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Pré-Publication, Document De Travail Année : 2022

Provable Nonnegative Tensor Ring Approximation: Stability and Optimality

Résumé

Tensor Ring (TR) decomposition provides a flexible and powerful format to represent multi-way data. However, most current works based on TR point out that the performance of TR degrades with a choice of large ranks and/or with few available entries. To alleviate this issue, optimal TR ranks selection strategies or adding regularization on TR cores are proposed. In this paper, we explain that the above issue is caused by the fact that tensor representations in TR format with predefined bounds on TR-ranks do not form a closed set, which makes the computation of approximate TR unstable. Based on this finding, we propose a non-negative TR (NTR) decomposition and theoretically show that an optimal approximate NTR always exists and can be obtained. Computer experiments show that TR is more prone to instability when there are fewer observations, or when the low rankness of missing data is not obvious. On the contrary, NTR is stable and performs well on image recovery. Furthermore, we discuss factor degeneracy of TR-cores, which is one of the reasons for the instability of the TR, and builds a new bridge to existing works, where adding a norm regularization on TR-cores is also a way to avoid factor degeneracy.
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Dates et versions

hal-03781546 , version 1 (20-09-2022)
hal-03781546 , version 2 (13-01-2023)

Identifiants

  • HAL Id : hal-03781546 , version 2

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Zhen Long, Ce Zhu, Jiani Liu, Yipeng Liu, Pierre Comon. Provable Nonnegative Tensor Ring Approximation: Stability and Optimality. 2022. ⟨hal-03781546v2⟩
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