Projection-based estimators for matrix/tensor-valued data

Abstract

A general approach for extending estimators to matrix- and tensor-valued data is proposed. The extension is based on using random projections to project out dimensions of a tensor and then computing a multivariate estimator for each projection. The mean of the obtained set of estimates is used as the final, joint estimate. In some basic cases, the resulting estimator can be given a closed form, and particular ones are shown to coincide with existing methodology. We derive sufficient conditions for the consistency and limiting normality of the resulting estimators under weak assumptions. In particular, limiting normality is retained as soon as the number of projections grows super-linearly in the sample size, and consistency is achieved regardless of the growth rate. Comparisons with competing methods show that the extensions prove useful in extracting components for classification and yield an efficient estimator for sufficient dimension reduction.