Two For The Price Of One: Curvature-Aware Pruning Preserves Influence Functions Too

Post-training pruning enables efficient deployment, but conventional methods do not explicitly preserve internal-gradient alignment between sparse and dense models. We study novel connections between downstream performance of compressed models and the ability to retain gradient information via influence-function preservation. Central to this connection is a curvature-aware objective that accounts for both layer activations and backpropagated gradients. Despite its promise, this objective remains underexplored in pruning due to a lack of efficient solvers. We introduce textttCACTUS, a novel ADMM-based optimizer supporting both pruning and sparse-plus-low-rank decompositions of pre-trained weights. Compared to LLM-Surgeon which targets the same objective, textttCACTUS can achieve over an order-of-magnitude speedup. At 50% pruning of Qwen3-4B, textttCACTUS reduces the C4 perplexity gap to the dense model by 45%, improves compressed-to-dense per-sample gradient cosine similarity on WikiText-2 from 0.670 to 0.755, and runs over 28× faster than LLM-Surgeon on substantially weaker hardware. By preserving influence functions, textttCACTUS supports effective data selection. On Llama-3.2-3B, fine-tuning a fresh dense model on the 5% of data selected by textttCACTUS achieves 49.44% accuracy on BIG-Bench Hard, compared with 47.22% using ALPS and 49.54% using dense-model selection. Finally, integrating textttCACTUS with LoGRA, a recent gradient-projection strategy, on Qwen3-4B with MLP layers pruned to 2:4 sparsity and a rank-64 correction retains a 0.711 cosine alignment with dense LoGRA-projected C4 gradients.