Cross
| swMATH ID: | 30282 |
| Software Authors: | Zhang, Anru |
| Description: | Cross: efficient low-rank tensor completion. The completion of tensors, or high-order arrays, attracts significant attention in recent research. Current literature on tensor completion primarily focuses on recovery from a set of uniformly randomly measured entries, and the required number of measurements to achieve recovery is not guaranteed to be optimal. In addition, the implementation of some previous methods are NP-hard. In this article, we propose a framework for low-rank tensor completion via a novel tensor measurement scheme that we name Cross. The proposed procedure is efficient and easy to implement. In particular, we show that a third-order tensor of Tucker rank-((r_{1},r_{2},r_{3})) in (p_{1})-by-(p_{2})-by-(p_{3}) dimensional space can be recovered from as few as (r_{1}r_{2}r_{3}+r_{1}(p_{1}-r_{1})+r_{2}(p_{2}-r_{2})+r_{3}(p_{3}-r_{3})) noiseless measurements, which matches the sample complexity lower bound. In the case of noisy measurements, we also develop a theoretical upper bound and the matching minimax lower bound for recovery error over certain classes of low-rank tensors for the proposed procedure. The results can be further extended to fourth or higher-order tensors. Simulation studies show that the method performs well under a variety of settings. Finally, the procedure is illustrated through a real dataset in neuroimaging. |
| Homepage: | https://projecteuclid.org/euclid.aos/1547197244 |
| Keywords: | cross tensor measurement; denoising; minimax rate-optimal; neuroimaging; tensor completion |
| Related Software: | Wirtinger Flow; sparcl; MultiwayRegression; FinTS; msda; CHIME; gamair; glmnet; CRAN; CLEMM; TensorClustering; ROPTLIB; SOFAR; astsa; RobustSP; PhaseLift; TensorReg; MultiCluster; rTensor; cross2D |
| Cited in: | 20 Documents |
Standard Articles
| 1 Publication describing the Software, including 1 Publication in zbMATH | Year |
|---|---|
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Cross: efficient low-rank tensor completion. Zbl 1416.62298 Zhang, Anru |
2019
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Cited by 42 Authors
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Cited in 12 Serials
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