Bayesian compressive principal component analysis

Principal component analysis(PCA)is a widely used method for multivariate data analysis that projects the original high-dimensional data onto a low-dimensional subspace with maximum variance.However,in practice,we would be more likely to obtain a few compressed sensing(CS)measurements than the complete high-dimensional data due to the high cost of data acquisition and storage.In this paper,we propose a novel Bayesian algorithm for learning the solutions of PCA for the original data just from these CS measurements.To this end,we utilize a generative latent variable model incorporated with a structure prior to model both sparsity of the original data and effective dimensionality of the latent space.The proposed algorithm enjoys two important advantages:1)The effective dimensionality of the latent space can be determined automatically with no need to be pre-specified;2)The sparsity modeling makes us unnecessary to employ multiple measurement matrices to maintain the original data space but a single one,thus being storage efficient.Experimental results on synthetic and real-world datasets show that the proposed algorithm can accurately learn the solutions of PCA for the original data,which can in turn be applied in reconstruction task with favorable results.