The Optimal Hard Threshold for Singular Values is 4 / √ 3

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 MSE (AMSE) in a framework where the matrix size is large compared to 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·ymed when σ is unknown, where ymed is the median empirical singular value. For nonsquare m by n matrices with m 6= n the thresholding coefficients 4/ √ 3 and 2.858 are replaced with different provided constants that depend on m/n. Asymptotically, this thresholding rule adapts to unknown rank and unknown noise level in an optimal manner: it is always better than hard thresholding at any other value, and is always better than ideal Truncated SVD (TSVD), which truncates at the true rank of the low-rank matrix we are trying to recover. Hard thresholding at the recommended value to recover an n-by-n matrix of rank r guarantees an AMSE at most 3nrσ. In comparison, the guarantees provided by TSVD, optimally tuned singular value soft thresholding and the best guarantee achievable by any shrinkage of the data singular values are 5nrσ, 6nrσ, and 2nrσ, respectively. The recommended value for hard threshold also offers, among hard thresholds, the best possible AMSE guarantees for recovering matrices with bounded nuclear norm. Empirical evidence suggests that performance improvement over TSVD and other popular shrinkage rules can be substantial, for different noise distributions, even in relatively small n.