Supersymmetric Tensor Research Articles

Axial anomalies of maximally supersymmetric tensor theories

We revisit anomalies of (4, 0) and (3, 1) maximally supersymmetric tensor theories in d = 6. A (4, 0) on-shell tensor multiplet descends to that of the d = 5 maximal supergravity upon a dimensional reduction, hypothesized to offer a strong-coupled UV completion of the latter in the same sense of (2, 0) theories as the UV completion of d = 5 N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 pure Yang-Mills. The gravitational anomalies, found to be nonvanishing, had been computed, although its relevance in the absence of the d = 6 metric is not obvious. We perform a comprehensive anomaly computation for (4, 0) and (3, 1) tensor supermultiplets, respectively, for Sp(4) and Sp(3) × Sp(1) R-symmetry anomalies and the mixed R-gravitational anomaly thereof, and find that anomalies involving R-symmetries cancel out identically. We close with questions on how to address the anomaly in this class of theories with no general covariance.

An adaptive cubic regularization algorithm for computing H- and Z-eigenvalues of real even-order supersymmetric tensors

In this paper, we compute the H- and Z-eigenvalues of real even-order supersymmetric tensors by using the adaptive cubic regularization algorithm. First, the equation of eigenvalues of the tensor is represented by a spherically constrained optimization problem. Owing to the nice geometry of the spherical constraint, we minimize the objective function and preserve the constraint in an alternating way. The objective function of the optimization model is approximated by a cubic function with a tunable parameter, which is solved inexactly to obtain a trial step. Then the Cayley transform is applied to the trial step. Based on the ratio of actual and predicted reductions, a parameter is regulated to make sure that the cubic function is a good estimation of the original objective function. Finally we obtain an adaptive cubic regularization algorithm for computing an eigenvalue of a tensor (ACRCET). Furthermore, we prove that the sequence of iterations generated by ACRCET converges to an eigenvalue of a given tensor globally. In order to improve the computational efficiency, we propose a fast computing skill for Txr−2 which is the matrix-valued product of a hypergraph related tensor T and a vector x. Numerical experiments illustrate that the fast computing skill for Txr−2 is efficient and ACRCET is effective when computing H- and Z-eigenvalues of real even-order supersymmetric tensors.

Eigenproblem of tensors - a geometrical viewpoint

The classical eigenproblem focuses on eigenvalues and eigenvectors of linear operators acting on a vector space. The matrix representation of the problem has been extended towards multidimensional arrays, with various applications. Another extension addresses invariant subspaces of multilinear operators in Banach spaces. The generalization of the eigenproblem for tensors is still a challenging issue. We investigate eigenproblems of supersymmetric tensors on Riemannian manifolds, emerging from the initial proper definitions, with the proposed extensions.

Power Function Method for Finding the Spectral Radius of Weakly Irreducible Nonnegative Tensors

Since the eigenvalue problem of real supersymmetric tensors was proposed, there have been many results regarding the numerical algorithms for computing the spectral radius of nonnegative tensors, among which the NQZ method is the most studied. However, the NQZ method is only suitable for calculating the spectral radius of a special weakly primitive tensor, or a weakly irreducible primitive tensor that satisfies certain conditions. In this paper, by means of diagonal similarrity transformation of tensors, we construct a numerical algorithm for computing the spectral radius of nonnegative tensors with the aid of power functions. This algorithm is suitable for the calculation of the spectral radius of all weakly irreducible nonnegative tensors. Furthermore, it is efficient and can be widely applied.

A large-N tensor model with four supercharges

We study a supersymmetric tensor model with four supercharges and O(N)3 global symmetry. The model is based on a chiral scalar superfield with three indices and quartic tetrahedral interaction in the superpotential, which is relevant below three dimensions. In the large-N limit the model is dominated by melonic diagrams. We solve the Dyson-Schwinger equations in superspace for generic d and extract the dimension of the chiral field and the dimensions of bilinear operators transforming in various representations of O(N)3. We find that all operator dimensions are real and above the unitarity bound for 1 < d < 3. Our results also agree with perturbative results in 3 − ε expansion. Finally, we extract the large spin behaviour of bilinear operators and discuss the connection with lightcone bootstrap.

