The Optimal Hard Threshold for Singular Values is <inline-formula> <tex-math notation="TeX">\(4/\sqrt {3}\) </tex-math></inline-formula>

We consider recovery of low-rank matrices from noisy data by hard thresholding of singular values, in which empirical singular values below a threshold λ are set to 0. We study the asymptotic mean squared error (AMSE) in a framework, where the matrix size is large compared with the rank of the matrix to be recovered, and the signal-to-noise ratio of the low-rank piece stays constant. The AMSE-optimal choice of hard threshold, in the case of n-by-n matrix in white noise of level σ, is simply (4/√3)√nσ ≈ 2.309√nσ when σ is known, or simply 2.858 · y med when σ is unknown, where y med is the median empirical singular value. For nonsquare, m by n matrices with m ≠ n the thresholding coefficients 4/√3 and 2.858 are…