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

Hybrid Kronecker Product Decomposition and Approximation