This paper focuses on estimating probability distributions over the set of 3D rotations (SO(3)) using deep neural networks. Learning to regress models to the set of rotations is inherently difficult due to differences in topology between RN and SO(3). We overcome this issue by using a neural network to output the parameters for a matrix Fisher distribution since these parameters are homeomorphic toR9. By using a negative log likelihood loss for this distribution we get a loss which is convex with respect to the network outputs. By optimizing this loss we improve state-of-the-art on several challenging applicable datasets, namely Pascal3D+, ModelNet10-SO(3).

This paper focuses on estimating probability distributions over the set of 3D rotations (SO(3)) using deep neural networks. Learning to regress models to the set of rotations is inherently difficult due to differences in topology between R^N and SO(3). We overcome this issue by using a neural network to output the parameters for a matrix Fisher distribution since these parameters are homeomorphic to R^9 . By using a negative log likelihood loss for this distribution we get a loss which is convex with respect to the network outputs. By optimizing this loss we improve state-of-the-art on several challenging applicable datasets, namely Pascal3D+, ModelNet10-SO(3). Our code is available at https://github.com/Davmo049/Public

Diffusion and flow matching models have recently emerged as promising approaches for peptide binder design. Despite their progress, these models still face two major challenges. First, categorical sampling of discrete residue types collapses their continuous parameters into onehot assignments, while continuous variables (e.g., atom positions) evolve smoothly throughout the generation process. This mismatch disrupts the update dynamics and results in suboptimal performance. Second, current models assume unimodal distributions for side-chain torsion angles, which conflicts with the inherently multimodal nature of side chain rotameric states and limits prediction accuracy. To address these limitations, we introduce PepBFN, the first Bayesian flow network for full atom peptide design that directly models parameter distributions in fully continuous space. Specifically, PepBFN models discrete residue types by learning their continuous parameter distributions, enabling joint and smooth Bayesian updates with other continuous structural parameters. It further employs a novel Gaussian mixture based Bayesian flow to capture the multimodal side chain rotameric states and a Matrix Fisher based Riemannian flow to directly model residue orientations on the $\mathrm{SO}(3)$ manifold. Together, these parameter distributions are progressively refined via Bayesian updates, yielding smooth and coherent peptide generation. Experiments on side chain packing, reverse folding, and binder design tasks demonstrate the strong potential of PepBFN in computational peptide design.

Diffusion models are a popular class of generative models trained to reverse a noising process starting from a target data distribution. Training a diffusion model consists of learning how to denoise noisy samples at different noise levels. When training diffusion models for point clouds such as molecules and proteins, there is often no canonical orientation that can be assigned. To capture this symmetry, the true data samples are often augmented by transforming them with random rotations sampled uniformly over $SO(3)$. Then, the denoised predictions are often rotationally aligned via the Kabsch-Umeyama algorithm to the ground truth samples before computing the loss. However, the effect of this alignment step has not been well studied. Here, we show that the optimal denoiser can be expressed in terms of a matrix Fisher distribution over $SO(3)$. Alignment corresponds to sampling the mode of this distribution, and turns out to be the zeroth order approximation for small noise levels, explaining its effectiveness. We build on this perspective to derive better approximators to the optimal denoiser in the limit of small noise. Our experiments highlight that alignment is often a `good enough' approximation for the noise levels that matter most for training diffusion models.

Advances of deep learning techniques have resulted in improvements in estimation of 3D orientation. However, precise orientation estimation remains an open problem.

This paper focuses on estimating probability distributions over the set of 3D ro- tations (SO(3)) using deep neural networks. Learning to regress models to the set of rotations is inherently difficult due to differences in topology between R N and SO(3). We overcome this issue by using a neural network to out- put the parameters for a matrix Fisher distribution since these parameters are homeomorphic to R 9 . By using a negative log likelihood loss for this distri- bution we get a loss which is convex with respect to the network outputs. By optimizing this loss we improve state-of-the-art on several challenging applica- ble datasets, namely Pascal3D, ModelNet10-SO(3).

Estimating the 3DoF rotation from a single RGB image is an important yet challenging problem. As a popular approach, probabilistic rotation modeling additionally carries prediction uncertainty information, compared to single-prediction rotation regression. For modeling probabilistic distribution over SO(3), it is natural to use Gaussian-like Bingham distribution and matrix Fisher, however they are shown to be sensitive to outlier predictions, e.g. $180^\circ$ error and thus are unlikely to converge with optimal performance. In this paper, we draw inspiration from multivariate Laplace distribution and propose a novel rotation Laplace distribution on SO(3). Our rotation Laplace distribution is robust to the disturbance of outliers and enforces much gradient to the low-error region that it can improve. In addition, we show that our method also exhibits robustness to small noises and thus tolerates imperfect annotations. With this benefit, we demonstrate its advantages in semi-supervised rotation regression, where the pseudo labels are noisy. To further capture the multi-modal rotation solution space for symmetric objects, we extend our distribution to rotation Laplace mixture model and demonstrate its effectiveness. Our extensive experiments show that our proposed distribution and the mixture model achieve state-of-the-art performance in all the rotation regression experiments over both probabilistic and non-probabilistic baselines.