For discrete perturbation sets between the hypercube and the odd integer grid, the focused width overestimates the detectability radius by at least a √(log n) factor in the Gaussian model and by n^{1/4} for Laplace data.
Tony Cai, Jiashun Jin, and Mark G
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Focused Width in Adversarial Fake Detection: A Separation
For discrete perturbation sets between the hypercube and the odd integer grid, the focused width overestimates the detectability radius by at least a √(log n) factor in the Gaussian model and by n^{1/4} for Laplace data.