Positional Encodings Anchor Spatial Structure in Vision Transformers: A Geometric Perspective on Robustness

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.

Positional encoding is a fundamental component of Transformer architectures, as it injects information about the spatial or sequential arrangement of inputs. Among recent alternatives to standard absolute and sinusoidal encodings, similarity-based positional encoding (simPE) has emerged as a flexible framework for representing positional structure through pairwise relations. simPE was originally designed for medical imaging applications, where geometric robustness is especially relevant: small rotations naturally arise during image acquisition, induced by imaging instruments, patient positioning, or slight acquisition misalignments. Despite its empirical promise, the theoretical behavior of simPE under geometric perturbations has not been fully characterized. In this paper, we study the robustness of simPE with respect to rotations, combining formal theoretical analysis with experimental validation. We first show that simPE is generally not rotation-invariant. We then prove that, under mild Lipschitz assumptions on the elementary components, simPE is stable under rotational perturbations and derive explicit perturbation bounds in Frobenius norm. We validate these findings experimentally on four controlled datasets--a synthetic Arrow dataset, a synthetic Shapes dataset (four geometric shape categories), a synthetic Digits dataset, and a benchmark image classification dataset (FashionMNIST)--in which training and validation images are kept in a fixed canonical orientation while test images are subjected to increasing rotation angles. Across all datasets, simPE consistently outperforms standard learned positional encoding in terms of accuracy, F1 score, precision, and recall under rotation, particularly in the small-to-moderate angle regime, corroborating the theoretical stability guarantees.

Vision Transformers (ViTs) commonly rely on injected positional mechanisms to address self-attention's permutation invariance. Motivated by the spatial regularities of natural images, we ask whether spatial organization can be induced from data rather than explicitly injected. Under controlled, matched from-scratch training, we propose Active Spatial Guidance (Guidance), a training-only objective that disables positional injection and applies an auxiliary 2D coordinate-regression loss to the final-layer patch tokens. The guidance head is used only during training and removed for inference; the deployed model consists of a positional-injection-free ViT encoder and the task-specific prediction module. Using DINOv3 ViT backbones, Guidance consistently improves performance on ImageNet-100 classification, ADE20K semantic segmentation, and Hypersim monocular depth estimation, outperforming strong injected baselines such as learned absolute positional embeddings and rotary positional embeddings under identical training protocols. On ImageNet-100, broader comparisons against representative injected positional designs further support Guidance's effectiveness. Guidance also improves robustness under resolution transfer, and multi-resolution training further strengthens accuracy across input sizes. Overall, our results suggest that spatial inductive bias in ViTs need not be architecturally injected, but can be shaped through training-time supervision. The code used for training and evaluation is publicly available in https://github.com/cloudlc/asg.