RODR: Riemannian Orthogonally Decoupled Regularization for Disentangled Manifold Representation

Point cloud denoising is essentially a geometric recovery task that aims to reconstruct the intrinsic structure of a smooth 2D Riemannian manifold embedded in R^3 from noisy, discrete ambient-space samples. Despite the remarkable progress of modern manifold-aware encoders and generative transport models in geometric representation learning, a fundamental objective-geometry mismatch remains underexplored. Theoretically, we identified that this mismatched coupling leads to geometric gradient interference, where conflicting optimization objectives result in structural degradation and point clustering. We introduce Riemannian Orthogonally Decoupled Regularization (RODR) to reformulate the optimization trajectory by disentangling the normal (fitting) and tangential (distribution) components. Guided by a vector-attention and entropy-aware adaptive strategy, RODR effectively preserves high-fidelity geometric details while maintaining sampling uniformity. Experiments demonstrate that RODR reaches performance comparable to state-of-the-art baselines and suggests improved distribution regularity and reduced local aggregation effectively. Our work establishes a generic and interpretable framework for disentangled geometric optimization in point cloud processing.

Geometric 2D Scene Graph Generation

In production processes for consumer products, assembly instructions are essential not only for planning but also for executing the production process. Likewise in robotics, it is crucial for an assembly robot to understand how components fit together and can be assembled. To facilitate these tasks, we contribute a method for constructing scene graphs to represent and characterize assembly relationships between components. Our approach does not rely on semantic data and is capable of handling a very small dataset. To realize this, the output of a Faster R-CNN model is used to create geometric representations, which are then processed by a transformer architecture to generate an adjacency matrix. This matrix serves as input to a Siamese network that uses message passing based on an attentional graph convolutional network (aGCN) architecture to characterize the connections between the components. We validate our method on a study dataset of toy model components which can be assembled into transportation vehicles.

Understanding Geometric Representations in Self-Supervised Vision Transformers via Subspace Intervention

We introduce a controlled subspace intervention framework to investigate how self-supervised Vision Transformers (ViTs) encode dense geometric information. While linear probing is widely used to assess geometric representations, it treats features as a black box, failing to disentangle the underlying topology. To address this issue, we decompose the weights of converged linear probes to isolate the low-rank subspaces containing explicit geometric signals using Singular Value Decomposition (SVD). Our perspective yields three key insights: (1) Pre-training objectives determine how features are encoded. DINOv2 aligns spatial features for efficient linear extraction, while Masked Autoencoders (MAE) tend to disperse these signals, requiring a broader spatial context. (2) Explicit geometric representations are highly compressible, suggesting dense predictive heads could potentially be constrained to low-rank subspaces with minimal performance loss. (3) The layer-wise task affinity suggests that geometric precision peaks at intermediate layers before yielding to semantic abstraction in the final layers. By connecting internal encoding mechanics with downstream performance, these findings provide a basis for effective feature selection and lightweight decoder design. The source code is available at https://github.com/Zhou-Weichen/Geosubprobe.