Schatten-budgeted low-rank update aggregation

Integrate the method into LoRA or adapter merging, exactly where m rank-one updates are summed into a base weight matrix W. Write each update as A_k=u_k v_k^T; if an adapter has rank r, either split it into r outer products or run the method on each component group. Choose p=2 for the first implementation, since it requires only Frobenius/eigenvalue calculations, and then test p=4 for stronger sensitivity to large singular directions. For each rank-one update, avoid a dense matrix square root: |u_kv_k^T|=(||u_k||_2/||v_k||_2)v_kv_k^T. Accumulate S=sum_k |A_k| and compute s=(Tr(S^p))^{1/p} from the eigenvalues of the small d-by-d Gram representation or with randomized trace-power estimation for large layers. Given a merge budget tau, set alpha=min(1,tau/max(s,epsilon)) and apply DeltaW=alpha sum_k A_k, then W<-W+DeltaW. Solve x^p-2x-(m-1)=0 by bisection on [1, max(2,m+1)] and compute C_TZ; log the realized ratio ||sum A_k||_p/||S||_p and the theorem's upper bound. What comes from the paper is the rank-one inequality and scalar constant; eigenvalues and realized ratios are estimated computationally. First experiment: merge four independently trained rank-one adapters into a ViT-Tiny on CIFAR-10 or into a small language model on a held-out instruction task. Compare no clipping, per-update Frobenius clipping, global Frobenius clipping, and Schatten budgeting at equal merge budgets. Success is fewer post-merge loss spikes, lower validation loss, or a measured merged Schatten norm below C_TZ tau. Test p=3/2 only as a deliberately non-certified control.