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

arXiv: https://arxiv.org/abs/2610.19246 · HTML: https://mlsanity.com/paper/2610.19246/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 15 references that are in the map.
- First posted on arXiv 13 Oct 2026.

## Builds on (most cited first, 5 of 5)

- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs](https://mlsanity.com/paper/1603.09320/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)

## Abstract (arXiv)

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 mi/4 and one-sided constant mi/2, (ii) the minimum-variance (BLUE) replay gain γ^* and the resulting feasibility crossover, (iii) an ℓ_∞ trust-region projection that preserves convergence, and (iv) an analytic near-tie spacing law E[m] ≈ rk/(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, projected basis) and validated under a matched-conditions protocol (identical seed, MSA depth and trajectory family, only the topology mode differs). L2 reproduces the PB edge set exactly (D = 0.0000 at every boundary), L1 stays within 0.6% and L3 interpolates monotonically: L2 reproduces the PB edge set exactly when the trust region is removed, and L3 interpolates between the frozen and the PB topology. The recorded trajectory itself yields a measured 1.5× Pass-1 and 1.76× epoch speed-up with no measurable loss gap. An open-source implementation of the runtime and of all the experiments reported here is available at https://github.com/AlbertHuang96/ProteinPredictMachineLearning.