報告人: 王建軍 教授
講座日期：2020-11-19
講座時間：15:00
報告地點：騰訊會議（768 415 831）
主辦單位：數學與信息科學學院
講座人簡介：
王建軍，西南大學教授，巴渝學者特聘教授，重慶市創新創業領軍人才，重慶工業與應用數學學會副理事長，CSIAM全國大數據與人工智能專家委員會委員，美國數學評論評論員，曾獲重慶市自然科學獎勵三等獎。主要研究方向為：高維數據建模與挖掘、深度學習、壓縮感知與張量恢復、函數逼近論等。在神經網絡(深度學習)逼近復雜性和高維數據稀疏建模等方面有一定的學術積累。多次出席國際、國內重要學術會議，并應邀做大會特邀報告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，中國科學(A、F輯), 數學學報, 計算機學報，電子學報等知名專業期刊發表90余篇學術論文。主持國家自然科學基金5項，教育部科學技術重點項目1項，重慶市自然科學基金1項，主研8項國家自然、社會科學基金；現主持國家自然科學基金面上項目2項，參與國家重點基礎研究發展973計劃1項。
講座簡介：
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.
