KoPA: Automated Kronecker Product Approximation
KoPA: Automated Kronecker Product Approximation
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发表时间：
2019-12
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通讯作者：
Chencheng Cai;Rong Chen;Han Xiao
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作者：
Chencheng Cai;Rong Chen;Han Xiao
考虑了由克罗内克积分解引起的矩阵逼近和去噪问题。具体地说，我们建议用几个矩阵的克罗内克积的和来近似给定的矩阵，我们称之为克罗内克积近似（KoPA）。由于Kronecker积是外积从向量到矩阵的扩展，KoPA扩展了低秩矩阵近似，并将其作为一个特例。与后者相比，KoPA还提供了更大的灵活性，因为它允许用户选择配置，即形成Kronecker积的两个较小矩阵的尺寸。另一方面，要使用的配置通常是未知的，需要从数据中确定，以便在准确性和简约性之间实现最佳平衡。我们建议使用扩展信息标准来选择配置。在高维分析范式下，我们证明了该方法能够在合适的信噪比条件下选择概率趋于1的真构型。我们通过数值研究和几个基准图像实例证明了KoPA比低秩近似的优越性。
We consider the problem of matrix approximation and denoising induced by the Kronecker product decomposition. Specifically, we propose to approximate a given matrix by the sum of a few Kronecker products of matrices, which we refer to as the Kronecker product approximation (KoPA). Because the Kronecker product is an extension of the outer product from vectors to matrices, KoPA extends the low rank matrix approximation, and includes it as a special case. Comparing with the latter, KoPA also offers a greater flexibility, since it allows the user to choose the configuration, which are the dimensions of the two smaller matrices forming the Kronecker product. On the other hand, the configuration to be used is usually unknown, and needs to be determined from the data in order to achieve the optimal balance between accuracy and parsimony. We propose to use extended information criteria to select the configuration. Under the paradigm of high dimensional analysis, we show that the proposed procedure is able to select the true configuration with probability tending to one, under suitable conditions on the signal-to-noise ratio. We demonstrate the superiority of KoPA over the low rank approximations through numerical studies, and several benchmark image examples.