Low-Rank Latent Atlases for Atomistic Relaxation

Learned atomistic latent variables are often treated as if they were ordinary Euclidean coordinates. For structural relaxation this assumption can be badly misleading: the decoder may be redundant, its local sensitivity can be highly anisotropic, and re-encoding after a physical move changes the coordinate chart. We study a fixed-size latent representation with 512 scalar degrees of freedom and reinterpret its decoder as a sequence of \emph{local charts} over atomic configuration space. At each relaxation step, the decoder Jacobian defines a physical tangent image; global translations are removed and a truncated singular-value decomposition retains only directions that are both physically responsive and numerically stable. On a pilot benchmark of perturbed MPtrj structures with a frozen MACE-MP oracle, the retained tangent has mean rank only about 2.1 while capturing roughly 86--90\% of the physical force direction. Spectral geometric FIRE reaches 91.7\% convergence across three seeds with 25.65 oracle evaluations on average. A reduced Riemannian BFGS solver reveals a much faster successful regime (14.85 evaluations on successful trajectories) but loses robustness; guarding the curvature model restores FIRE-level success while erasing most of the speed gain. A learned two-force macro-solver subsequently improves robustness on a disjoint holdout (84.4\% versus 65.6\% for matched FIRE) but nearly doubles oracle cost. Further dimensional diagnostics show that adding a third temporal force direction gives only a 0.0028 mean oracle-confidence gain and is essentially uncorrelated with the third instantaneous singular mode. These results support a local low-rank geometric view of latent atomistic relaxation while showing that low dimensionality alone is not yet sufficient for acceleration.