Quantifying the noise sensitivity of the Wasserstein metric for images

Referee: [§4, Theorem 4.1] §4, Theorem 4.1: the O(sqrt(sigma)) upper bound on |W_signed(I+N, J+N) - W_signed(I,J)| is the load-bearing claim; the derivation must show explicitly how independence of the pixel-wise Gaussian entries produces a variance term whose square root yields the scaling after expectation or concentration, without a linear total-variation penalty reappearing from the signed-measure extension.

Authors: We thank the referee for this observation. The proof of Theorem 4.1 begins from the dual formulation and expresses the difference |W_signed(I+N, J+N) - W_signed(I,J)| as the supremum over 1-Lipschitz test functions of the inner product between the noise field and the difference of the optimal dual potentials. Because the noise entries are independent Gaussians, the variance of this inner product equals sigma^2 times the squared L2-norm of the potential difference, which remains bounded independently of sigma by the Lipschitz constraint. A standard sub-Gaussian concentration inequality then produces a high-probability deviation of order sqrt(sigma); taking the expectation preserves the same scaling. The signed-measure extension does not reintroduce a linear total-variation term because the truncation operator used to restore non-negativity after noise addition is 1-Lipschitz with respect to the Wasserstein metric, and the quadratic cost structure absorbs any first-order contributions into higher-order remainders. In the revised manuscript we will insert an expanded paragraph immediately after the statement of Theorem 4.1 that isolates these variance and truncation calculations. revision: yes

Referee: [§2.2] §2.2: the precise definition of the signed 2-Wasserstein distance (dual formulation, lifting, or truncation of negative masses) is not stated with sufficient detail to confirm that the stability bound remains uniform in grid size and does not introduce an extra O(sigma) term when noise creates negative pixel values.

Authors: We agree that the definition merits greater explicitness. Section 2.2 defines the signed 2-Wasserstein distance via the Kantorovich dual: W_2^2(mu, nu) = sup {int f d mu - int f d nu : Lip(f) <= 1}, extended to signed measures by allowing negative parts and applying a truncation operator T that projects each pixel value onto the non-negative reals. The operator T is a contraction in the 2-Wasserstein metric, so the perturbation introduced by truncation is at most the L1 norm of the negative part, which is absorbed into the sqrt(sigma) term already present from the concentration step. Uniformity in grid size follows directly from the fact that the dual functions remain 1-Lipschitz independently of the number of pixels and the noise is added coordinate-wise with per-pixel variance sigma^2. In the revision we will replace the current informal description in §2.2 with the full dual statement, add a short lemma establishing the contraction property of T, and include a remark confirming that the resulting bound is uniform in the grid cardinality. revision: yes