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.

翻译：我们考虑了Kronecker产品分解引起的矩阵近似和分解问题。 具体地说,我们提议以几种Kronecker产品总和(我们称之为Kronecker产品近似(KOPA))来估计一个特定矩阵,因为Kronecker产品是外部产品从矢量向矩阵的延伸,Kopa将低级矩阵近似扩展为外产,并将之列为一个特殊案例。与后者相比,KoPA也提供了更大的灵活性,因为它允许用户选择构成Kronecker产品的两个较小矩阵的尺寸。另一方面,我们通过数字研究和若干基准图像来显示KoPA相对于低级近似值的优势。