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The traditional definition of co-salient object detection (CoSOD) task is to segment the common salient objects in a group of relevant images. This definition is based on an assumption of group consensus consistency that is not always reasonable in the open-world setting, which results in robustness issue in the model when dealing with irrelevant images in the inputting image group under the open-word scenarios. To tackle this problem, we introduce a group selective exchange-masking (GSEM) approach for enhancing the robustness of the CoSOD model. GSEM takes two groups of images as input, each containing different types of salient objects. Based on the mixed metric we designed, GSEM selects a subset of images from each group using a novel learning-based strategy, then the selected images are exchanged. To simultaneously consider the uncertainty introduced by irrelevant images and the consensus features of the remaining relevant images in the group, we designed a latent variable generator branch and CoSOD transformer branch. The former is composed of a vector quantised-variational autoencoder to generate stochastic global variables that model uncertainty. The latter is designed to capture correlation-based local features that include group consensus. Finally, the outputs of the two branches are merged and passed to a transformer-based decoder to generate robust predictions. Taking into account that there are currently no benchmark datasets specifically designed for open-world scenarios, we constructed three open-world benchmark datasets, namely OWCoSal, OWCoSOD, and OWCoCA, based on existing datasets. By breaking the group-consistency assumption, these datasets provide effective simulations of real-world scenarios and can better evaluate the robustness and practicality of models.

In this paper, we present a method to certify the approximation quality of a low rank tensor to a given third order symmetric tensor. Under mild assumptions, best low rank approximation is attained if a control parameter is zero or quantified quasi-optimal low rank approximation is obtained if the control parameter is positive.This is based on a primal-dual method for computing a low rank approximation for a given tensor. The certification is derived from the global optimality of the primal and dual problems, and is characterized by easily checkable relations between the primal and the dual solutions together with another rank condition. The theory is verified theoretically for orthogonally decomposable tensors as well as numerically through examples in the general case., Comment: 46pages

Due to the multi-linearity of tensors, most algorithms for tensor optimization problems are designed based on the block coordinate descent method. Such algorithms are widely employed by practitioners for their implementability and effectiveness. However, these algorithms usually suffer from the lack of theoretical guarantee of global convergence and analysis of convergence rate. In this paper, we propose a block coordinate descent type algorithm for the low rank partially orthogonal tensor approximation problem and analyse its convergence behaviour. To achieve this, we carefully investigate the variety of low rank partially orthogonal tensors and its geometric properties related to the parameter space, which enable us to locate KKT points of the concerned optimization problem. With the aid of these geometric properties, we prove without any assumption that: (1) Our algorithm converges globally to a KKT point; (2) For any given tensor, the algorithm exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual $O(1/k)$ for first order methods in nonconvex optimization; {(3)} For a generic tensor, our algorithm converges $R$-linearly., Comment: 60 pages

In this article, nondegeneracy of singular vector tuples, Z-eigenvectors and eigenvectors of tensors is studied. They have found many applications in diverse areas. The main results are: (i) each (Z-)eigenvector/singular vector tuple of a generic tensor is nondegenerate, and (ii) each nonzero Z-eigenvector/singular vector tuple of an orthogonally decomposable tensor is nondegenerate., Comment: Sci China Math, to appear

Low rank orthogonal tensor approximation (LROTA) is an important problem in tensor computations and their applications. A classical and widely used algorithm is the alternating polar decomposition method (APD). In this article, an improved version iAPD of the classical APD is proposed. For the first time, all the following four fundamental properties are established for iAPD: (i) the algorithm converges globally and the whole sequence converges to a KKT point without any assumption; (ii) it exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual $O(1/k)$ for first order methods in optimization; (iii) more importantly, it converges $R$-linearly for a generic tensor without any assumption; (iv) for almost all LROTA problems, iAPD reduces to APD after finitely many iterations if it converges to a local minimizer., Comment: 39 pages

Biquadratic tensors play a central role in many areas of science. Examples include elasticity tensor and Eshelby tensor in solid mechanics, and Riemann curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse.

