Towards a Data-Parameter Correspondence for LLMs: A Preliminary Discussion

Large language model optimization has historically bifurcated into isolated data-centric and model-centric paradigms: the former manipulates involved samples through selection, augmentation, or poisoning, while the latter tunes model weights via masking, quantization, or low-rank adaptation. This paper establishes a unified \emph{data-parameter correspondence} revealing these seemingly disparate operations as dual manifestations of the same geometric structure on the statistical manifold $\mathcal{M}$. Grounded in the Fisher-Rao metric $g_{ij}(\theta)$ and Legendre duality between natural ($\theta$) and expectation ($\eta$) parameters, we identify three fundamental correspondences spanning the model lifecycle: 1. Geometric correspondence: data pruning and parameter sparsification equivalently reduce manifold volume via dual coordinate constraints; 2. Low-rank correspondence: in-context learning (ICL) and LoRA adaptation explore identical subspaces on the Grassmannian $\mathcal{G}(r,d)$, with $k$-shot samples geometrically equivalent to rank-$r$ updates; 3. Security-privacy correspondence: adversarial attacks exhibit cooperative amplification between data poisoning and parameter backdoors, whereas protective mechanisms follow cascading attenuation where data compression multiplicatively enhances parameter privacy. Extending from training through post-training compression to inference, this framework provides mathematical formalization for cross-community methodology transfer, demonstrating that cooperative optimization integrating data and parameter modalities may outperform isolated approaches across efficiency, robustness, and privacy dimensions.