Optical fiber intrusion signal unmixing by nonorthogonal principal skewness analysis

The optical fiber intrusion signal detection technology adopts the distributed optical fiber as sensor to monitor and identify perimeter intrusion signals. Due to the diversity of perimeter intrusion, the optical fiber intrusion signals are composed of various types of pure signals. Therefore, direct identification on mixed signals will cause the system performance degradation. A more effective method is to first unmix the mixed signal to obtain each pure signal component, and then perform signal identification. In this paper, nonorthogonal principal skewness analysis (NPSA) based unmixing algorithm is proposed. By introducing supersymmetric tensors, the mixed fiber signal unmixing problem is transformed into the skewness analysis problem, and then the non-orthogonal solution is further solved which significantly improves the accuracy of solving pure signal components. The effectiveness of the proposed algorithm is verified by actual data experiments.

Analysis of Hypergraph Signals via High-Order Total Variation

Beyond pairwise relationships, interactions among groups of agents do exist in many real-world applications, but they are difficult to capture by conventional graph models. Generalized from graphs, hypergraphs have been introduced to describe such high-order group interactions. Inspired by graph signal processing (GSP) theory, an existing hypergraph signal processing (HGSP) method presented a spectral analysis framework relying on the orthogonal CP decomposition of adjacency tensors. However, such decomposition may not exist even for supersymmetric tensors. In this paper, we propose a high-order total variation (HOTV) form of a hypergraph signal (HGS) as its smoothness measure, which is a hyperedge-wise measure aggregating all signal values in each hyperedge instead of a pairwise one in most existing work. Further, we propose an HGS analysis framework based on the Tucker decomposition of the hypergraph Laplacian induced by the aforementioned HOTV. We construct an orthonormal basis from the HOTV, by which a new spectral transformation of the HGS is introduced. Then, we design hypergraph filters in both vertex and spectral domains correspondingly. Finally, we illustrate the advantages of the proposed framework by applications in label learning.

Webs of type Q

Howe dualities lead to diagrammatic categories which describe the representations of Lie-type objects as a monoidal category (that is, via generators and relations). Applying this philosophy to the type Q Howe duality of Cheng–Wang and Sergeev, we introduce diagrammatic web supercategories of type Q via generators and relations and show they describe the full subcategory of supermodules for the Lie superalgebra of type Q given by the tensor products of supersymmetric tensor powers of the natural supermodule.

Odd-dimensional self-duality for non-Abelian tensor-multiplet in D = 3 + 2 as master theory of integrable-systems

We present N=2 supersymmetric non-Abelian duality-symmetry between a tensor multiplet and an extra vector multiplet in D=3+2 dimensions. Our system has the Yang-Mills (YM) vector multiplet (AμI,λI), a tensor-multiplet (BμνI,χI,φI), and an extra vector multiplet (CμI,ρI,σI). The index I=1,2,⋯,dimG is for the adjoint representation of a non-Abelian group G. The AμI is the conventional YM gauge field, BμνI is a non-Abelian tensor field, while φI and σI are scalar fields. The λI,χI and ρI are Majorana fermions in the 2 of Sp(1). The BμνI and CμI-fields have their respective field-strengths defined by GμνρI≡+3D⌊⌈μBνρ⌋⌉I+3fIJKF⌊⌈μνJCρ⌋⌉K and HμνI≡+2D⌊⌈μCν⌋⌉I+mBμνI+fIJKϕJHμνK−fIJKσJFμνK. The duality relationship is HμνI=(1/6)ϵμνρστGρστI−(1/2)fIJK(λ‾JγμνχK), with its super-partner relationships: φI=−σI,χI=−ρI. Since HστI contains mBστI linearly, this is a ‘massive’ self-dual relationship. Interestingly, the closure of supersymmetries shows the intrinsic global scale symmetry: δζ(BμνI,χI,φI,CμI,ρI,σI)=+mζ(BμνI,χI,φI,CμI,ρI,σI). By certain dimensional-reduction scheme into D=2+2, we show that self-dual supersymmetric tensor multiplet is generated. We deduce that our present theory in D=3+2 can serve as the underlying ‘Master Theory’ of a similar system in D=2+2.

NPSA: Nonorthogonal Principal Skewness Analysis.

