Superpixels partition an image into perceptually coherent regions, reducing the cost of downstream vision tasks. Modern deep learning methods excel at superpixel generation but often yield irregular boundaries and isolated pixels, necessitating non-differentiable post-processing to enforce connectivity. This undermines the end-to-end learning capabilities. We propose a simple, fully differentiable graph-Laplacian loss that encourages spatial regularity and connectivity during training. The loss is model-agnostic and can be seamlessly integrated into the training of existing architectures to improve the quality of superpixels. In addition, we introduce two novel metrics, the average stray pixel count and excess component count, to measure the quality of superpixels. We demonstrate both qualitative and quantitative improvements over state-of-the-art methods with and without enforced connectivity. Our approach represents a significant step toward eliminating non-differentiable post-processing.

Recent advances in pretraining 3D point cloud encoders (e.g., Point-BERT, Point-MAE) have produced powerful models, whose abilities are typically evaluated on geometric or semantic tasks. At the same time, topological descriptors have been shown to provide informative summaries of a shape's multiscale structure. In this paper we pose the question whether topological information can be derived from features produced by 3D encoders. To address this question, we first introduce DONUT, a synthetic benchmark with controlled topological complexity, and propose FILTR (Filtration Transformer), a learnable framework to predict persistence diagrams directly from frozen encoders. FILTR adapts a transformer decoder to treat diagram generation as a set prediction task. Our analysis on DONUT reveals that existing encoders retain only limited global topological signals, yet FILTR successfully leverages information produced by these encoders to approximate persistence diagrams. Our approach enables, for the first time, data-driven extraction of persistence diagrams from raw point clouds through an efficient learnable feed-forward mechanism.

This work proposes a new formulation to the long-standing problem of convex decomposition through learning feature fields, enabling the first feed-forward model for open-world learning of convex decomposition. Our method produces high-quality decompositions of 3D shapes into a union of convex bodies, which are essential to accelerate collision detection in physical simulation, amongst many other applications.The key insight is to adopt a feature learning approach and learn a continuous feature field that can later be clustered to yield a good convex decomposition via our self-supervised, purely-geometric objective derived from the classical definition of convexity.Our formulation can be used for single shape optimization, but more importantly, feature prediction unlocks scalable, self-supervised learning on large datasets resulting in the first learned open-world for convex decomposition.Experiments show that our decompositions are higher-quality than alternatives and generalize across open-world objects as well as across representations to meshes, CAD models, and even Gaussian splats.

We introduce a novel framework that directly learns a spectral basis for shape and manifold analysis from unstructured data, eliminating the need for traditional operator selection, discretization, and eigensolvers.Grounded in optimal-approximation theory, we train a network to decompose an implicit approximation operator by minimizing the reconstruction error in the learned basis over a chosen distribution of probe functions. For suitable distributions, they can be seen as an approximation of the Laplacian operator and its eigendecomposition, which are fundamental in geometry processing. Furthermore, our method recovers in a unified manner not only the spectral basis, but also the implicit metric's sampling density and the eigenvalues of the underlying operator. Notably, our unsupervised method makes no assumption on the data manifold, such as meshing or manifold dimensionality, allowing it to scale to arbitrary datasets of any dimension.On point clouds lying on surfaces in 3D and high-dimensional image manifolds, our approach yields meaningful spectral bases, that can resemble those of the Laplacian, without explicit construction of an operator. By replacing the traditional operator selection, construction, and eigendecomposition with a learning-based approach, our framework offers a principled, data-driven alternative to conventional pipelines. This opens new possibilities in geometry processing for unstructured data, particularly in high-dimensional spaces.

The escalating parameter counts in modern deep learningmodels pose a fundamental challenge to efficient trainingand resolution of overfitting. We address this by introducingthe Mapping Networks which replace the high dimensionalweight space by a compact, trainable latent vector based onthe hypothesis that the trained parameters of large networksreside on smooth, low-dimensional manifolds. Henceforth,the Mapping Theorem enforced by a dedicated MappingLoss, shows the existence of a mapping from this latentspace to the target weight space both theoretically and inpractice. Mapping Networks significantly reduce overfittingand achieve comparable to better performance than target network across complex vision and sequence tasks, including Image Classification, Deepfake Detection etc., with99.5%, i.e., around 500× reduction in trainable parameters.