The use of self-adaptive principal components in PCA-based denoising

PCA-based denoising usually implies either discarding a number of high-index principal components (PCs) of a data matrix or their attenuation according to a regularization model. This work introduces an alternative, model-free, approach to high-index PC attenuation that seeks to average values of PC vectors as if they were expected from noise perturbation of data. According to the perturbation theory, the average PCs are attenuated versions of the clean PCs of noiseless data - the higher the noise-related content in a PC vector, the lower is its average's norm. This enables a regularization of the PC expansion of data where the PC terms are self-adapted to their noise content. To approximate the average PC vectors, the data matrix is randomly sampled several times to obtain numerous pseudo-random PC sets. The PCs of same ranks are then used to reconstruct the full-data PCs of that rank. A numerical algorithm of the reconstruction and its implementation in Python are provided. The proposed automatic adaptation to data offers a convenient solution for those who face with a problem of scaling or discarding PCs in PCA-based denoising. Questions of optimal sampling schedule and sampling amount remain issues that future work must address.