Jevon Twitty

Viresh Pati, Yubin Kim, Vinh Pham et al.

This paper presents $\textbf{CAPS}$ (Clock-weighted Aggregation with Prefix-products and Softmax), a structured attention mechanism for time series forecasting that decouples three distinct temporal structures: global trends, local shocks, and seasonal patterns. Standard softmax attention entangles these through global normalization, while recent recurrent models sacrifice long-term, order-independent selection for order-dependent causal structure. CAPS combines SO(2) rotations for phase alignment with three additive gating paths -- Riemann softmax, prefix-product gates, and a Clock baseline -- within a single attention layer. We introduce the Clock mechanism, a learned temporal weighting that modulates these paths through a shared notion of temporal importance. Experiments on long- and short-term forecasting benchmarks surpass vanilla softmax and linear attention mechanisms and demonstrate competitive performance against seven strong baselines with linear complexity. Our code implementation is available at https://github.com/vireshpati/CAPS-Attention.

Jevon Twitty, Vinh Pham, Nitiwith Rotchanarak et al.

A persistent paradox in time-series forecasting is that structurally simple MLP and linear models often outperform high-capacity Transformers. We argue that this gap arises from a mismatch in the sequence-modeling primitive: while many time-series dynamics are governed by global temporal operators (e.g., filtering and harmonic structure), standard attention forms each output as a convex combination of inputs. This restricts its ability to represent signed and oscillatory transformations that are fundamental to temporal signal processing. We formalize this limitation as a simplex-constrained mixing bottleneck in softmax attention, which becomes especially restrictive for operator-driven time-series tasks. To address this, we propose $\textbf{Temporal Operator Attention (TOA)}$, a framework that augments attention with explicit, learnable sequence-space operators, enabling direct signed mixing across time while preserving input-dependent adaptivity. To make dense $N \times N$ operators practical, we introduce Stochastic Operator Regularization, a high-variance dropout mechanism that stabilizes training and prevents trivial memorization. Across forecasting, anomaly detection, and classification benchmarks, TOA consistently improves performance when integrated into standard backbones such as PatchTST and iTransformer, with particularly strong gains in reconstruction-heavy tasks. These results suggest that explicit operator learning is a key ingredient for effective time-series modeling.

Yubin Kim, Viresh Pati, Jevon Twitty et al.

Transformer architectures have established strong baselines in time series forecasting, yet they typically rely on positional encodings that assume uniform, index-based temporal progression. However, real-world systems, from shifting financial cycles to elastic biological rhythms, frequently exhibit "time-warped" dynamics where the effective flow of time decouples from the sampling index. In this work, we first formalize this misalignment and prove that rotary position embedding (RoPE) is mathematically incapable of representing non-affine temporal warping. To address this, we propose Symplectic Positional Embeddings (SyPE), a learnable encoding framework derived from Hamiltonian mechanics. SyPE strictly generalizes RoPE by extending the rotation group $\mathrm{SO}(2)$ to the symplectic group $\mathrm{Sp}(2,\mathbb{R})$, modulated by a novel input-dependent adaptive warp module. By allowing the attention mechanism to adaptively dilate or contract temporal coordinates end-to-end, our approach captures locally varying periodicities without requiring pre-defined warping functions. We implement this mechanism in StretchTime, a multivariate forecasting architecture that achieves state-of-the-art performance on standard benchmarks, demonstrating superior robustness on datasets exhibiting non-stationary temporal dynamics.