Principal skewness analysis (PSA) has been introduced for feature extraction in hyperspectral imagery. As a thirdorder generalization of principal component analysis (PCA), its solution of searching for the local maximum skewness direction is transformed into the problem of calculating the eigenpairs (the eigenvalues and the corresponding eigenvectors) of a coskewness tensor. By combining a fixed-point method with an orthogonal constraint, the new eigenpairs are prevented from converging to the same previously determined maxima. However, in general, the eigenvectors of the supersymmetric tensor are not inherently orthogonal, which implies that the results obtained by the search strategy used in PSA may unavoidably deviate from the actual eigenpairs. In this paper, we propose a new nonorthogonal search strategy to so lve this problem and the new algorithm is named nonorthogonal principal skewness analysis (NPSA). The contribution of NPSA lies in the finding that the search space of the eigenvector to be determined can be enlarged by using the orthogonal complement of the Kronecker product of the previous eigenvector with itself, instead of its orthogonal complement space. We also give a detailed theoretical proof on why we can obtain the more accurate eigenpairs through the new search strategy by comparison with PSA. In addition, after some algebraic derivations, the complexity of the presented algorithm is also greatly reduced. Experiments with both simulated data and real multi/hyperspectral imagery demonstrate its validity in feature extraction.

An algorithm for arbitrary–order cumulant tensor calculation in a sliding window of data streams

Abstract High-order cumulant tensors carry information about statistics of non-normally distributed multivariate data. In this work we present a new efficient algorithm for calculation of cumulants of arbitrary orders in a sliding window for data streams. We show that this algorithm offers substantial speedups of cumulant updates compared with the current solutions. The proposed algorithm can be used for processing on-line high-frequency multivariate data and can find applications, e.g., in on-line signal filtering and classification of data streams. To present an application of this algorithm, we propose an estimator of non-Gaussianity of a data stream based on the norms of high order cumulant tensors. We show how to detect the transition from Gaussian distributed data to non-Gaussian ones in a data stream. In order to achieve high implementation efficiency of operations on super-symmetric tensors, such as cumulant tensors, we employ a block structure to store and calculate only one hyper-pyramid part of such tensors.

An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensors

A new eigenvalue localization set for tensors is given, and proved to be tighter than those in [Qi L. Eigenvalues of a real supersymmetric tensor. J. Symbolic Comput. 2005;40:1302–1324] and [Li CQ, Li YT, Kong X. New eigenvalue inclusion sets for tensors. Numer. Linear Algebra Appl. 2014;21:39–50]. Based on this set, we give two checkable sufficient conditions of the positive definiteness of tensors, and two checkable sufficient conditions of the positive semi-definiteness of tensors.

A new eigenvalue inclusion set for tensors and its applications

A new tensor eigenvalue inclusion set is given, and proved to be tighter than those in L.Q. Qi (2005) [18] and C.Q. Li, Y.T. Li, X. Kong (2014) [12]. In addition, we study the eigenvalues lying on the boundary of the eigenvalue inclusion set provided by Qi (2005) for weakly irreducible tensors. As applications, we give new bounds for the spectral radius of nonnegative tensors, some sufficient conditions for a tensor to be a strong M-tensor and some inequalities to identify the positive definiteness for an even-order real supersymmetric tensor.

Probability Bounds for Polynomial Functions in Random Variables

Random sampling is a simple but powerful method in statistics and in the design of randomized algorithms. In a typical application, random sampling can be applied to estimate an extreme value, say maximum, of a function f over a set S ⊆ ℝn. To do so, one may select a simpler (even finite) subset S0 ⊆ S, randomly take some samples over S0 for a number of times, and pick the best sample. The hope is to find a good approximate solution with reasonable chance. This paper sets out to present a number of scenarios for f, S and S0 where certain probability bounds can be established, leading to a quality assurance of the procedure. In our setting, f is a multivariate polynomial function. We prove that if f is a d-th order homogeneous polynomial in n variables and F is its corresponding super-symmetric tensor, and ξi (i = 1, 2, …, n) are i.i.d. Bernoulli random variables taking 1 or −1 with equal probability, then Prob{f(ξ1, ξ2, …, ξn) ≥ τn−d/2 ‖F‖1} ≥ θ, where τ, θ &gt; 0 are two universal constants and ‖·‖1 denotes the summation of the absolute values of all its entries. Several new inequalities concerning probabilities of the above nature are presented in this paper. Moreover, we show that the bounds are tight in most cases. Applications of our results in optimization are discussed as well.

