Shenglong Hu

Shenglong Hu, Zheng-Hai Huang, Liqun Qi

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.

Shenglong Hu, Zheng-Hai Huang, Chen Ling et al.

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.

Shenglong Hu, Liqun Qi

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.

Yang Wu, Shenglong Hu, Huihui Song et al.

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.

Shenglong Hu

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.