报告人: 王建军 教授
讲座日期：2020-11-19
讲座时间：15:00
报告地点：腾讯会议（768 415 831）
主办单位：数学与信息科学学院
讲座人简介：
王建军，西南大学教授，巴渝学者特聘教授，重庆市创新创业领军人才，重庆工业与应用数学学会副理事长，CSIAM全国大数据与人工智能专家委员会委员，美国数学评论评论员，曾获重庆市自然科学奖励三等奖。主要研究方向为：高维数据建模与挖掘、深度学习、压缩感知与张量恢复、函数逼近论等。在神经网络(深度学习)逼近复杂性和高维数据稀疏建模等方面有一定的学术积累。多次出席国际、国内重要学术会议，并应邀做大会特邀报告22余次。已在IEEE Transactions on Pattern Analysis and Machine Intelligence, Applied and Computational Harmonic Analysis, Inverse Problems, Neural Networks, Signal Processing, IEEE Signal Processing letters, Journal of Computational and Applied Mathematics，ICASSP，中国科学(A、F辑), 数学学报, 计算机学报，电子学报等知名专业期刊发表90余篇学术论文。主持国家自然科学基金5项，教育部科学技术重点项目1项，重庆市自然科学基金1项，主研8项国家自然、社会科学基金；现主持国家自然科学基金面上项目2项，参与国家重点基础研究发展973计划1项。
讲座简介：
This talk focuses on the recovery of low-tubal-rank tensors from binary measurements based on tensor-tensor product (or t-product) and tensor Singular Value Decomposition (t-SVD). Two types of recovery models are considered; one is the tensor hard singular tube thresholding and the other is the tensor nuclear-norm minimization. In the case no random dither exists in the measurements, our research shows that the direction of tensor $\mathcal{X} \in \R^{n_1\times n_2\times n_3}$ with tubal rank r can be well approximated from $\Omega((n_1+n_2)n_3r)$ random Gaussian measurements. In the case nonadaptive adaptive dither exists in the measurements, it is proved that both the direction and the magnitude of $\mathcal{X}$ can be simultaneously recovered. As we will see, under the nonadaptive adaptive measurement scheme, the recovery errors of two reconstruction procedures decay at the rate of polynomial of the oversampling factor $\lambda:=m/(n_1+n_2)n_3r$,i.e., $\mathcal{O}(\lambda^{-1/6})$ and $\mathcal{O}(\lambda^{-1/4})$, respectively. In order to obtain faster decay rate, we introduce a recursive strategy and allow the dithers in quantization adaptive to previous measurements for each iterations. Under this quantization scheme, two iterative recovery algorithms are proposed which establish recovery errors decaying at the rate of exponent of the oversampling factor, i.e., $\exp(-\mathcal{O}(\lambda))$. Numerical experiments on both synthetic and real-world data sets are conducted and demonstrate the validity of our theoretical results and the superiority of our algorithms.
