We formalize the optimization problem for the preliminary noise reduction step. Given a noisy STEM-EDX data M P R CˆN , we assume that the ideally denoised data X ˚(A1) should have nonnegative values; (A2) should deviate from the observation as little as possible; (A3) should be comprised of a small number of chemical components, described with the rank of X ˚; and (A4) should have spectra that vary smoothly over energy. Regarding the assumption (A3), however, optimizing the matrix rank is NP-hard. Therefore, the nuclear norm is generally used as its convex surrogate for low-rank optimization . Then, before projecting onto the low-dimensional subspace to obtain X ˚, we have the following optimization problem (we call this low-rank regularized spectral smoothing as smoothing is its main function) to obtain the moderately denoised data X ˚1: