Efficient Nonnegative Tucker Decompositions: Algorithms and Uniqueness
Efficient Nonnegative Tucker Decompositions: Algorithms and Uniqueness
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DOI：
10.1109/tip.2015.2478396
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发表时间：
2014-04
影响因子：
10.6
通讯作者：
Guoxu Zhou;A. Cichocki;Qibin Zhao;Shengli Xie
中科院分区：
文献类型：
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作者：
Guoxu Zhou;A. Cichocki;Qibin Zhao;Shengli Xie
非负Tucker分解（NTD）是一种从高维张量数据中提取基于非负部分和物理上有意义的潜在成分，同时保持数据的自然多线性结构的强大工具。然而，由于数据张量往往具有多模态且规模较大，现有的NTD算法在存储和计算时间上的计算复杂度都非常高，这是NTD实际应用的主要障碍之一。为了克服这些缺点，我们展示了张量的低（多线性）秩近似（LRA）如何能够显着简化成本函数梯度的计算，并在此基础上开发了一系列有效的一阶NTD算法。除了显著降低存储复杂度和运行时间外，新算法还具有相当的灵活性和对噪声的鲁棒性，因为任何成熟的LRA方法都可以应用。我们还展示了包含稀疏性的非负性如何极大地提高了唯一性，并部分缓解了Tucker分解的维数诅咒。综合数据和实际数据的仿真结果验证了NTD算法的有效性和高效性。
Nonnegative Tucker decomposition (NTD) is a powerful tool for the extraction of nonnegative parts-based and physically meaningful latent components from high-dimensional tensor data while preserving the natural multilinear structure of data. However, as the data tensor often has multiple modes and is large scale, the existing NTD algorithms suffer from a very high computational complexity in terms of both storage and computation time, which has been one major obstacle for practical applications of NTD. To overcome these disadvantages, we show how low (multilinear) rank approximation (LRA) of tensors is able to significantly simplify the computation of the gradients of the cost function, upon which a family of efficient first-order NTD algorithms are developed. Besides dramatically reducing the storage complexity and running time, the new algorithms are quite flexible and robust to noise, because any well-established LRA approaches can be applied. We also show how nonnegativity incorporating sparsity substantially improves the uniqueness property and partially alleviates the curse of dimensionality of the Tucker decompositions. Simulation results on synthetic and real-world data justify the validity and high efficiency of the proposed NTD algorithms.