PRISM: A Geometric Risk Bound that Decomposes Drift into Scale, Shape, and Head
Authors: Chieh-Yen Lin, Shao-Hua Sun
Summary
The authors propose PRISM, a method to diagnose how post-training modifications (quantization, LoRA, distillation) change an LLM, not just whether they degrade it. They derive a mathematical upper bound on the quality gap between original and modified models by exploiting the linear output head and near-isometric backbone structure, then decompose the drift into three independent dimensions: scale mismatch, shape distortion, and head divergence. Each dimension corresponds to a different failure mode and suggests specific fixes, and the shape term can be used as a regularizer during training to prevent forgetting.
Main takeaways:
- Existing similarity metrics (CKA, SVCCA) flag degradation but don't explain what went wrong or suggest remedies
- PRISM decomposes model drift into three axes: scale (magnitude mismatch), shape (geometric distortion), and head (output-layer divergence)
- Each axis maps to specific problems: shape breaks under low-bit quantization, scale separates under LoRA forgetting, head diverges under k-quantization
- Ranks model variants with ~0.82 Spearman correlation to actual performance, helping choose which variant to deploy
- The shape term is differentiable and works as a training regularizer, outperforming experience replay at preventing catastrophic forgetting
Relevance
Relevant to my installation-path equivalence work comparing fine-tuning vs. steering vectors—PRISM's decomposition could help me understand whether different installation methods (prompting, activation steering, fine-tuning) produce the same behavioral changes at the representation level, or if they differ in scale/shape/head dimensions. The geometry analysis might also explain why SFT collapses persona geometry to cos ≥0.97 in my experiments.
Threat model
Potential threat/caveat for clean result "Fine-tuning one persona on a two-marker chunk and another on the start marker plants the end marker at every donor answer's end, not chained to the start (LOW confidence)": this item discusses failure, benchmark.
Abstract
arXiv:2605.11608v1 Announce Type: new Abstract: Comparing post-training LLM variants, such as quantized, LoRA-adapted, and distilled models, requires a diagnostic that identifies how a variant has drifted, not only whether it has degraded. Existing similarity scores such as CKA and SVCCA can flag degradation, but they do not directly link representation drift to risk or mechanism. We propose PRISM, Proxy Risk Inference via Structural Mapping, which exploits the linear output head of LLMs and the empirically near-isometric structure of their backbones to derive a closed-form upper bound on the cross-entropy risk gap between a target model and a post-training variant. The bound is calibrated for variant ranking and decomposes drift into three independently measurable axes: scale mismatch, shape mismatch, and head divergence. Each axis corresponds to a distinct failure mode, including shape distortion under low-bit quantization, scale separability under LoRA forgetting, and head divergence under GGUF k-quantization. As a result, the dominant axis suggests a remediation direction rather than merely raising a degradation flag. Because the shape term is differentiable, the same geometry can also serve as a training-time regularizer against catastrophic forgetting. Across two model families and five benchmarks, PRISM ranks variants with mean Spearman correlations of 0.820 for post-training quantization and 0.831 for LoRA forgetting, and its axis-guided shape regularizer outperforms experience replay in aggregate at mitigating downstream forgetting.