在TT-格式下基于稳定ALS近似的秩自适应张量补全
Stable ALS Approximation in the TT-Format for Rank-Adaptive Tensor Completion
AI总结：
本文研究了在TT-格式下通过稳定ALS近似实现的秩自适应张量补全问题，提出了一种基于奇异值的正则化方法，以提高交替最优化的稳定性，并推导出一种自然的半隐式秩自适应策略。
摘要：
低秩张量补全是一个高度病态的逆问题，特别是当数据模型不准确时，需要某种形式的正则化来解决。本文聚焦于数据模型的校准。在交替优化中，我们发现现有的秩自适应方法无法在不同秩的流形之间实现连续过渡。我们称这种特性为“不稳定性（在截断下）”。由于这一性质，迭代中的任意小变化可能会对后续的重建产生任意大的影响。因此，我们向标准交替最小二乘（ALS）方法引入了基于奇异值的正则化，其灵感来自于微步平均。我们证明了其稳定性，并推导出一种自然的半隐式秩自适应策略。我们进一步证明，标准ALS微步在补全问题中仅在固定秩的流形上稳定，并且仅在具有我们定义为“内部张量受限等距性质（iTRIP）”的点附近稳定。最后，提供了数值实验，显示新的稳定ALS近似（SALSA）在重建质量上相比标准ALS和已知的黎曼优化方法RTTC提高了数量级。
英文摘要：
Low rank tensor completion is a highly ill-posed inverse problem, particularly when the data model is not accurate, and some sort of regularization is required in order to solve it. In this article we focus on the calibration of the data model. For alternating optimization, we observe that existing rank adaption methods do not enable a continuous transition between manifolds of different ranks. We denote this characteristic as $\textit{instability (under truncation)}$. As a consequence of this property, arbitrarily small changes in the iterate can have arbitrarily large influence on the further reconstruction. We therefore introduce a singular value based regularization to the standard alternating least squares (ALS), which is motivated by averaging in microsteps. We prove its $\textit{stability}$ and derive a natural semi-implicit rank adaption strategy. We further prove that the standard ALS microsteps for completion problems are only stable on manifolds of fixed ranks, and only around points that have what we define as $\textit{internal tensor restricted isometry property, iTRIP}$. In conclusion, numerical experiments are provided that show improvements of the reconstruction quality up to orders of magnitude in the new Stable ALS Approximation (SALSA) compared to standard ALS and the well known Riemannian optimization RTTC.
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