An unknown m by n matrix X0 is to be estimated from noisy measurements Y = X0 + Z, where the noise matrix Z has i.i.d. Gaussian entries. A popular matrix denoising scheme solves the nuclear norm penalization problem minX |Y-X|2 F/2+λ|X|∗, where |X|∗ denotes the nuclear norm (sum of singular values). This is the analog, for matrices, of l1 penalization in the vector case. It has been empirically observed that if X0 has low rank, it may be recovered quite accurately from the noisy measurement Y. In a proportional growth framework where the rank rn, number of rows mn and number of columns n all tend to ∞ proportionally to each other (rn/mn → ρ, mn/n → β), we evaluate the asymptotic minimax MSE M(ρ,β) = limmn,n→∞ infλ sup rank(X)≤rn MSE(X0, Xλ). Our formulas involve incomplete moments of the quarter-and semi-circle laws (β = 1, square case) and the Marčenko-Pastur law (β < 1, nonsquare case). For finite m and n, we show that MSE increases as the nonzero singular values of X0 grow larger. As a result, the finite-n worst-case MSE, a quantity which can be evaluated numerically, is achieved when the signal X0 is "infinitely strong". The nuclear norm penalization problem is solved by applying soft thresholding to the singular values of Y. We also derive the minimax threshold, namely the value λ∗.(ρ), which is the optimal place to threshold the singular values. All these results are obtained for general (nonsquare, nonsymmetric) real matrices. Comparable results are obtained for square symmetric nonnegativedefinite matrices.
Bibliographical notePublisher Copyright:
© 2014 Institute of Mathematical Statistics.
- Matrix completion from Gaussian measurements
- Matrix denoising
- Monotonicity of power functions of multivariate tests
- Nuclear norm minimization
- Optimal threshold
- Phase transition
- Singular value thresholding
- Stein unbiased risk estimate