Xitong Hu

Zhihang Xin, Rui Wang, Xitong Hu et al.

Vision Transformers have achieved remarkable success in computer vision, but their common use of learnable one-dimensional positional encodings weakens the inherent two-dimensional spatial structure of images after patch flattening. Existing positional encodings often lack geometric constraints and do not preserve a monotonic relationship between Euclidean spatial distances and sequential index distances, limiting ViTs' ability to exploit spatial proximity priors. Motivated by the usefulness of periodicity in positional encoding, we propose Weierstrass elliptic Positional Encoding (WePE), a mathematically grounded method for encoding two-dimensional coordinates in the complex domain. WePE maps normalized 2D patch coordinates onto the complex plane and constructs compact four-dimensional positional features using the Weierstrass elliptic function and its derivative. The double periodicity provides a principled representation of 2D positions, and its intrinsic lattice structure naturally matches the regular geometry of image patch grids. Its nonlinear geometric properties help model spatial distance relationships more faithfully, while the algebraic addition formula enables relative positional information between arbitrary patch pairs to be derived directly from their absolute encodings. WePE is plug-and-play and resolution-agnostic, allowing seamless integration into existing ViTs. Extensive experiments show that WePE brings consistent performance gains in most settings. With precomputed lookup tables, these improvements introduce no noticeable computational or memory overhead. Additional analyses and ablation studies further validate the effectiveness of the proposed method.

Zhihang Xin, Xitong Hu, Rui Wang

Vision Transformers have demonstrated remarkable success in computer vision tasks, yet their reliance on learnable one-dimensional positional embeddings fundamentally disrupts the inherent two-dimensional spatial structure of images through patch flattening procedures. Traditional positional encoding approaches lack geometric constraints and fail to establish monotonic correspondence between Euclidean spatial distances and sequential index distances, thereby limiting the model's capacity to leverage spatial proximity priors effectively. We propose Weierstrass Elliptic Function Positional Encoding (WEF-PE), a mathematically principled approach that directly addresses two-dimensional coordinates through natural complex domain representation, where the doubly periodic properties of elliptic functions align remarkably with translational invariance patterns commonly observed in visual data. Our method exploits the non-linear geometric nature of elliptic functions to encode spatial distance relationships naturally, while the algebraic addition formula enables direct derivation of relative positional information between arbitrary patch pairs from their absolute encodings. Comprehensive experiments demonstrate that WEF-PE achieves superior performance across diverse scenarios, including 63.78\% accuracy on CIFAR-100 from-scratch training with ViT-Tiny architecture, 93.28\% on CIFAR-100 fine-tuning with ViT-Base, and consistent improvements on VTAB-1k benchmark tasks. Theoretical analysis confirms the distance-decay property through rigorous mathematical proof, while attention visualization reveals enhanced geometric inductive bias and more coherent semantic focus compared to conventional approaches.The source code implementing the methods described in this paper is publicly available on GitHub.