用群表示理论构建张量半定规划，解决低秩张量补全问题
Tensor-Tensor Products, Group Representations, and Semidefinite Programming
- 通过矩阵M定义张量乘积，关联群表示理论
- 建立张量半定规划框架，求解低秩补全问题
- 适用于对称性结构的优化与非负二次型分析
星号族张量-张量乘积（$\ar{M}$-product）将线性代数的许多性质推广到三阶张量。本文研究在该乘积下的正定性与半定规划问题，关键发现是矩阵M的选择与底层群作用的表示理论密切相关。借助此框架，带有$\ar{M}$-乘积的三阶张量成为研究不变半定规划的自然设定。作为应用，本文刻画了某些非负二次型，并解决了低秩张量补全问题。
原文摘要 · Abstract (English)
The $\star_M$-family of tensor-tensor products is a framework which generalizes many properties from linear algebra to third order tensors. Here, we investigate positive semidefiniteness and semidefinite programming under the $\star_M$-product. Critical to our investigation is a connection between the choice of matrix M in the $\star_M$-product and the representation theory of an underlying group action. Using this framework, third order tensors equipped with the $\star_M$-product are a natural setting for the study of invariant semidefinite programs. As applications of the M-SDP framework, we provide a characterization of certain nonnegative quadratic forms and solve low-rank tensor completion problems.
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