Uniqueness of Nonnegative Tensor Approximations

We show that for a nonnegative tensor, a best nonnegative rank-r approximation is almost always unique, its best rank-one approximation may always be chosen to be a best nonnegative rank-one approximation, and that the set of nonnegative tensors with non-unique best rank-one approximations form an algebraic hypersurface. We show that the last part holds true more generally for real tensors and thereby determine a polynomial equation so that a real or nonnegative tensor which does not satisfy this equation is guaranteed to have a unique best rank-one approximation. We also establish an analogue for real or nonnegative symmetric tensors. In addition, we prove a singular vector variant of the Perron--Frobenius Theorem for positive tensors and apply it to show that a best nonnegative rank-r approximation of a positive tensor can never be obtained by deflation. As an aside, we verify that the Euclidean distance (ED) discriminants of the Segre variety and the Veronese variety are hypersurfaces and give defining equations of these ED discriminants

Studying the bound state of the BKˉB\bar{K} system in the Bethe-Salpeter formalism

In this work, we study the $B\bar{K}$ molecule in the Bethe-Salpeter (BS) equation approach. With the kernel containing one-particle-exchange diagrams and introducing two different form factors (monopole form factor and dipole form factor) in the vertex, we solve the BS equation numerically in the covariant instantaneous approximation. We investigate the isoscalar and isovector $B\bar{K}$ systems, and we find $X(5568)$ cannot be a $B\bar{K}$ molecule

SFCN-OPI: Detection and Fine-grained Classification of Nuclei Using Sibling FCN with Objectness Prior Interaction

Cell nuclei detection and fine-grained classification have been fundamental yet challenging problems in histopathology image analysis. Due to the nuclei tiny size, significant inter-/intra-class variances, as well as the inferior image quality, previous automated methods would easily suffer from limited accuracy and robustness. In the meanwhile, existing approaches usually deal with these two tasks independently, which would neglect the close relatedness of them. In this paper, we present a novel method of sibling fully convolutional network with prior objectness interaction (called SFCN-OPI) to tackle the two tasks simultaneously and interactively using a unified end-to-end framework. Specifically, the sibling FCN branches share features in earlier layers while holding respective higher layers for specific tasks. More importantly, the detection branch outputs the objectness prior which dynamically interacts with the fine-grained classification sibling branch during the training and testing processes. With this mechanism, the fine-grained classification successfully focuses on regions with high confidence of nuclei existence and outputs the conditional probability, which in turn benefits the detection through back propagation. Extensive experiments on colon cancer histology images have validated the effectiveness of our proposed SFCN-OPI and our method has outperformed the state-of-the-art methods by a large margin.Comment: Accepted at AAAI 201