13. Sufficient Dimension Reduction for Tensor Data

With the rapid development of science and technology, a large volume of array data has been collected in areas such as genomics, finance, image processing, and Internet search. How to extract useful information from massive data becomes the key issue nowadays. In spite of the urgent need for statistical tools to deal with such data, there are limited methods that can fully address the high-dimensional problem. In this chapter, we review the general setting of sufficient dimension reduction framework and its generalization to tensor data. Tensor is a multi-way array, and its usage is becoming more and more important with the advancement of social and behavioral science, chemistry, and imaging technology. The vector-based statistical methods can be applied to tensor data by vectorizing a tensor into a vector. However, vectorized tensor usually has a large dimension which may largely exceed the number of samples. To preserve the tensor structure and reduce the dimensionality simultaneously, we revisit the tensor sufficient dimension reduction model and apply it to colorimetric sensor arrays. Tensor sufficient dimension reduction method is simple but powerful and exhibits a competent empirical performance in real data analysis.