When Can We Trust the Sparse Lens? A Certification Framework for SAE Faithfulness

Sparse autoencoders (SAEs) are increasingly used to study internal representations in language models (LMs), but it remains unclear when an SAE-based representation reliably preserves the underlying model's predictive behavior. We introduce a post-hoc certification framework for this purpose. At a chosen layer, we replace the model's hidden activation with its pretrained SAE reconstruction and derive a bound on the expected loss of the original frozen LM through this SAE-based proxy. The resulting certificate depends on three quantities measured on held-out data: the proxy risk, the reconstruction loss gap, and support mismatch between the calibration feature pool and new inputs. We further derive a more conservative exact-P extension whose proxy-risk concentration term is uniform over all feature pools of the same size. Empirically, the certificates become non-vacuous for GPT-2 Small, Gemma-2B, and Llama-3-8B. A detailed layerwise study of Llama-3-8B shows that later layers are easier to certify, driven mainly by improved reconstruction fidelity and weaker downstream amplification of reconstruction error. GPT-2 Small shows much weaker layer dependence, indicating that this pattern is not universal across model-SAE pairs. Overall, the framework provides a practical way to determine whether an SAE representation preserves enough of a frozen LM's predictive behavior to support certification of the original model. Code is available at: https://github.com/newcodevelop/sparse-lens-certification.