We establish two inequalities for the nuclear norm and the spectral norm of tensor products. The first inequality indicates that the nuclear norm of the square matrix is a matrix norm. We extend the concept of matrix norm to tensor norm. We show that the $1$-norm, the Frobenius norm and the nuclear norm of tensors are tensor norms, but the infinity norm and the spectral norm of tensors are not tensor norms. We introduce the cubic power for a general third order tensor, and show that a Gelfand formula holds for a general third order tensor. In that formula, for any norm, a common spectral radius-like limit exists for that third order tensor. We call such a limit the Gelfand limit. The Gelfand limit is zero if the third order tensor is nilpotent, and is one or zero if the third order tensor is idempotent. The Gelfand limit is not greater than any tensor norm of that third order tensor, and the cubic power of that third order tensor tends to zero as the power increases to infinity if and only if the Gelfand limit is less than one. The cubic power and the Gelfand limit can be extended to any higher odd order tensors.

The spectral norm and the nuclear norm of a third order tensor play an important role in the tensor completion and recovery problem. We show that the spectral norm of a third order tensor is equal to the square root of the spectral norm of three positive semi-definite biquadratic tensors, and the square roots of the nuclear norms of those three positive semi-definite biquadratic tensors are lower bounds of the nuclear norm of that third order tensor. This provides a way to estimate and to evaluate the spectral norm and the nuclear norm of that third order tensor. Some upper and lower bounds for the spectral norm and nuclear norm of a third order tensor, by spectral radii and nuclear norms of some symmetric matrices, are presented.

In 1973, Gauger proposed a generator-relation method and a duality theory for two-step nilpotent Lie algebras. Based upon these, he classified two-step nilpotent Lie algebras of dimension $8$. In 1999, Galitski and Timashev continued this approach to classify two-step nilpotent Lie algebras of dimension $9$. Their results were partially improved by Ren and Zhu in 2011, Yan and Deng in 2013. Some decomposable two-step nilpotent Lie algebras were excluded in the case of dimension $8$. In this paper, we define generating hypergraph for a two-step nilpotent Lie algebra. The two-step nilpotent Lie algebra is decomposable if and only if its generating hypergraph is not connected under certain bases. Using this result, we identify some decomposable two-step nilpotent Lie algebras in dimension $9$. We give a direct proof that the five two-step nilpotent Lie algebras for dimension $8$, classified by Ren and Zhu in 2011, are all indecomposable. We also introduce a conventional nomenclature for two-step nilpotent Lie algebras of dimension $n = 8, 9$, classified by Ren, Zhu, Yan and Deng, etc.

In this paper, we study the polynomial optimization problem of multi-forms over the intersection of the multi-spheres and the nonnegative orthants. This class of problems is NP-hard in general, and includes the problem of finding the best nonnegative rank-one approximation of a given tensor. A Positivstellensatz is given for this class of polynomial optimization problems, based on which a globally convergent hierarchy of doubly nonnegative (DNN) relaxations is proposed. A (zero-th order) DNN relaxation method is applied to solve these problems, resulting in linear matrix optimization problems under both the positive semidefinite and nonnegative conic constraints. A worst case approximation bound is given for this relaxation method. Then, the recent solver SDPNAL+ is adopted to solve this class of matrix optimization problems. Typically, the DNN relaxations are tight, and hence the best nonnegative rank-one approximation of a tensor can be revealed frequently. Extensive numerical experiments show that this approach is quite promising., Comment: 27 pages

In this article, we give a proof for that the cardinality of a function basis of the invariants for a finite dimensional real vector space by a compact group is lower bounded by the intuitive difference of the dimensions of the vector space and the group. An application is given to the space of third order three dimensional symmetric and traceless tensors, showing that each minimal integrity basis is an irreducible function basis, which solves a problem in applied mechanics., Comment: arXiv admin note: substantial text overlap with arXiv:1712.02087

Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless tensor has four invariants with degrees two, four, six and ten respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no polynomial syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.

