SO(3)-RoPE for Spherical Transformers

Relative position embedding has become a standard mechanism for encoding positional information in Transformers. However, existing formulations are typically limited to a fixed geometric space, namely 1D sequences or regular 2D/3D grids, which restricts their applicability to many computer vision tasks that require geometric reasoning across camera views or between 2D and 3D spaces. To address this limitation, we propose URoPE, a universal extension of Rotary Position Embedding (RoPE) to cross-view or cross-dimensional geometric spaces. For each key/value image patch, URoPE samples 3D points along the corresponding camera ray at predefined depth anchors and projects them into the query image plane. Standard 2D RoPE can then be applied using the projected pixel coordinates. URoPE is a parameter-free and intrinsics-aware relative position embedding that is invariant to the choice of global coordinate systems, while remaining fully compatible with existing RoPE-optimized attention kernels. We evaluate URoPE as a plug-in positional encoding for transformer architectures across a diverse set of tasks, including novel view synthesis, 3D object detection, object tracking, and depth estimation, covering 2D-2D, 2D-3D, and temporal scenarios. Experiments show that URoPE consistently improves the performance of transformer-based models across all tasks, demonstrating its effectiveness and generality for geometric reasoning. Our code is available on our project website: https://urope-pe.github.io/.

We introduce Hyperbolic Symplectic Positional Encoding (HySPE), grounding positional attention in non-compact symplectic transformations. While canonical Rotary Position Embedding (RoPE) parameterizes the compact, elliptic branch of $\Sp(2,\R)$ via rotations, HySPE operationalizes its hyperbolic branch via a damped symmetric composition of dual shears, yielding a conformally symplectic contraction with two spectral decay rates per channel pair. To eliminate the exponential representation drift inherent to naive absolute factorizations, we diagonalize the operator in its invariant eigenbasis and introduce blockwise coordinate rebasing with adaptive centered execution. This guarantees length-independent numerical bounds while matching cached RoPE forward latency (7.21,ms on an RTX 4090). On TinyShakespeare, HySPE-UltraLong maintains an invariant perplexity of 4.810 up to $16\times$ zero-shot extrapolation ($L=4096$), whereas RoPE degrades to 131.198. Scaled to a 51M-parameter subword Transformer on WikiText-103 ($L_{\text{train}}=512$), HySPE closely matches RoPE in-domain while robustly extrapolating to length 8192, reducing tail perplexity by 83.9% over RoPE. While these controlled experiments establish HySPE's extrapolation robustness and numerical stability, evaluating its scaling behavior on large-scale foundation models remains an important direction for future investigation.

We present a new class of efficient attention mechanisms applying universal 3D Relative Positional Encoding (RPE) methods given by arbitrary integrable modulation functions $f$. They lead to the new class of 3D-Transformer models, called \textit{RelFlexformers}, flexibly integrating those RPEs, and characterized by the $O(L \log L)$ time complexity of the attention computation for the $L$-length input sequences. RelFlexformers builds on the theory of the Non-Uniform Fourier Transform (NU-FFT), naturally generalizing several existing efficient RPE-attention methods from structured settings with tokens homogeneously embedded in unweighted grids into general non-structured heterogeneous scenarios, where tokens' positions are arbitrarily distributed in the corresponding 3D spaces. As such, RelFlexformers can be applied in particular to model point clouds. Our extensive empirical evaluation on a large portfolio of 3D datasets confirms quality improvements provided by the NU-FFT-driven attention modulation techniques in the RelFlexformers.