Factorized Gradients for Scalable Highly-expressive Parametric Diffeomorphisms

Continuous Piecewise-Affine Based (CPAB) transformations, introduced by Freifeld et al. (2015, 2017), are flows obtained by integrating continuous piecewise-affine (CPA) velocity fields. These expressive transformations have proven useful in applications including nonlinear temporal warping and trainable activation functions in 1D, as well as nonrigid spatial deformations in 2D and 3D. While computational costs were not an issue in 1D, they originally limited the practical expressiveness of CPAB transformations in higher dimensions, especially in 3D. Our recent work (Aflalo et al., 2026) removed this bottleneck, enabling efficient optimization with much finer tessellations and substantially greater expressiveness. There is, however, a mathematical subtlety that was overlooked in the original CPAB papers (Freifeld et al., 2015, 2017). CPAB transformations are always bi-Lipschitz homeomorphisms, implying that both the transformations and their inverses are differentiable almost everywhere. They are not, however, necessarily diffeomorphisms. This note remedies the omission by identifying sufficient regularity conditions under which a CPAB transformation is guaranteed to be a diffeomorphism. We explain how differentiability can fail and how the proposed conditions prevent this failure. In 1D and 2D, we show that these conditions can be easily enforced and, for commonly used tessellations, hold with probability one under standard sampling assumptions. We expect analogous behavior in 3D.

Persistence-based topological optimization deforms a point cloud $X \subset \mathbb{R}^d$ by minimizing objectives of the form $L(X) = \ell(\mathrm{Dgm}(X))$, where $\mathrm{Dgm}(X)$ is a persistence diagram. In practice, optimization is limited by two coupled issues: persistent homology is typically computed on subsamples, and the resulting topological gradients are highly sparse, with only a few anchor points receiving nonzero updates. Motivated by diffeomorphic interpolation, which extends sparse gradients to smooth ambient vector fields via Reproducing Kernel Hilbert Space (RKHS) interpolation, we propose a more scalable pipeline that improves both subsampling and gradient extension. We introduce subsampling via random slicing, a lightweight scheme that promotes iteration-wise geometric coverage and mitigates density bias. We further replace the costly kernel solve with a fast Nadaraya-Watson (NW) Gaussian convolution, producing a globally defined smooth update field at a fraction of the computational cost, while being more suited for topological optimization tasks. We provide theoretical guarantees for NW smoothing, including anchor approximation bounds and global Lipschitz estimates. Experiments in $2$D and $3$D show that combining random slicing with NW smoothing yields consistent speedups and improved objective values over other baselines on common persistence losses.

Contact-rich inverse problems require efficient gradients and reliable enforcement of non-penetration. Existing differentiable methods provide fast simulation and rigid-deformable coupling, but contact treatments based on penalties or velocity-level constraints can permit penetration or positional drift. Incremental Potential Contact (IPC) maintains non-penetration through geometric barriers and {continuous collision detection (CCD)}. However, repeated forward and backward evaluations remain costly in full-space finite-element (FEM) implementations, particularly when nearly rigid components retain many vertex {degrees of freedom (DoFs)}. We present DiffGIPC, a GPU-accelerated differentiable IPC framework for optimizing interacting deformable and nearly rigid bodies. Our affine-body derivatives enable {rest-shape} design, initial-state optimization, and articulated control while retaining only twelve motion DoFs per affine body. These gradients account for how changes in geometry, placement, and actuation affect other objects through collision and friction. We preserve these interactions in one coupled FEM-affine adjoint system per frame, reducing the backward-system size without discarding the dependencies that drive the inverse problem. GPU-optimized backward computation further reduces the cost of repeated gradient evaluations. A reusable PyTorch interface connects supported physical parameters to user-defined parameter maps and losses across design, identification, motion, placement, and articulated control. Experiments demonstrate peak speedups over DiffIPC of approximately 150.0x with full-space FEM and 4133.3x with affine bodies for combined forward simulation and backward differentiation.