In 1907, Oskar Perron showed that a positive square matrix has a unique largest positive eigenvalue with a positive eigenvector. This result was extended to irreducible nonnegative matrices by Geog Frobenius in 1912, and to irreducible nonnegative tensors and weakly irreducible nonnegative tensors recently. This result is a fundamental result in matrix theory and has found wide applications in probability theory, internet search engines, spectral graph and hypergraph theory, etc. In this paper, we give a necessary and sufficient condition for the existence of such a positive eigenvector, i.e., a positive Perron vector, for a nonnegative tensor. We show that every nonnegative tensor has a canonical nonnegative partition form, from which we introduce strongly nonnegative tensors. A tensor is called strongly nonnegative, if the spectral radius of each genuine weakly irreducible block is equal to the spectral radius of the tensor, which is strictly larger than the spectral radius of any other block. We prove that a nonnegative tensor has a positive Perron vector if and only if it is strongly nonnegative. The proof is nontrivial. Numerical results for finding a positive Perron vector are reported., Comment: A slightly revised version to the second

A tensor $\mathcal T\in \mathbb T(\mathbb C^n,m+1)$, the space of tensors of order $m+1$ and dimension $n$ with complex entries, has $nm^{n-1}$ eigenvalues (counted with algebraic multiplicities). The inverse eigenvalue problem for tensors is a generalization of that for matrices. Namely, given a multiset $S\in \mathbb C^{nm^{n-1}}/\mathfrak{S}(nm^{n-1})$ of total multiplicity $nm^{n-1}$, is there a tensor in $\mathbb T(\mathbb C^n,m+1)$ such that the multiset of eigenvalues of $\mathcal{T}$ is exact $S$? The solvability of the inverse eigenvalue problem for tensors is studied in this paper. With tools from algebraic geometry, it is proved that the necessary and sufficient condition for this inverse problem to be generically solvable is $m=1,\ \text{or }n=2,\ \text{or }(n,m)=(3,2),\ (4,2),\ (3,3)$., Comment: 20 pages

We study in this article multiplicities of eigenvalues of tensors. There are two natural multiplicities associated to an eigenvalue $\lambda$ of a tensor: algebraic multiplicity $\operatorname{am}(\lambda)$ and geometric multiplicity $\operatorname{gm}(\lambda)$. The former is the multiplicity of the eigenvalue as a root of the characteristic polynomial, and the latter is the dimension of the eigenvariety (i.e., the set of eigenvectors) corresponding to the eigenvalue. We show that the algebraic multiplicity could change along the orbit of tensors by the orthogonal linear group action, while the geometric multiplicity of the zero eigenvalue is invariant under this action, which is the main difficulty to study their relationships. However, we show that for a generic tensor, every eigenvalue has a unique (up to scaling) eigenvector, and both the algebraic multiplicity and geometric multiplicity are one. In general, we suggest for an $m$-th order $n$-dimensional tensor the relationship \[ \operatorname{am}(\lambda)\geq \operatorname{gm}(\lambda)(m-1)^{\operatorname{gm}(\lambda)-1}. \] We show that it is true for serveral cases, especially when the eigenvariety contains a linear subspace of dimension $\operatorname{gm}(\lambda)$ in coordinate form. As both multiplicities are invariants under the orthogonal linear group action in the matrix counterpart, this generalizes the classical result for a matrix: the algebraic mutliplicity is not smaller than the geometric multiplicity.

For a $3$-tensor of dimensions $I_1\times I_2\times I_3$, we show that the nuclear norm of its every matrix flattening is a lower bound of the tensor nuclear norm, and which in turn is upper bounded by $\sqrt{\min\{I_i : i\neq j\}}$ times the nuclear norm of the matrix flattening in mode $j$ for all $j=1,2,3$. The results can be generalized to $N$-tensors with any $N\geq 3$. Both the lower and upper bounds for the tensor nuclear norm are sharp in the case $N=3$. A computable criterion for the lower bound being tight is given as well.

Yuan's theorem of the alternative is an important theoretical tool in optimization, which provides a checkable certificate for the infeasibility of a strict inequality system involving two homogeneous quadratic functions. In this paper, we provide a tractable extension of Yuan's theorem of the alternative to the symmetric tensor setting. As an application, we establish that the optimal value of a class of nonconvex polynomial optimization problems with suitable sign structure (or more explicitly, with essentially non-positive coefficients) can be computed by a related convex conic programming problem, and the optimal solution of these nonconvex polynomial optimization problems can be recovered from the corresponding solution of the convex conic programming problem. Moreover, we obtain that this class of nonconvex polynomial optimization problems enjoy exact sum-of-squares relaxation, and so, can be solved via a single semidefinite programming problem., Comment: acceted by Journal of Optimization Theory and its application, UNSW preprint, 22 pages

