  Dimension reduction plays a pivotal role in analysing high-dimensional data.
However, observations with missing values present serious difficulties in
directly applying standard dimension reduction techniques. As a large number of
dimension reduction approaches are based on the Gram matrix, we first
investigate the effects of missingness on dimension reduction by studying the
statistical properties of the Gram matrix with or without missingness, and then
we present a bias-corrected Gram matrix with nice statistical properties under
heterogeneous missingness. Extensive empirical results, on both simulated and
publicly available real datasets, show that the proposed unbiased Gram matrix
can significantly improve a broad spectrum of representative dimension
reduction approaches.
