Embodied perception requires positional representations that capture both physical geometry and the spatial and temporal extent of observations. We introduce POD-RoPE (Physical Observation Distribution RoPE), which represents token observation footprints as distributions over camera centers, viewing directions, and sampling times. Under independent sampling, expected rotations encode relative relationships between distributions; when each distribution collapses to a single address, the construction recovers pointwise RoPE. The same rotation statistics support Q/K matching and coherence-dependent V/O transformations, allowing observation geometry to guide both information selection and content aggregation. Per-token pre-aggregation avoids increasing sequence length or processing pairs of observation nodes. We evaluate POD-RoPE across four robot policy architectures, simulation and real-world manipulation, and complementary reconstruction, depth-estimation, and novel-view-synthesis tasks. Integrating POD-RoPE into Fast-WAM improves mean success from 91.83% to 92.88% across all 50 RoboTwin 2.0 tasks, and from 32.5% to 52.5% on two real-robot tasks on Sharpa North. Controlled reconstruction ablations support the contributions of distribution encoding and V/O transformations. In H20 model-block benchmarks, POD-RoPE achieves median forward-plus-backward speedups of $1.91\times$ over P-RoPE and $5.34\times$ over RayRoPE, with a median latency overhead of 17.4% over Standard RoPE. These results support physical observation distributions as an effective and computationally efficient basis for geometry-aware attention.
Jia-Nan Wang, Zixun Huang, Kairui Li, Zilin Wang, Lei Wu
We study when and how momentum improves large-batch training in the one-pass regime, using power-law kernel regression as a tractable setting. We first characterize stochastic stability through the \emph{critical learning rate}, defined as the largest learning rate for stable training, and obtain $η_{\mathrm{SGD}}^{\mathrm{crit}}\eqsim 1$, $η_{\mathrm{Polyak}}^{\mathrm{crit}}\eqsim \min\{1,B(1-β)\}$, and $η_{\mathrm{Nesterov}}^{\mathrm{crit}}\eqsim \min\{1,B^q(1-β)\}$, where $B$ is the batch size, $β\in[0,1)$ is the momentum factor, and \(q>1\) is the capacity exponent governing spectral decay. Within the stable region, we derive scaling laws for the full risk dynamics, capturing the progression from an early transient through power-law decay to a noise floor. We then optimize the final-step risk under a fixed data budget, obtaining a three-regime phase diagram that reveals how the benefits of momentum change with batch size. Notably, Polyak enlarges the critical batch size, the largest batch size preserving the optimal small-batch data-scaling exponent, thereby allowing fewer sequential updates without sacrificing data efficiency. In contrast, Nesterov improves the optimal data-scaling exponent in the large-batch regime through look-ahead-induced suppression of noise amplification. Experiments on PLK regression quantitatively support the theoretical scaling predictions, while neural-network experiments provide qualitative evidence for the predicted batch-size dependence and stability behavior.