In this paper, we consider convergence properties of a second order Markov chain. Similar to a column stochastic matrix is associated to a Markov chain, a so called {\em transition probability tensor} $P$ of order 3 and dimension $n$ is associated to a second order Markov chain with $n$ states. For this $P$, define $F_P$ as $F_P(x):=Px^{2}$ on the $n-1$ dimensional standard simplex $\Delta_n$. If 1 is not an eigenvalue of $\nabla F_P$ on $\Delta_n$ and $P$ is irreducible, then there exists a unique fixed point of $F_P$ on $\Delta_n$. In particular, if every entry of $P$ is greater than $\frac{1}{2n}$, then 1 is not an eigenvalue of $\nabla F_P$ on $\Delta_n$. Under the latter condition, we further show that the second order power method for finding the unique fixed point of $F_P$ on $\Delta_n$ is globally linearly convergent and the corresponding second order Markov process is globally $R$-linearly convergent., Comment: 16 pages, 3 figures

We give some graph theoretical formulas for the trace $Tr_k(\mathbb {T})$ of a tensor $\mathbb {T}$ which do not involve the differential operators and auxiliary matrix. As applications of these trace formulas in the study of the spectra of uniform hypergraphs, we give a characterization (in terms of the traces of the adjacency tensors) of the $k$-uniform hypergraphs whose spectra are $k$-symmetric, thus give an answer to a question raised in [3]. We generalize the results in [3, Theorem 4.2] and [5, Proposition 3.1] about the $k$-symmetry of the spectrum of a $k$-uniform hypergraph, and answer a question in [5] about the relation between the Laplacian and signless Laplacian spectra of a $k$-uniform hypergraph when $k$ is odd. We also give a simplified proof of an expression for $Tr_2(\mathbb {T})$ and discuss the expression for $Tr_3(\mathbb {T})$., Comment: 23 pages

In this paper, we introduce the class of cored hypergraphs and power hypergraphs, and investigate the properties of their Laplacian H-eigenvalues. From an ordinary graph, one may generate a $k$-uniform hypergraph, called the $k$th power hypergraph of that graph. Power hypergraphs are cored hypergraphs, but not vice versa. Hyperstars, hypercycles, hyperpaths are special cases of power hypergraphs, while sunflowers are a subclass of cored hypergraphs, but not power graphs in general. We show that the largest Laplacian H-eigenvalue of an even-uniform cored hypergraph is equal to its largest signless Laplacian H-eigenvalue. Especially, we find out these largest H-eigenvalues for even-uniform sunflowers. Moreover, we show that the largest Laplacian H-eigenvalue of an odd-uniform sunflower, hypercycle and hyperpath is equal to the maximum degree, i.e., 2. We also compute out the H-spectra of the class of hyperstars. When $k$ is odd, the H-spectra of the hypercycle of size 3 and the hyperpath of length 3 are characterized as well., Comment: 27 pages, 3 figures. arXiv admin note: text overlap with arXiv:1304.1315

In this paper, we show that the largest Laplacian H-eigenvalue of a $k$-uniform nontrivial hypergraph is strictly larger than the maximum degree when $k$ is even. A tight lower bound for this eigenvalue is given. For a connected even-uniform hypergraph, this lower bound is achieved if and only if it is a hyperstar. However, when $k$ is odd, it happens that the largest Laplacian H-eigenvalue is equal to the maximum degree, which is a tight lower bound. On the other hand, tight upper and lower bounds for the largest signless Laplacian H-eigenvalue of a $k$-uniform connected hypergraph are given. For a connected $k$-uniform hypergraph, the upper (respectively lower) bound of the largest signless Laplacian H-eigenvalue is achieved if and only if it is a complete hypergraph (respectively a hyperstar). The largest Laplacian H-eigenvalue is always less than or equal to the largest signless Laplacian H-eigenvalue. When the hypergraph is connected, the equality holds here if and only if $k$ is even and the hypergraph is odd-bipartite., Comment: 26 pages, 3 figures

