带线性约束的张量核范数最小化

中文摘要 AI 辅助

我们研究了带一般线性约束的张量核范数最小化问题，这是低秩张量优化的一个基本模型，涵盖了计算机视觉、推荐系统、收益管理和交通研究中的许多实际应用。尽管张量核范数是张量秩的凸替代，但由于评估核范数本身是NP难的，优化它仍然具有计算挑战性。我们提出了一种半定规划框架，该框架揭示了这个问题中隐藏的有限结构，尽管它看起来具有无限多个约束。特别地，我们证明了张量核范数具有一个精确的SDP表示，该表示由单位球面上的有限个点支持。基于这一表示，我们开发了两种互补的求解方法。第一种方法将未知的支持集替换为有限球面离散化，从而得到一个具有可证明的最坏情况近似保证的解。第二种方法自适应地选择球面上的点，在有限终止时返回精确最优解，否则渐近收敛到最优解。在合成张量补全实例和真实高速公路交通数据上的数值实验表明，自适应方法相对于现有方法实现了有竞争力的、且通常显著提高的恢复精度。

英文摘要

We study the tensor nuclear norm minimization with general linear constraints, a fundamental model for low-rank tensor optimization that includes many practical applications in computer vision, recommendation systems, revenue management, transportation research. Although the tensor nuclear norm is a convex surrogate for the tensor rank, optimizing it remains computationally challenging because evaluating the nuclear norm itself is NP-hard. We propose a semidefinite programming framework that exposes a finite structure hidden in this problem, even though it appears to possess infinitely many constraints. In particular, we show that the tensor nuclear norm admits an exact SDP representation supported by a finite number of points on the unit sphere. Based on this representation, we develop two complementary solution approaches. The first replaces the unknown support set by a finite sphere discretization which yields a solution with a provable worst-case approximation guarantee. The second selects spherical points adaptively, returns an exact optimum upon finite termination, and otherwise converges asymptotically to an optimal solution. Numerical experiments on synthetic tensor completion instances and real highway traffic data demonstrate that the adaptive method achieves competitive, and often substantially improved, recovery accuracy relative to existing methods.

发表机构

• The Chinese University of Hong Kong（香港中文大学）
• University of Western Ontario（韦仕敦大学）
• Shanghai University of Finance and Economics（上海财经大学）

机构由 AI 辅助整理，请以论文原文为准。