Minimax Risk and Least-Favorable Subspace Characterization of Matrix Denoising by Singular Value Soft Thresholding

⚠️ Curated Limitations & Negative Results

• Factor-of-3 suboptimality in the very-low-rank limit rho -> 0: Because SVST shrinks large singular values by a constant bias lambda_* -> (1 + sqrt(beta)) sigma sqrt(n), its low-rank minimax risk slope is 2(1 + sqrt(beta) + beta) rho (6 rho at beta = 1), which is up to 3 times larger than the optimal continuous singular-value shrinker slope 2 r / m and 2 times larger than optimal hard thresholding at 4/sqrt(3) (3 r / m) (Theorem 8-9, pp. 14-16).
• Requires known rank bound r (or rank fraction rho = r/m) and known Gaussian noise scale sigma = n^{-1/2} to compute the minimax threshold lambda_*(m, n, r); when r is unknown, one must either optimize SURE_lambda(Y) empirically over lambda (Candès et al. 2013) or adopt the rank-independent bulk-edge threshold (1 + sqrt(beta)) sigma sqrt(n) at the cost of suboptimal risk for moderate rho (Section 1.2.1, p. 3).
• Exact finite-sample singular-value monotonicity (Theorem 1, Lemma 4) relies on orthogonal invariance and log-concavity (unimodality) of the i.i.d. Gaussian noise density via Anderson's theorem; under heavy-tailed or non-orthogonal structured noise, finite-sample monotonicity in the singular values need not hold.
• Primary-source typographical audit of arXiv:1304.2085v4 (DOI:10.1214/14-AOS1257): (1) on p. 10 (Theorem 3, Eqs. 2.10-2.11 and line below) and p. 24 (Lemma 9), the upper bound on Lambda is printed '0 <= Lambda <= gamma_+' (and 'Lambda in [0, gamma_+]') instead of '0 <= Lambda <= sqrt(gamma_+)' (cf. Lemma 7 on p. 23), and line 7 below Theorem 3 repeats 'rho |-> M(rho, beta | Mat)' twice instead of 'M(rho, 1 | Mat)' and 'M(rho | Sym)'; (2) on p. 12 (Theorem 5, Eq. 2.16), 'Q_2(Lambda) - 2 lambda Q_1(Lambda) + Lambda^2 Q_0(Lambda)' prints lowercase 'lambda' instead of uppercase 'Lambda'; (3) on p. 14 (Theorem 7), line 3 below Figure 4 prints 'M(rho, rho | Mat)' instead of 'M(rho, 1 | Mat)', and the symmetric parametric formula prints 'rho(theta) = 1 - (g(theta) - pi/2)/(g(theta) + pi/2)' (which equals 1 - rho(theta)) instead of 'rho(theta) = (g(theta) - pi/2)/(g(theta) + pi/2) = 1 - pi/(g(theta) + pi/2)' when inverting Eq. (2.21) g(theta) = (pi/2)(1 + rho)/(1 - rho); (4) on p. 20 (Lemma 3, SURE formula), the interaction sum prints '(min{y_j, lambda} y_j - min{y_i, lambda} y_j)/(y_j^2 - y_i^2)' with 'y_j' instead of 'y_i' in the second numerator term 'min{y_i, lambda} y_i'; and (5) on p. 22 (Proof of Theorem 2, last line), 'M_n(Lambda; r, 1/2)' omits the third argument 'n', while on p. 27 (Proof of Theorem 9, last display), 'rho + beta rho + beta rho^2' prints '+ beta rho^2' instead of '- beta rho^2' in the denominator.