Construct a mode-mixing layer from a few sphere-preserving vector fields and shift operators rather than a dense learned Fourier convolution. A commutator of two low-complexity shifted rotations produces a new interaction at the sum of their offsets, allowing long-range or high-frequency mode coupling to be synthesized with only a small number of primitive operators. The layer can be used whenever each feature vector is normalized to the sphere, or more generally as a norm-preserving block on…
Replace a dense Haar or Gaussian random projection with a streamed product of random two-coordinate rotations followed by coordinate subsampling. The transform is exactly orthogonal before subsampling, requires only a list of rotation triples, and the paper's pseudo-mixing result predicts that degree-two statistics relevant to norm preservation and Johnson–Lindenstrauss embeddings become Haar-like after only O(n polylog(n)) rotations.
Use a small number of learned low-mode controls and a fixed bank of Lie words to generate structured high-mode updates. This gives a parameter-efficient adapter for spectral operators or sequence models: the trainable degrees of freedom live only in the low modes, while commutator compositions provide deterministic propagation paths to larger offsets. The design is especially suitable for fine-tuning a pretrained Fourier or state-space model under a strict parameter budget.
Replace a dense unconstrained channel-mixing matrix with a differentiable product of exponentials of a few skew-symmetric generators and their iterated commutators. The resulting layer is exactly orthogonal, preserves feature norms, and can express rotations in directions not explicitly stored as independent parameters. This is especially suitable for residual MLP blocks, recurrent state transitions, and networks processing rotation- or pose-valued features.
Paper: Nonlinear Controllability and the Propagation of Local Information: From the Kalman Family to Lie Brackets, Rotation Groups, and Reachable SubgroupsarXiv:2608.20094
Use the paper's explicitly solved SU(2)-based extremal flow as a structured recurrent transition instead of learning an unconstrained dense recurrent matrix. The transition has only two scalar parameters, a radius/frequency r and phase phi, while its rotating coefficient pattern continuously mixes four real state coordinates and can be integrated with a norm-preserving Cayley transform.
Replace an unconstrained low-rank adapter or similarity projection with a learned subspace carrying a prescribed signed metric. The module learns an orthonormal basis U for a k=p+q dimensional subspace, forces the compressed form U^*I_{m,n}U to have p positive and q negative eigenvalues, and uses the resulting pseudo-inner product for signed attention or retrieval scores.
Represent a population of N circular latent states using a three-parameter Möbius transformation applied to fixed uniform reference phases, rather than learning N unrelated angles. The resulting states remain on the circle by construction and can model concentrated or nearly uniform phase populations through a single concentration parameter.