Subspace Clifford Networks

In this work, we propose Subspace Clifford Networks, a reformulation that replaces global cyclic rolling with grouped subspace interactions. Instead of sampling sparse cyclic diagonals across the entire $D$-dimensional channel space, we project the features into an aligned space $\R^{\tilde{D}} = \bigoplus_{g=1}^{G} V_g$ partitioned into $G$ low-dimensional groups $V_g \cong \R^k$ with $k \in \{2, 3, 4\}$. Within each $k$-dimensional group, we compute all $\binom{k}{2}$ wedge coordinates and concatenate them with $\operatorname{SiLU}$-activated coordinate-wise products, while sandwiching the grouped interaction between learnable $1\times 1$ linear projections to enable cross-group channel mixing. This design reduces intermediate feature expansion from $2|\mathcal{S}|D$ (typically $10D$ for 5 shifts) down to $1.5\tilde{D}$ ($k=2$), $2.0\tilde{D}$ ($k=3$), or $2.5\tilde{D}$ ($k=4$). Furthermore, exploiting the small, fixed group size $k \in \{2, 3, 4\}$, we implement a fused CUDA kernel that evaluates the forward interaction and a recomputation-based analytical backward pass using thread-local variables without allocating global-memory buffers for elementwise intermediates. Finally, we extend the backbone from an isotropic columnar layout to a multi-resolution Hierarchical Clifford Network with configurable per-stage subspace dimensions. Experiments on CIFAR-100 show that the proposed models substantially reduce parameter count and training time across both regimes, while the hierarchical variants also improve classification accuracy over the evaluated shift-based hierarchical baselines.