In this paper, we show that the eigenvectors of the zero Laplacian and signless Lapacian eigenvalues of a $k$-uniform hypergraph are closely related to some configured components of that hypergraph. We show that the components of an eigenvector of the zero Laplacian or signless Lapacian eigenvalue have the same modulus. Moreover, under a {\em canonical} regularization, the phases of the components of these eigenvectors only can take some uniformly distributed values in $\{\{exp}(\frac{2j\pi}{k})\;|\;j\in [k]\}$. These eigenvectors are divided into H-eigenvectors and N-eigenvectors. Eigenvectors with minimal support is called {\em minimal}. The minimal canonical H-eigenvectors characterize the even (odd)-bipartite connected components of the hypergraph and vice versa, and the minimal canonical N-eigenvectors characterize some multi-partite connected components of the hypergraph and vice versa., Comment: 22 pages, 3 figures

We first show that the eigenvector of a tensor is well-defined. The differences between the eigenvectors of a tensor and its E-eigenvectors are the eigenvectors on the nonsingular projective variety $\mathbb S=\{\mathbf x\in\mathbb P^n\;|\;\sum\limits_{i=0}^nx_i^2=0\}$. We show that a generic tensor has no eigenvectors on $\mathbb S$. Actually, we show that a generic tensor has no eigenvectors on a proper nonsingular projective variety in $\mathbb P^n$. By these facts, we show that the coefficients of the E-characteristic polynomial are algebraically dependent. Actually, a certain power of the determinant of the tensor can be expressed through the coefficients besides the constant term. Hence, a nonsingular tensor always has an E-eigenvector. When a tensor $\mathcal T$ is nonsingular and symmetric, its E-eigenvectors are exactly the singular points of a class of hypersurfaces defined by $\mathcal T$ and a parameter. We give explicit factorization of the discriminant of this class of hypersurfaces, which completes Cartwright and Strumfels' formula. We show that the factorization contains the determinant and the E-characteristic polynomial of the tensor $\mathcal T$ as irreducible factors., Comment: 17 pages

The geometric measure of entanglement of a pure state, defined by its distance to the set of pure separable states, is extended to multipartite mixed states. We characterize the nearest disentangled mixed state to a given mixed state with respect to this measure by a system of equations. The entanglement eigenvalue for a mixed state is introduced. For a given mixed state, we show that its nearest disentangled mixed state is associated with its entanglement eigenvalue., Comment: 12 pages

The geometric measure of entanglement for a symmetric pure state with nonnegative amplitudes has attracted much attention. On the other hand, the spectral theory of nonnegative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of nonnegative tensors can be applied to the study of the geometric measure of entanglement for a pure state with nonnegative amplitudes. Especially, an elimination method for computing the geometric measure of entanglement for symmetric pure multipartite qubit or qutrit states with nonnegative amplitudes is given. For symmetric pure multipartite qudit states with nonnegative amplitudes, a numerical algorithm with randomization is presented and proven to be convergent. We show that for the geometric measure of entanglement for pure states with nonnegative amplitudes, the nonsymmetric ones can be converted to the symmetric ones., Comment: 21 pages

In this paper, we introduce a new class of nonnegative tensors --- strictly nonnegative tensors. A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa. We show that the spectral radius of a strictly nonnegative tensor is always positive. We give some sufficient and necessary conditions for the six well-conditional classes of nonnegative tensors, introduced in the literature, and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors. We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility. We show that for a nonnegative tensor T, there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible; and the spectral radius of T can be obtained from those spectral radii of the induced tensors. In this way, we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption. The preliminary numerical results demonstrate the feasibility and effectiveness of the proposed algorithm., Comment: 25 pages, 1 figue

We generalize the concept of the symmetric hyperdeterminants for symmetric tensors to the E-determinants for general tensors. We show that the E-determinant inherits many properties of the determinant of a matrix. These properties include: solvability of polynomial systems, the E-determinat of the composition of tensors, product formula for the E-determinant of a block tensor, Hadamard's inequality, Gersgrin's inequality and Minikowski's inequality. As a simple application, we show that if the leading coefficient tensor of a polynomial system is a triangular tensor with nonzero diagonal elements, then the system definitely has a solution. We investigate the characteristic polynomial of a tensor through the E-determinant. Explicit formulae for the coefficients of the characteristic polynomial are given when the dimension is two.