Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector

Let A = \pmatrix A_{11} & A_{12} \cr A_{21} & A_{22}\cr\pmatrix \in M_n, where $A_{11} \in M_m$ with $m \le n/2$, be such that the numerical range of $A$ lies in the set \{e^{i\varphi} z \in \IC: |\Im z| \le (\Re z) \tan \alpha\}, for some $\varphi \in [0, 2\pi)$ and $\alpha \in [0, \pi/2)$. We obtain the optimal containment region for the generalized eigenvalue $\lambda$ satisfying \lambda \pmatrix A_{11} & 0 \cr 0 & A_{22}\cr\pmatrix x = \pmatrix 0 & A_{12} \cr A_{21} & 0\cr\pmatrix x \quad \hbox{for some nonzero} x \in \IC^n, and the optimal eigenvalue containment region of the matrix $I_m - A_{11}^{-1}A_{12} A_{22}^{-1}A_{21}$ in case $A_{11}$ and $A_{22}$ are invertible. From this result, one can show $|\det(A)| \le \sec^{2m}(\alpha) |\det(A_{11})\det(A_{22})|$. In particular, if $A$ is a accretive-dissipative matrix, then $|\det(A)| \le 2^m |\det(A_{11})\det(A_{22})|$. These affirm some conjectures of Drury and Lin.Comment: 6 pages, to appear in Journal of Mathematical Analysi

Investigating Linguistic Pattern Ordering in Hierarchical Natural Language Generation

Natural language generation (NLG) is a critical component in spoken dialogue system, which can be divided into two phases: (1) sentence planning: deciding the overall sentence structure, (2) surface realization: determining specific word forms and flattening the sentence structure into a string. With the rise of deep learning, most modern NLG models are based on a sequence-to-sequence (seq2seq) model, which basically contains an encoder-decoder structure; these NLG models generate sentences from scratch by jointly optimizing sentence planning and surface realization. However, such simple encoder-decoder architecture usually fail to generate complex and long sentences, because the decoder has difficulty learning all grammar and diction knowledge well. This paper introduces an NLG model with a hierarchical attentional decoder, where the hierarchy focuses on leveraging linguistic knowledge in a specific order. The experiments show that the proposed method significantly outperforms the traditional seq2seq model with a smaller model size, and the design of the hierarchical attentional decoder can be applied to various NLG systems. Furthermore, different generation strategies based on linguistic patterns are investigated and analyzed in order to guide future NLG research work.Comment: accepted by the 7th IEEE Workshop on Spoken Language Technology (SLT 2018). arXiv admin note: text overlap with arXiv:1808.0274

Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations

Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of closed half planes (of complex numbers). As a result, it is always a convex set in $\mathcal C$. Moreover, the higher rank numerical range of a normal matrix is a convex polygon determined by the eigenvalues. These two consequences confirm the conjectures of Choi et al. on the subject. In addition, the results are used to derive a formula for the optimal upper bound for the dimension of a totally isotropic subspace of a square matrix, and verify the solvability of certain matrix equations.Comment: 10 pages. To appear in Proceedings of the American Mathematical Societ