Intrinsic Geometry Matching: learning observation-equivariant process geometry

High-dimensional datasets often concentrate near low-dimensional structures, but estimating their geometry from samples typically relies on graphs and kernels that scale poorly with dataset size and dimension. We propose Riemannian metric matching: a denoising probabilistic framework for learning the Riemannian geometry of data using neural networks. Specifically, we learn the carré du champ operator, which, using diffusion geometry, gives us access to the Riemannian geometry toolkit for downstream machine learning and statistical tasks. Our key observation is that the carré du champ operator can be formulated as a conditional expectation over random perturbations of the data, which can be exploited for sample-wise training and constant cost, amortized inference without explicit kernel construction. Empirically, metric matching rivals or improves the accuracy of $k$-NN-based diffusion geometry estimators, while enabling amortized inference that is up to $400\times$ faster, and supports graph-free geometric analysis on high-dimensional images where nearest neighbors break down.

Geometric analysis fundamentally distinguishes between extrinsic and intrinsic perspectives. The dominant paradigm in current 3D representation learning relies on either extrinsic spatial structures or high-level semantics, struggling to capture the essence of shape identity and underlying manifold topology. To bridge this gap, we introduce a novel 3D representation learning paradigm, namely PRISM, for Pre-training, which learns isometric embeddings by Recovering the Intrinsic Surface geodesic Metric. PRISM incorporates a topology-enforcing objective that explicitly constrains the structure of latent space, alongside a specialized two-stage training recipe mitigating sample imbalance inherent in the distribution of geodesic distances. Experiments demonstrate that our approach shows satisfactory accuracy, robustness, and high efficiency in geodesic distance prediction and achieves superior performance across diverse downstream tasks, including shape recognition, surface parameterization, and non-rigid correspondence. The code will be publicly available at https://github.com/AidenZhao/PRISM.

Recovering metric 3D geometry from images is fundamentally ill-posed due to perspective projection ambiguity. Current geometry models incorporate supplied or predicted intrinsics during inference, but do not use them to supervise model adaptation. To refine these predictions, test-time adaptation (TTA) typically relies on external depth measurements. In this paper, we instead use camera intrinsics as an output-side geometric verifier. Comparing the point cloud's viewing rays with those defined by the known intrinsics yields a direct measure of geometric inconsistency. We propose IntriAdapt, a TTA framework that minimizes this closed-form geometric residual. By optimizing a lightweight LoRA adapter on a frozen geometry foundation model, our method aligns the prediction with the specific camera model without depth sensors or external priors. We further extend this approach to video sequences, demonstrating that adapting a short sequence generalizes to other sequences captured by the same camera. The adapted model processes held-out images in a feed-forward pass, without repeating optimization for each frame. Experiments on ten single-image and seven sequence benchmarks across multiple base models show that IntriAdapt consistently outperforms both unadapted and oracle-intrinsics baselines in metric accuracy and relative geometry.