Distance Preserving Dimension Reduction Using the QR Factorization or the Cholesky Factorization

Dimension reduction plays an important role in handling the massive quantity of high dimensional data such as biomedical text data, gene expression data, and mass spectrometry data, and so forth. In this paper, we introduce distance preserving dimension reduction (DPDR) based on the QR factorization (DPDR/QR) or the Cholesky factorization (DPDR/C). DPDR generates lower dimensional representations of the high-dimensional data, which can exactly preserve Euclidean distances and cosine similarities between any pair of data points in the original dimensional space. After projecting data points to the lower dimensional space obtained from DPDR, one can execute other data analysis algorithms. DPDR can substantially reduce the computing time and/or memory requirement of a given data analysis algorithm, especially when we need to run the data analysis algorithm many times for estimating parameters or searching for a better solution.