A sequential subspace projection method for extreme Z‐eigenvalues of supersymmetric tensors

SummaryZ‐eigenvalues of tensors, especially extreme ones, are quite useful and are related to many problems, such as automatic control, quantum physics, and independent component analysis. For supersymmetric tensors, calculating the smallest/largest Z‐eigenvalue is equivalent to solving a global minimization/maximization problem of a homogenous polynomial over the unit sphere. In this paper, we utilize the sequential subspace projection method (SSPM) to find extreme Z‐eigenvalues and the corresponding Z‐eigenvectors. The main idea of SSPM is to form a 2‐dimensional subspace at the current point and then solve the original optimization problem in the subspace. SSPM benefits from the fact that the 2‐dimensional subproblem can be solved by a direct method. Global convergence and linear convergence are established for supersymmetric tensors under certain assumptions. Preliminary numerical results over several testing problems show that SSPM is very promising. Besides, the globalization strategy of random phase can be easily incorporated into SSPM, which promotes the ability to find extreme Z‐eigenvalues. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Criterions for the positive definiteness of real supersymmetric tensors

The positive definiteness of an even-degree homogeneous polynomial form f ( x ) plays an important role in the stability study of nonlinear autonomous systems via Lyapunov's direct method in automa...

SuperMatching: Feature Matching Using Supersymmetric Geometric Constraints

Feature matching is a challenging problem at the heart of numerous computer graphics and computer vision applications. We present the SuperMatching algorithm for finding correspondences between two sets of features. It does so by considering triples or higher order tuples of points, going beyond the pointwise and pairwise approaches typically used. SuperMatching is formulated using a supersymmetric tensor representing an affinity metric that takes into account feature similarity and geometric constraints between features: Feature matching is cast as a higher order graph matching problem. SuperMatching takes advantage of supersymmetry to devise an efficient sampling strategy to estimate the affinity tensor, as well as to store the estimated tensor compactly. Matching is performed by an efficient higher order power iteration approach that takes advantage of this compact representation. Experiments on both synthetic and real data show that SuperMatching provides more accurate feature matching than other state-of-the-art approaches for a wide range of 2D and 3D features, with competitive computational cost.

Criterions for the positive definiteness of real supersymmetric tensors

The positive definiteness of an even-degree homogeneous polynomial form f(x) plays an important role in the stability study of nonlinear autonomous systems via Lyapunov’s direct method in automatic control, and the positive definiteness of f(x) is equivalent to that of an even-order supersymmetric tensor which defines f(x). In this paper, we provide some criterions for identifying the positive definiteness of an even-order real supersymmetric tensor. Moreover, an iterative algorithm for identifying the positive definiteness of an even-order real supersymmetric tensor is obtained. Numerical examples are given to verify the corresponding results.

New eigenvalue inclusion sets for tensors

Two new eigenvalue inclusion sets for tensors are established. It is proved that the new eigenvalue inclusion sets are tighter than that in Qi's paper “Eigenvalues of a real supersymmetric tensor”. As applications, upper bounds for the spectral radius of a nonnegative tensor are obtained, and it is proved that these upper bounds are sharper than that in Yang's paper “Further results for Perron–Frobenius theorem for nonnegative tensors”. And some sufficient conditions of the positive definiteness for an even-order real supersymmetric tensor are given. Copyright © 2012 John Wiley & Sons, Ltd.

Self-dual non-Abelian [formula omitted] tensor multiplet in [formula omitted] dimensions

We present a self-dual non-Abelian N=1 supersymmetric tensor multiplet in D=2+2 space–time dimensions. Our system has three on-shell multiplets: (i) The usual non-Abelian Yang–Mills multiplet (AμI,λI), (ii) a non-Abelian tensor multiplet (BμνI,χI,φI), and (iii) an extra compensator vector multiplet (CμI,ρI). Here the index I is for the adjoint representation of a non-Abelian gauge group. The duality symmetry relations are GμνρI=−ϵμνρσ∇σφI, FμνI=+(1/2)ϵμνρσFρσI, and HμνI=+(1/2)ϵμνρσHρσI, where G and H are respectively the field strengths of B and C. The usual problem with the coupling of the non-Abelian tensor is avoided by non-trivial Chern–Simons terms in the field strengths GμνρI and HμνI. For an independent confirmation, we re-formulate the component results in superspace. As applications of embedding integrable systems, we show how the N=2, r=3 and N=3, r=4 flows of generalized Korteweg–de Vries equations are embedded into our system.