»ã±¨±êÌâ (Title)£ºThe Rank-1Tensor Completion Problem£¨ÖÈÒ»ÕÅÁ¿²¹È«ÎÊÌ⣩

»ã±¨ÈË (Speaker)£ºJiawang Nie£¨University of California, San Diego£©

»ã±¨¹¦·ò (Time)£º2024Äê07ÔÂ29ÈÕ(ÖÜÒ») 10£º00

»ã±¨µØÖ· (Place)£ºÐ£±¾²¿GJ403

Ô¼ÇëÈË(Inviter)£ºÖÜ°²ÍÞ

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»ã±¨ÌáÒª£º We discuss the rank-1 tensor completion problem for cubic order tensors. First of all, we show that this problem is equivalent to a special rank-1 matrix recovery problem. We propose both nuclear norm relaxation and moment relaxation methods for solving the resulting rank-1 matrix recovery problem. The nuclear norm relaxation sometimes get a rank- tensor completion, while sometimes it does not. When it fails, we apply the moment hierarchy of semidefinite programming relaxations to solve the rank- matrix recovery problem. The moment hierarchy can always get a rank- tensor completion, or detect its nonexistence. In particular, when the tensor is strongly rank-1 completable, we show that the problem is equivalent to a rank-1 matrix completion problem and it can be solved by an iterative formula. Therefore, much larger size problems can be solved efficiently for strongly rank-completable tensors. Numerical experiments are shown to demonstrate the efficiency of these proposed methods.