The paper studies the rank-1 tensor completion problem for cubic order tensors. It shows that this problem is equivalent to a special rank-1 matrix recovery problem. Two main approaches are proposed to solve this matrix recovery problem:
Nuclear norm relaxation: This sometimes gets a rank-1 tensor completion, but may fail in other cases.
Moment relaxation: This can always get a rank-1 tensor completion or detect its nonexistence. The moment hierarchy of semidefinite programming relaxations is used for this.
For a special class of "strongly rank-1 completable" tensors, the problem can be further reduced to a rank-1 matrix completion problem. If the corresponding bipartite graph is connected, this can be solved efficiently using an iterative formula.
The paper also discusses the case of symmetric tensors, where the rank-1 tensor completion problem has additional attractive properties.
Numerical experiments are provided to demonstrate the efficiency of the proposed methods.
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ข้อมูลเชิงลึกที่สำคัญจาก
by Jinling Zhou... ที่ arxiv.org 04-15-2024
https://arxiv.org/pdf/2404.08171.pdfสอบถามเพิ่มเติม