Estimation of low-rank tensors via convex optimization
Estimation of low-rank tensors via convex optimization
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发表时间：
2010-10
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通讯作者：
Ryota Tomioka;K. Hayashi;H. Kashima
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作者：
Ryota Tomioka;K. Hayashi;H. Kashima
本文提出了三种基于局部观测的多路阵列（张量）Tucker分解估计方法。所有的方法都被表述为凸最小化问题。因此，最小值保证是唯一的。提出的方法可以通过优化自动估计因子的数量（秩）。因此，不需要事先指定秩。我们采用的关键技术是迹范数正则化，这是一种常用的估计低秩矩阵的方法。此外，我们提出了一个简单的启发式来提高所得到的分解的可解释性。通过在合成数据集和真实数据集上的数值实验，论证了三种方法的优缺点。我们表明，所提出的基于凸优化的方法在预测性能上更准确，在恢复已知的多线性结构方面比传统方法更快，更可靠。
In this paper, we propose three approaches for the estimation of the Tucker decomposition of multi-way arrays (tensors) from partial observations. All approaches are formulated as convex minimization problems. Therefore, the minimum is guaranteed to be unique. The proposed approaches can automatically estimate the number of factors (rank) through the optimization. Thus, there is no need to specify the rank beforehand. The key technique we employ is the trace norm regularization, which is a popular approach for the estimation of low-rank matrices. In addition, we propose a simple heuristic to improve the interpretability of the obtained factorization. The advantages and disadvantages of three proposed approaches are demonstrated through numerical experiments on both synthetic and real world datasets. We show that the proposed convex optimization based approaches are more accurate in predictive performance, faster, and more reliable in recovering a known multilinear structure than conventional approaches.