TOGEARI: Interaction-Space Preconditioning for Condensed Finite-Element Systems with IPC Contact

What the math gives to ML

The paper's transferable contribution is an interaction-space construction for low-rank preconditioning: instead of selecting parameter-space directions directly, it selects directions after measuring how they propagate through a reusable core inverse. For a system of the form M+U^TU, the operator Gram matrix G_M=UM^{-1}U^T identifies interaction combinations that are poorly handled by the core-preconditioned system, and a truncated Woodbury correction targets those directions. In neural networks, the same mechanism can build a low-rank curvature or gradient-interaction correction on top of a diagonal or block-diagonal optimizer preconditioner, with an explicit rank-versus-conditioning tradeoff.