependent delays conform to the true situation of multi-subject fMRI data. The proposed approach is suitable for decomposing multi-subject fMRI data with large inter-subject temporal and spatial variability. [ABSTRACT FROM AUTHOR]

In Part I of the study of the tensor deflation for CANDECOMP/PARAFAC, we have shown that the rank-1 tensor deflation is applicable under some conditions. Part II of the study presents several initialization algorithms suitable for the algorithm proposed in Part I. In addition, Part II contains an algorithm for the case when one or more factor matrices in the estimated model is constrained to be orthogonal. Finally, Part II provides an error analysis of the tensor deflation algorithm, which shows that there is a marginal loss of accuracy of the deflation algorithm compared to the ordinary CP decomposition. [ABSTRACT FROM PUBLISHER]

CANDECOMP/PARAFAC (CP) approximates multiway data by sum of rank-1 tensors. Unlike matrix decomposition, the procedure which estimates the best rank-R tensor approximation through R sequential best rank-1 approximations does not work for tensors, because the deflation does not always reduce the tensor rank. In this paper, we propose a novel deflation method for the problem. When one factor matrix of a rank-R CP decomposition is of full column rank, the decomposition can be performed through (R-1) rank-1 reductions. At each deflation stage, the residue tensor is constrained to have a reduced multilinear rank. For decomposition of order-3 tensors of size R\times R\times R and rank-R, estimation of one rank-1 tensor has a computational cost of \cal O(R^3) per iteration which is lower than the cost \cal O(R^4) of the ALS algorithm for the overall CP decomposition. The method can be extended to tracking one or a few rank-one tensors of slow changes, or inspect variations of common patterns in individual datasets. [ABSTRACT FROM PUBLISHER]

We consider countable so-called rich subsemigroups of [ABSTRACT FROM AUTHOR]

We find conditions that guarantee that a decomposition of a generic third-order tensor in a minimal number of rank-1 tensors (canonical polyadic decomposition (CPD)) is unique up to a permutation of rank-1 tensors. Then we consider the case when the tensor and all its rank-1 terms have symmetric frontal slices (INDSCAL). Our results complement the existing bounds for generic uniqueness of the CPD and relax the existing bounds for INDSCAL. The derivation makes use of algebraic geometry. We stress the power of the underlying concepts for proving generic properties in mathematical engineering. [ABSTRACT FROM AUTHOR]

For any given positive integer $m$ , a necessary and sufficient condition for the existence of Type-I $m$ -adic constacyclic codes is given. Furthermore, for any given integer $s$ , a necessary and sufficient condition for $s$ to be a multiplier of a Type-I polyadic constacyclic code is given. As an application, some optimal codes from Type-I polyadic constacyclic codes, including generalized Reed–Solomon codes and alternant maximum distance separable codes, are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we discuss new efficient algorithms for nonnegative matrix factorization (NMF) with smoothness constraints imposed on nonnegative components or factors. Such constraints allow us to alleviate certain ambiguity problems, which facilitates better physical interpretation or meaning. In our approach, various basis functions are exploited to flexibly and efficiently represent the smooth nonnegative components. For noisy input data, the proposed algorithms are more robust than the existing smooth and sparse NMF algorithms. Moreover, we extend the proposed approach to the smooth nonnegative Tucker decomposition and smooth nonnegative canonical polyadic decomposition (also called smooth nonnegative tensor factorization). Finally, we conduct extensive experiments on synthetic and real-world multi-way array data to demonstrate the advantages of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The coupled canonical polyadic decomposition (CPD) is an emerging tool for the joint analysis of multiple data sets in signal processing and statistics. Despite their importance, linear algebra based algorithms for coupled CPDs have not yet been developed. In this paper, we first explain how to obtain a coupled CPD from one of the individual CPDs. Next, we present an algorithm that directly takes the coupling between several CPDs into account. We extend the methods to single and coupled decompositions in multilinear rank-(Lr,n, Lr,n, 1) terms. Finally, numerical experiments demonstrate that linear algebra based algorithms can provide good results at a reasonable computational cost. [ABSTRACT FROM AUTHOR]

Looking at the operation of forming neat $$\alpha $$ -reducts as a functor, with $$\alpha $$ an infinite ordinal, we investigate when such a functor obtained by truncating $$\omega $$ dimensions, has a right adjoint. We show that the neat reduct functor for representable cylindric algebras does not have a right adjoint, while that of polyadic algebras is an equivalence. We relate this categorial result to several amalgamation properties for classes of representable algebras. We show that the variety of cylindric algebras of infinite dimension, endowed with the merry go round identities, fails to have the amalgamation property answering a question of Németi's. We also study two variants of the so-called cylindric polyadic algebras introduced by Ferenczi (all are reducts of polyadic equality algebras, that are also varieties). We show that one is more cylindric than polyadic, and that the other is more polyadic than cylindric. Our classification is determined by results on neat embeddings and amalgamation expressed from the point of view of category theory, thereby witnessing, and, indeed, further emphasizing, the dichotomy between the cylindric and polyadic paradigms. For example, the first class does not have the unique neat embedding property, fails to have the amalgamation property and the neat reduct functor does not have a right adjoint, while the second class has the unique neat emdedding property, the superamalgamation property and the neat reduct functor is strongly invertible. Other results, like first order definability of the class of neat reducts and the class of completely representable algebras, confirming our classification along these lines are presented. [ABSTRACT FROM AUTHOR]

Let α be an ordinal and κ be a cardinal, both infinite, such that κ ≤ |α|. For τ ∈αα, letsup(τ) = {i ∈ α: τ(i) ≠ i}. LetGκ = {τ ∈αα: |sup(τ)| < κ}. We consider variants of polyadic equality algebras by taking cylindrifications on Γ ⊆ α, |Γ| < κ and substitutions restricted toGκ. Such algebras are also enriched with generalized diagonal elements. We show that for any varietyVcontaining the class of representable algebas and satisfying a finite schema of equations,Vfails to have the amalgamation property. In particular, many varieties of Halmos’ quasi-polyadic equality algebras and Lucas’ extended cylindric algebras (including that of the representable algebras) fail to have the amalgamation property. [ABSTRACT FROM AUTHOR]