Seminars on Numerical Algebra, Optimization and Data Sciences: Low-rank optimization on matrix and tensor varieties

：高斌（中国科学院）
：2024-05-31 09:00
：海韵园数理大楼686会议室

报告人：高斌（中国科学院）

时  间：2024年5月31日9:00

地  点：海韵园数理大楼686会议室

内容摘要:

In the realm of tensor optimization, low-rank tensor decomposition, particularly Tucker decomposition, stands as a pivotal technique for reducing the number of parameters and for saving storage. We embark on an exploration of Tucker tensor varieties—the set of tensors with bounded Tucker rank—in which the geometry is notably more intricate than the well-explored geometry of matrix varieties. We give an explicit parametrization of the tangent cone of Tucker tensor varieties and leverage its geometry to develop provable gradient-related line-search methods for optimization on Tucker tensor varieties. The search directions are computed from approximate projections of antigradient onto the tangent cone, which circumvents the calculation of intractable metric projections. To the best of our knowledge, this is the first work concerning geometry and optimization on Tucker tensor varieties. In practice, low-rank tensor optimization suffers from the difficulty of choosing a reliable rank parameter. To this end, we incorporate the established geometry and propose a Tucker rank-adaptive method that is capable of identifying an appropriate rank during iterations while the convergence is also guaranteed. Numerical experiments on tensor completion with synthetic and real-world datasets reveal that the proposed methods are in favor of recovering performance over other state-of-the-art methods. Moreover, the rank-adaptive method performs the best across various rank parameter selections and is indeed able to find an appropriate rank. 

个人简介：

高斌，中国科学院数学与系统科学研究院计算数学所副研究员。曾先后赴比利时法语鲁汶大学，德国明斯特大学从事博士后研究。其主要研究兴趣是最优化计算方法的理论，分析和应用，研究内容包括矩阵流形上的优化算法，机器学习中的矩阵填充问题等。他的多篇论文发表在《SIAM Journal on Optimization》，《SIAM Journal on Scientific Computing》，《SIAM Journal on Matrix Analysis and Applications》等学术期刊上。曾获中国数学会颁发的钟家庆数学奖，并于2023年获得海外优青项目。

 

联系人：黄文