Chencheng Cai

Chencheng Cai, Rong Chen, Han Xiao

Discovering the underlying low dimensional structure of high dimensional data has attracted a significant amount of researches recently and has shown to have a wide range of applications. As an effective dimension reduction tool, singular value decomposition is often used to analyze high dimensional matrices, which are traditionally assumed to have a low rank matrix approximation. In this paper, we propose a new approach. We assume a high dimensional matrix can be approximated by a sum of a small number of Kronecker products of matrices with potentially different configurations, named as a hybird Kronecker outer Product Approximation (hKoPA). It provides an extremely flexible way of dimension reduction compared to the low-rank matrix approximation. Challenges arise in estimating a hKoPA when the configurations of component Kronecker products are different or unknown. We propose an estimation procedure when the set of configurations are given and a joint configuration determination and component estimation procedure when the configurations are unknown. Specifically, a least squares backfitting algorithm is used when the configuration is given. When the configuration is unknown, an iterative greedy algorithm is used. Both simulation and real image examples show that the proposed algorithms have promising performances. The hybrid Kronecker product approximation may have potentially wider applications in low dimensional representation of high dimensional data

Chencheng Cai, Rong Chen, Han Xiao

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.

Chencheng Cai, Rong Chen, Han Xiao

A matrix completion problem is to recover the missing entries in a partially observed matrix. Most of the existing matrix completion methods assume a low rank structure of the underlying complete matrix. In this paper, we introduce an alternative and more general form of the underlying complete matrix, which assumes a low Kronecker rank instead of a low regular rank, but includes the latter as a special case. The extra flexibility allows for a much more parsimonious representation of the underlying matrix, but also raises the challenge of determining the proper Kronecker product configuration to be used. We find that the configuration can be identified using the mean squared error criterion as well as a modified cross-validation criterion. We establish the consistency of this procedure under suitable conditions on the signal-to-noise ratio. A aggregation procedure is also proposed to deal with special missing patterns and complex underlying structures. Both numerical and empirical studies are carried out to demonstrate the performance of the new method.