HAMON: Passive Optical Sequence Mixing for Long-Horizon Forecasting

Transformer architectures have been widely adopted for time series forecasting, yet whether the representational mechanisms that make them powerful in NLP actually engage on time series data remains unexplored. The persistent competitiveness of simple linear models such as DLinear has fueled ongoing debate, but no mechanistic explanation for this phenomenon has been offered. We address this gap by applying sparse autoencoders (SAEs), a tool from mechanistic interpretability, to probe the internal representations of PatchTST. We first establish that a single-layer, narrow-dimensional transformer matches the forecasting performance of deeper configurations across commonly used benchmarks. We then train SAEs on the post-GELU intermediate FFN activations with dictionary sizes ranging from 0.5x to 4.0x the native dimensionality. Expanding the dictionary yields negligible downstream performance change (average 0.214%), with large portions of overcomplete dictionaries remaining inactive. Targeted causal interventions on dominant latent features produce minimal forecast perturbation. Across all evaluated settings, we observe no empirical evidence that the analyzed FFN representations rely on strong superposition. Instead, the representations remain sparse, stable under aggressive dictionary expansion, and largely insensitive to latent interventions. These results demonstrate that superposition is not necessary for competitive performance on standard forecasting benchmarks, suggesting they may not demand the rich compositional representations that drive transformer success in language modeling, and helping explain the persistent competitiveness of simple linear models

Forecasting multiple time-series with high-dimensional covariates presents a core challenge: unifying common temporal patterns while retaining meaningful series-specific information. We introduce Hopformer (Homogeneity-Pursuit Transformer), a two-stage framework that addresses this challenge. In the first stage, we perform a Sparsity Pattern Aggregation (SPA) scheme extracting a common low-variance trend that incorporates the covariates. This acts as a homogenization layer. In the second stage, a LoRA-fine-tuned Transformer models the remaining complex dependencies in the residual. Our method is theoretically grounded. We prove that SPA achieves a near-optimal bias-variance trade-off via an oracle inequality. We also provide generalization bounds for the second stage under dependent time series data. Hopformer sets a new state of the art, improving MASE by an average of 6.56% across synthetic and real-world forecasting benchmarks.

Adaptive patching is a recent and compelling proposal for time-series Transformers: allocate finer patches where the sequence looks locally informative. This paper asks under what conditions a content-adaptive patching operator should outperform a tuned uniform one. Local heterogeneity alone is not enough: under pointwise forecasting losses, a complex-looking region is not automatically one where finer patching reduces the loss. We model patching as a budgeted bitrate allocation and derive an explicit threshold that a dynamic patching rule must satisfy to beat a well-tuned uniform baseline, then bound the achievable improvement both locally (a quadratic surrogate) and globally (a strong-convexity bound under the model's assumptions). Two structural results follow: without a coupling constraint, scalar local complexity cannot produce a non-uniform optimum under a common loss landscape; and once the backbone is trained to its representation-aware optimum, the alignment gain collapses around a well-tuned uniform patch size. To test these predictions, we run a controlled isolation study on three representative architectures, replacing each adaptive mechanism with a uniform patch-size sweep while keeping the backbone, data, and training protocol fixed. On standard long-horizon forecasting benchmarks, the validation-selected uniform baseline is competitive with the dynamic counterpart, with per-setting effects concentrated near zero and no consistent directional advantage once results are aggregated by dataset. The larger gains we do observe are method- and dataset-specific. Adaptive patching should therefore be evaluated against a tuned uniform baseline; its value depends on whether a cheap and reliable routing signal can identify where finer patches actually reduce forecasting loss.