Low Rank Triple Decomposition and Tensor Recovery
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A simple approach for matrix completion and recovery is via low rank matrix decomposition. Recently, low rank matrix decomposition was extended to third order tensors, by T-product, then used in tensor completion and recovery. In this approach, a third order tensor is decomposed as the T-product of two low rank third order tensors. The three modes of the third order tensor are not balanced in this approach. This may cause some problems in applications where three modes have different meanings. The innovation of this paper is to decompose a third order tensor to three low rank third tensors in a balanced way. We call such a decomposition the triple decomposition, and the corresponding rank the triple rank. Numerical tests show that third order tensor data from practical applications such as internet traffic and video image are of low triple ranks. A tensor recovery method is proposed based on such low rank triple decomposition. Comparing with some main methods for tensor recovery, this method has its own merits, and can be a new choice for practical users.
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