HyPE: Attention with Hyperbolic Biases for Relative Positional Encoding

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.

The attention operation is naively position invariant. However, positional information is fundamental to natural language, and therefore a variety of explicit position encodings have been developed in transformer-based models, such as rotary position encoding (RoPE). Although explicit position encodings have long been assumed to be required, recent methods that interleave local mixing layers, such as sliding window attention (SWA) and gated linear attention, while not encoding position (NoPE) in global attention layers has recently been shown to be successful at scale. How and why this approach works is not well-understood. In this paper, we develop an explanation of how hybrid models of this sort can implicitly encode position at global NoPE layers. Supported by both theoretical and empirical evidence, our central argument is that SWA and gated linear attention induce a recency bias in the residual stream that propagates to, and is selected by, the global attention logits. Moreover, in contrast to the implicit position encodings found in models with only global NoPE attention, in which positional information arises solely from the causal mask, the recency bias in hybrid models can be maintained across long sequences. In addition to deepening our understanding of how hybrid models encode position, these findings may provide insights for how to encode position in a way that can extrapolate to longer sequence lengths indefinitely.

Transformers with relative positional encodings often extrapolate to sequences longer than those seen during training, whereas transformers with learned absolute encodings typically do not. This is a robust empirical regularity, and the explanations offered for it so far are chiefly about expressivity, that is, about whether a length-generalizing solution exists. We give an optimization explanation. On a minimal fixed-offset retrieval task that isolates positional selection, the gap is governed by the implicit bias of the trained attention head: among the many solutions that fit short sequences, which one gradient descent actually selects. We prove that rotary encodings make the attention logit a function of relative offset alone, an exact equivariance, so whatever selection rule is learned at training lengths is reproduced verbatim at every longer length. Learned absolute encodings instead leave out-of-range positions unconstrained, and the trained head pins to a fixed absolute position inside the training range. We characterize the learned rotary rule as a low-rank ``carrier'' kernel aligned with the target offset, and we derive the resulting graceful accuracy decay as an attention-dilution law; both predictions are confirmed across seeds and offsets. A linear-attention control shows the mechanism is specific to softmax: without normalization, training selects a min-norm interpolant that does not extrapolate. The phenomenon, the equivariance, and the carrier all transfer to a multi-layer, multi-head transformer trained on a full-sequence length-generalization task. The account connects the implicit bias of attention, implicit bias for extrapolation in recurrent models, and the learning side of the RASP-L conjecture.