Convex Tensor Decomposition via Structured Schatten Norm Regularization
Convex Tensor Decomposition via Structured Schatten Norm Regularization
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发表时间：
2013-03
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通讯作者：
Ryota Tomioka;Taiji Suzuki
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作者：
Ryota Tomioka;Taiji Suzuki
我们讨论了张量分解的结构化Schatten范数，其中包括两个最近提出的基于凸优化的张量分解的范数（“重叠”和“潜在”），并将张量分解与结构化稀疏性的更广泛文献联系起来。基于结构化Schatten范数的性质，我们从数学上分析了张量分解的“潜伏”方法的性能，经验发现在某些情况下，这种方法的性能优于“重叠”方法。我们从理论上证明，情况确实如此。特别地，当未知真实张量在特定模式中是低秩时，该方法的性能与知道具有最小秩的模式一样好。沿着的方式，我们显示了一个新的对偶结构Schatten规范的结果，建立的一致性，并讨论这种方法的可识别性。我们通过数值模拟证实，我们的理论预测可以精确地预测标度行为的均方误差。
We discuss structured Schatten norms for tensor decomposition that includes two recently proposed norms ("overlapped" and "latent") for convex-optimization-based tensor decomposition, and connect tensor decomposition with wider literature on structured sparsity. Based on the properties of the structured Schatten norms, we mathematically analyze the performance of "latent" approach for tensor decomposition, which was empirically found to perform better than the "overlapped" approach in some settings. We show theoretically that this is indeed the case. In particular, when the unknown true tensor is low-rank in a specific mode, this approach performs as good as knowing the mode with the smallest rank. Along the way, we show a novel duality result for structures Schatten norms, establish the consistency, and discuss the identifiability of this approach. We confirm through numerical simulations that our theoretical prediction can precisely predict the scaling behavior of the mean squared error.