Efficient dimension reduction for high-dimensional matrix-valued data

Abstract

Collection of groups of high-dimensional matrix-valued data is becoming increasingly common in many modern applications such as imaging analysis. The massive size of such data creates challenges in terms of computing speed and computer memory. Numerical techniques developed for small or moderate-sized datasets simply do not translate to such massive datasets. The need to analyze such data effectively calls for the development of efficient dimension reduction techniques. We propose a novel dimension reduction approach that has nice approximation property, computes fast for high dimensionality, and also explicitly incorporates the intrinsic two-dimensional structure of the matrices. We approximate each matrix as the product of a group-level left basis matrix, a group-level right basis matrix, and an individual-level coefficient matrix, which are estimated through a two-stage singular value decomposition. We discuss the connection of our proposal with existing approaches, and compare them both numerically and theoretically. We also obtain theoretical upper bounds on the approximation error of our method. In the numerical studies, ours is much faster than the most accurate one, comparable to the near-optimal one both computationally and theoretically, and more precise than the one that requires the same amount of memory.