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The multidimensional discrete Fourier transform (MD-DFT) plays an important role in a growing number of signal processing applications. The fundamentals of its applicability as a unitary transform between discrete periodic sequences defined on multidimensional lattices stand on the Hermitian orthogonality of the vectors defining the MD-DFT matrix. A proof of the consistency of the MD-DFT formulation was first provided by Bernardini and Manduchi in 1994 using the Smith normal form theorem of integer matrices. In this reported work, a new proof is provided based on the nullity of the cardinal function on the nonzero cardinal points. [PUBLICATION ABSTRACT]

It is shown for the first time in this paper, that Kleinberg’s (2002) (self-contradictory) axiomatic system for distance-based clustering fails (that is one of the data transforming axioms, consistency axiom, turns out to be identity transformation) in fixed-dimensional Euclidean space due to the consistency axiom limitations and that its replacement with inner-consistency or outer consistency does not help if continuous data transformations are required. Therefore we formulate a new, sound axiomatic framework for cluster analysis in the fixed dimensional Euclidean space, suitable for k-means like algorithms. The system incorporates centric consistency axiom and motion consistency axiom which induce clustering preserving transformations useful e.g. for deriving new labelled sets for testing clustering procedures. It is suitable for continuous data transformations so that labelled data with small perturbations can be derived. Unlike Kleinberg’s consistency, the new axioms do not lead the data outside of Euclidean space nor cause increase in data dimensionality. Our cluster preserving transformations have linear complexity in data transformation and checking. They are in practice less restrictive, less rigid than Kleinberg’s consistency as they do not enforce inter-cluster distance increase and inner cluster distance decrease when performing clustering preserving transformation.

Two fundamental axioms in social choice theory are consistency with respect to a variable electorate and consistency with respect to components of similar alternatives. In the context of traditional non-probabilistic social choice, these axioms are incompatible with each other. We show that in the context of probabilistic social choice, these axioms uniquely characterize a function proposed by Fishburn (1984). Fishburn's function returns so-called maximal lotteries, that is, lotteries that correspond to optimal mixed strategies in the symmetric zero-sum game induced by the pairwise majority margins. Maximal lotteries are guaranteed to exist due to von Neumann's Minimax Theorem, are almost always unique, and can be efficiently computed using linear programming.

For the average trajectory consistency test, the density is good, which leads to the problem of poor consistency. This paper proposes to use the Mahalanobis distance method to process the test data of two kinds of ammunition. First, each pair of values in a set of tests is obtained through experiments. The obtained data is processed by the Mahalanobis distance method for relative distance processing. Then the mean and variance of the distance data of the two ammunition are calculated. In the simulation test, the data of tests which are processed by the Mahalanobis distance method. The consistency of the two ammunition is checked according to the trajectory consistency test method. The test data using the Mahalanobis distance method can better than the data of the two ammunition in the consistency. The test data processed by the Mahalanobis distance method can improve the situation of good density and poor consistency in the consistency test, and provide technical support for the subsequent average trajectory consistency test.

This paper surveys recent theoretical advances in convex optimization approaches for community detection. We introduce some important theoretical techniques and results for establishing the consistency of convex community detection under various statistical models. In particular, we discuss the basic techniques based on the primal and dual analysis. We also present results that demonstrate several distinctive advantages of convex community detection, including robustness against outlier nodes, consistency under weak assortativity, and adaptivity to heterogeneous degrees. This survey is not intended to be a complete overview of the vast literature on this fast-growing topic. Instead, we aim to provide a big picture of the remarkable recent development in this area and to make the survey accessible to a broad audience. We hope that this expository article can serve as an introductory guide for readers who are interested in using, designing, and analyzing convex relaxation methods in network analysis.

A well-regarded as well as powerful method named the ‘analytic hierarchy process’ (AHP) uses mathematics and psychology for making and analysing complex decisions. This article aims to present a brief review of the consistency measure of the judgments in AHP. Judgments should not be random or illogical. Several researchers have developed different consistency measures to identify the rationality of judgments. This article summarises the consistency measures which have been proposed so far in the literature. Moreover, this paper describes briefly the functional relationships established in the literature among the well-known consistency indices. At last, some thoughtful research directions that can be helpful in further research to develop and improve the performance of AHP are provided as well.

