Many imaging problems require computing spatial transformations induced by spatially varying intensity, feature, or density fields. Canonical examples include distortion correction, deformable image registration, atlas-based segmentation, and deformation-driven image analysis. These tasks can be formulated as geometric mapping problems in which the transformation is constrained to preserve local structure, control boundary behavior, or regulate angular distortion. Such formulations typically lead to variational models, diffusion processes, or elliptic partial differential equations. However, repeatedly solving high-resolution systems becomes computationally expensive when the underlying parameter fields vary across instances. In this work, we propose a resolution-free neural surrogate for geometric parameterization and mapping problems. Given a spatially varying parameter field p:Ω→Rm and query locations {xi}i=1N⊂Ω, the model predicts mapped locations {u(xi)}i=1N on arbitrary structured or unstructured point sets. To avoid dependence on a fixed grid, we use a multi-resolution geometric encoding strategy that conditions the network on coordinate-augmented samples of the parameter field. The model is trained without labeled solution data by enforcing geometry-aware constraints derived from variational energies, diffusion-based density equalization, and quasi-conformal theory. Experimental results on quasi-conformal mapping and density-equalizing mapping problems are presented to demonstrate the effectiveness of our proposed method.
Neural networks can often be trained or fine-tuned through random low-dimensional reparameterization, where a small latent vector is mapped into a full parameter update by a frozen random map. This raises a practical question: how large must the latent search space be to reach a low-loss region? We first express the known accessibility transition in an equivalent conic form, centered for compact convex targets at the statistical dimension of the polar cone. Our main theoretical contribution is an orientation-resolved quadratic master formula that predicts the random-slice residual from both the curvature spectrum and the reference-to-solution displacement profile. It yields a self-consistent isotropic-orientation predictor and, in a conservative radius-only specialization, recovers the earlier Gaussian-width quadratic bound. Building on this analysis, we introduce Random Mapping Networks (RaMaN), which instantiate the predicted latent dimension using structured Hadamard or seed-regenerated Gaussian maps. These constructions avoid the O(dP) storage of dense random maps and reduce optimizer-state memory from O(P) to O(d). We also develop matrix-free curvature approximations and sweep-free dimension selection. Across controlled quadratic and neural-curvature experiments, the orientation-resolved predictor closely tracks measured transition locations and outperforms orientation-agnostic approximations when displacement direction matters. End-to-end experiments further show sharp, protocol-dependent training transitions across image and language models.
Recent feedforward 3D reconstruction methods predict point maps and estimate global 3D geometry remarkably well. However, their predictions still exhibit inaccurate local surface geometry, which is clearly visible qualitatively but only weakly reflected in common metrics. To make these errors more explicit in evaluation, we introduce a point map normal metric that evaluates the local surface orientation induced by neighboring 3D predictions. To reduce these errors, we propose two complementary components: a point gradient matching loss that supervises depth-normalized 3D finite differences, and a Neighborhood Attention Decoder (NAD) that progressively upsamples features and uses Neighborhood Attention for local feature mixing. Across eight zero-shot monocular geometry benchmarks, our model, SurGe, achieves the best average rank for global point map AbsRel and consistently improves local point map and point map normal evaluations.
Karim Knaebel, Gonzalo Martin Garcia, Christian Schmidt +4
Geographic measurements are often sparse, leaving large areas without labels for the quantities we want to map. Geographic implicit neural representations (INRs) address this by learning smooth, general-purpose embeddings that can be queried at any coordinate. Downstream models combine these embeddings with sparse labels to predict target values at unsampled locations without satellite imagery at inference. However, generalization to distant regions remains largely unexplored, despite its importance for remote sensing applications. We introduce Matryoshka Implicit Neural Distillation (MIND), which distills embeddings from specialist pretrained geospatial models into a single generalist coordinate embedding with adjustable spatial granularity. MIND uses nested supervision at several embedding dimensions, which define a series of contiguous chunks. In our experiments, early chunks capture coarser geographic variation, while later chunks add more fine-grained details. A downstream predictor can retain only leading chunks or be fitted with our Chunked Penalty to downweight later chunks while keeping the full embedding, without retraining the INR. To measure MIND and compare to existing approaches around the world, we introduce CoordBench, a large-scale INR evaluation suite of 52 datasets and 78 targets that aims to test both local interpolation and prediction in held-out regions at various spatial scales. MIND and its Chunked Penalty variant achieve the highest aggregate regression and classification scores among tested INRs, and the highest scores overall under regional holdout, setting a new state-of-the-art for geographic INRs.