Many projection-based imaging modalities reconstruct unknown objects from collections of noisy projections related by unknown in-plane rotations, making effective estimation dependent on rotationally consistent alignment and aggregation. In cryogenic electron microscopy (cryo-EM), the extremely low signal-to-noise ratio (SNR) makes such multi-image integration essential. We introduce the *polar transformer*, a permutation-invariant architecture for *sets* of images built around an $\mathrm{SO}(2)^K$-equivariant *angular attention* mechanism that jointly aligns and aggregates information across images. This architecture leverages a stable polar representation in which we define polar CNN feature extractors and evaluate attention weights over all relative angular shifts efficiently via FFT-based correlations, guaranteeing equivariance to independent in-plane rotations of each image in the set. We evaluate the polar transformer on simulated cryo-EM images using denoising as a quantitative proxy. Across different regimes, polar transformers outperform strong single-image and image-set baselines, achieving a reduction of up to 40% in relative MSE at an SNR of 0.02 and improving downstream ab initio reconstruction resolution by 20%.

Geometry is invariant to viewpoint, which makes any collection of images a redundant encoding of a single 3D state. Existing feed-forward reconstruction models fail to exploit this: per-view methods emit overlapping, unaligned pointmaps that grow linearly with input count, while global-latent methods commit to a fixed, low-resolution output. We introduce Surflo, which compresses a variable number of unposed RGB views into K latent tokens-one global state-and decodes oriented 3D surface points by independently transporting them from noise onto the surface via flow matching. This frees the output from any fixed grid or token budget: the same latent yields from a few thousand to a million points in a single forward pass. To suppress the local inconsistencies inherent to independent per-point decoding, an inference-time guidance term correlates nearby points by injecting a photometric gradient during ODE integration. Surflo matches or surpasses feed-forward baselines on surface metrics, run an order of magnitude faster than optimization-based methods that require hundreds of views, and is the only feed-forward approach to combine a global latent with arbitrary-resolution decoding.

Rotation averaging on the circle, equivalently robust angular synchronization, underlies a wide range of geometric estimation problems, for example, gravity-aligned Structure-from-Motion, multi-way point-cloud registration, and in-plane alignment in single-particle cryo-electron microscopy. We present the first algorithm that certifiably solves the robust maximum-consensus formulation of this problem to global optimality. Our approach discretises the circle into uniform bins, formulates a pairwise Markov random field and solves it via branch-and-bound with two key novelties: an ICM-guided label-pruning rule that collapses the branching factor to a handful of candidates, and a node-constrained relaxation bound orders of magnitude tighter than the standard per-edge bound. Together, they reduce the search from millions of nodes to a few hundred on typical instances. On all tested problems, the algorithm terminates with a proof of global optimality, typically in a few seconds. Across synthetic graphs and three real-world domains -- image-based SfM on IMC 2023/2024, multi-way LiDAR point-cloud registration on KITTI and NSS, and cryo-EM angular synchronization on EMPIAR-10166 -- the certified solution achieves sub-degree accuracy at outlier rates where other baselines fail. Our code will be made publicly available.

Current brain–machine interfaces (BMI) face fundamental limitations due to inherent latency from reliance on delayed motor cortical signals, and computational overhead, restricting their effectiveness in real-time applications such as rehabilitation therapy. Recent neuroscience indicates that prefrontal and sensory cortical activities precede motor execution, offering an opportunity for proactive intent prediction. However, challenges remain in acquiring multi-region neural data, efficiently decoding high-dimensional signals, and ensuring model interpretability. To address these, we developed a high-density electrocorticography (ECoG)-based paradigm based on marmosets and introduced an information-bottleneck-driven graph transformer (ECoG-IBGT), reframing neural decoding as graph classification. Our method achieves 99.29% accuracy up to 400 ms before action onset with inherent interpretability, laying the foundation for reliable, low-latency BMIs. The source code is available at https://anonymous.4open.science/r/ecog-ibgt-code-F2D2.