A bstract Transverse momentum dependent parton distributions (TMDPDFs) which appear in factorized cross sections involve infinite Wilson lines with edges on or close to the light-cone. Since these TMDPDFs are not directly calculable with a Euclidean path integral in lattice QCD, we study the construction of quasi-TMDPDFs with finite-length spacelike Wilson lines that are amenable to such calculations. We define an infrared consistency test to determine which quasi-TMDPDF definitions are related to the TMDPDF, by carrying out a one-loop study of infrared logarithms of transverse position b T  ∼ Λ QCD −1 , which must agree between them. This agreement is a necessary condition for the two quantities to be related by perturbative matching. TMDPDFs necessarily involve combining a hadron matrix element, which nominally depends on a single light-cone direction, with soft matrix elements that necessarily depend on two light-cone directions. We show at one loop that the simplest definitions of the quasi hadron matrix element, the quasi soft matrix element, and the resulting quasi-TMDPDF all fail the infrared consistency test. Ratios of impact parameter quasi-TMDPDFs still provide nontrivial information about the TMD-PDFs, and are more robust since the soft matrix elements cancel. We show at one loop that such quasi ratios can be matched to ratios of the corresponding TMDPDFs. We also introduce a modified “bent” quasi soft matrix element which yields a quasi-TMDPDF that passes the consistency test with the TMDPDF at one loop, and discuss potential issues at higher orders.

In this paper, the thermal consistency and electrochemical performance of batteries were comprehensively considered to improve the test and ensure the consistency of the power battery pack for automotive applications. At the same time, a safer and more efficient device IS established for testing and evaluation of battery consistency for automotive applications to achieve the real-time monitoring, assessment, prediction, and control of its consistency.

Consistency regularization is one of the most widely-used techniques for semi-supervised learning (SSL). Generally, the aim is to train a model that is invariant to various data augmentations. In this paper, we revisit this idea and find that enforcing invariance by decreasing distances between features from differently augmented images leads to improved performance. However, encouraging equivariance instead, by increasing the feature distance, further improves performance. To this end, we propose an improved consistency regularization framework by a simple yet effective technique, FeatDistLoss, that imposes consistency and equivariance on the classifier and the feature level, respectively. Experimental results show that our model defines a new state of the art across a variety of standard semi-supervised learning benchmarks as well as imbalanced semi-supervised learning benchmarks. Particularly, we outperform previous work by a significant margin in low data regimes and at large imbalance ratios. Extensive experiments are conducted to analyze the method, and the code will be published.

We study theoretical properties of regularized robust M-estimators, applicable when data are drawn from a sparse high-dimensional linear model and contaminated by heavy-tailed distributions and/or outliers in the additive errors and covariates. We first establish a form of local statistical consistency for the penalized regression estimators under fairly mild conditions on the error distribution: When the derivative of the loss function is bounded and satisfies a local restricted curvature condition, all stationary points within a constant radius of the true regression vector converge at the minimax rate enjoyed by the Lasso with sub-Gaussian errors. When an appropriate nonconvex regularizer is used in place of an ℓ₁-penalty, we show that such stationary points are in fact unique and equal to the local oracle solution with the correct support; hence, results on asymptotic normality in the low-dimensional case carry over immediately to the high-dimensional setting. This has important implications for the efficiency of regularized nonconvex M-estimators when the errors are heavy-tailed. Our analysis of the local curvature of the loss function also has useful consequences for optimization when the robust regression function and/or regularizer is nonconvex and the objective function possesses stationary points outside the local region. We show that as long as a composite gradient descent algorithm is initialized within a constant radius of the true regression vector, successive iterates will converge at a linear rate to a stationary point within the local region. Furthermore, the global optimum of a convex regularized robust regression function may be used to obtain a suitable initialization. The result is a novel two-step procedure that uses a convex M-estimator to achieve consistency and a nonconvex M-estimator to increase efficiency. We conclude with simulation results that corroborate our theoretical findings.