Influence functions (IFs) approximate the effect of removing or reweighting training data on a trained model, but their linear approximation often lacks the accuracy required for reliable post hoc model editing. We identify two culprits and propose generalized influence functions (GIFs) and pseudo-LiSSA (p-LiSSA) to address each in turn. First, classical IFs update all parameters indiscriminately, leading to unnecessary updates to data-irrelevant parameters. GIFs instead restrict edits to a data-relevant subspace and leave the remaining parameters fixed. We theoretically show that GIFs achieve a tighter error bound than classical IFs when the parameter subset is properly selected, and that effective subspace selection requires avoiding low-curvature directions while retaining a sufficiently large component of the target gradient. Second, LiSSA iterations are destabilized by negative Hessian eigenvalues, which existing methods handle via $\ell_2$ damping, at the cost of a biased inverse-Hessian-vector product (iHVP) estimate. p-LiSSA eliminates this bias by solving the restricted inverse-Hessian problem without such damping. We further show that the presence of low-curvature directions slows p-LiSSA convergence, and that the data-relevant subspace selected by GIFs mitigates this. Across five image and text unlearning benchmarks, GIFs update only 10% of the parameters yet preserve accuracy up to 2.12 percentage points better than IF-based baselines, while improving unlearning performance by up to 2.73 percentage points. Moreover, p-LiSSA improves restricted iHVP estimation by reducing the residual by 9.6$\times$ over existing iHVP baselines, while using 19.8% less memory with comparable computation time. Code is available at https://github.com/hslyu/GIF.

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