Sparsity loss (L1 penalty on feature activations) — trades off against → Reconstruction loss

Both terms sum into one training objective (Equation 4), so the optimizer is free to trade one against the other rather than driving each to its own minimum. The paper observes this trade directly: dictionaries trained with a larger sparsity coefficient alpha reach lower reconstruction accuracy, which shows up in the IOI activation-patching experiments (Section 4.2) as a higher floor on the minimum KL divergence that no amount of further patching can close. At the opposite extreme, a non-sparse dictionary (alpha=0) reconstructs well but loses the localization advantage over PCA that motivates the whole method -- Figure 3 shows its features need as many patches as PCA components to reach a given KL divergence, because sparsity rather than reconstruction fidelity is what makes the patched feature set small and precise. The two losses therefore pull in opposite directions along one smooth curve (Appendix B, Figures 6-7) rather than settling at a joint optimum, which is the same empirical basis behind the sparsity-reconstruction tradeoff's 'no knee' finding.