The Fréchet derivative of rank-truncated CUR is a sampling-induced oblique tangent projector, so perturbations in its kernel are removed to first order.
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Revisiting CUR Perturbation Analysis: A Local Tangent-Space Expansion
The Fréchet derivative of rank-truncated CUR is a sampling-induced oblique tangent projector, so perturbations in its kernel are removed to first order.