Refined Topology-Pass Scheduling: Damped Trust-Region Compensation for Dynamic k-NN Graphs in SE(3)-Equivariant Structure Prediction

Jiankai Huang, Hanwen Shen

Equivariant structure-prediction pipelines alternate coordinate updates with the construction of a k-nearest-neighbour (kNN) graph whose membership is a discrete, piecewise-constant function of the coordinates. Rebuilding that graph at every block boundary (PB) is the expensive part of an epoch, whereas reusing one frozen graph (F) is fast but leaves a topology gap. We study a two-pass design in which the cheap pass records a small amount of state (margins, kNN radii, measured per-boundary displacements) and a second pass compensates conservatively. We formulate the choice of the graph-construction basis as a damped fixed-point iteration whose unique fixed point is exactly PB, and derive (i) exact non-flip certificates with the two-sided constant $m_i/4$ and one-sided constant $m_i/2$, (ii) the minimum-variance (BLUE) replay gain $γ^*$ and the resulting feasibility crossover, (iii) an $\ell_\infty$ trust-region projection that preserves convergence, and (iv) an analytic near-tie spacing law $E[m] \approx r_k/(3k)$ that matches measurements to 1\%. Three compensation levels are implemented in C++ (L1: frozen set with live geometry; L2: margin-gated local repair; L3: damped,…