Invariance of principal components under low-dimensional random projection of the data

Algorithms that can efficiently recover principal components of high-dimensional data from compressive sensing measurements (e.g. low-dimensional random projections) of it have been an important topic of recent interest in the literature. In this paper, we show that, under certain conditions, normal principal component analysis (PCA) on such low-dimensional random projections of data actually returns the same result as PCA on the original data set would. In particular, as the number of data samples increases, the center of the randomly projected data converges to the true center of the original data (up to a known scaling factor) and the principal components converge to the true principal components of the original data as well, even if the dimension of each random subspace used is very low. Indeed, experimental results verify that this approach does estimate the original center and principal components very well for both synthetic and real-world datasets, including hyperspectral data. Its performance is even superior to that of other algorithms recently developed in the literature for this purpose.