On Tensor Train Rank Minimization : Statistical Efficiency and Scalable Algorithm
On Tensor Train Rank Minimization : Statistical Efficiency and Scalable Algorithm
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发表时间：
2017-08
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通讯作者：
M. Imaizumi;Takanori Maehara;K. Hayashi
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作者：
M. Imaizumi;Takanori Maehara;K. Hayashi
张量序列（TT）分解为高阶张量提供了一种空间有效的表示。尽管它的优势，我们面临着两个关键的限制，当我们应用TT分解机器学习问题：缺乏统计理论和可扩展的算法。在本文中，我们解决的局限性。首先，我们引入了一个凸松弛的TT分解问题，并推导出其误差界的张量完成任务。接下来，我们开发了一个交替优化方法与随机化技术，其中的时间复杂度是有效的空间复杂度。在实验中，我们在数值上证实了推导的界限和经验证明了我们的方法与真实的高阶张量的性能。
Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of statistical theory and of scalable algorithms. In this paper, we address the limitations. First, we introduce a convex relaxation of the TT decomposition problem and derive its error bound for the tensor completion task. Next, we develop an alternating optimization method with a randomization technique, in which the time complexity is as efficient as the space complexity is. In experiments, we numerically confirm the derived bounds and empirically demonstrate the performance of our method with a real higher-order tensor.