Recent research on graph embedding has achieved success in various applications. Most graph embedding methods preserve the proximity in a graph into a manifold in an embedding space. We argue an important but neglected problem about this proximity-preserving strategy: Graph topology patterns, while preserved well into an embedding manifold by preserving proximity, may distort in the ambient embedding Euclidean space, and hence to detect them becomes difficult for machine learning models. To address the problem, we propose curvature regularization, to enforce flatness for embedding manifolds, thereby preventing the distortion. We present a novel angle-based sectional curvature, termed ABS curvature, and accordingly three kinds of curvature regularization to induce flat embedding manifolds during graph embedding. We integrate curvature regularization into five popular proximity-preserving embedding methods, and empirical results in two applications show significant improvements on a wide range of open graph datasets.

Recent research on graph embedding has achieved success in various applications.

Neural signed distance functions (SDFs) have become a powerful representation for geometric reconstruction from point clouds, yet they often require both gradient- and curvature-based regularization to suppress spurious warp and preserve structural fidelity. FlatCAD introduced the Off-Diagonal Weingarten (ODW) loss as an efficient second-order prior for CAD surfaces, approximating full-Hessian regularization at roughly half the computational cost. However, FlatCAD applies a fixed ODW weight throughout training, which is suboptimal: strong regularization stabilizes early optimization but suppresses detail recovery in later stages. We present scheduling strategies for the ODW loss that assign a high initial weight to stabilize optimization and progressively decay it to permit fine-scale refinement. We investigate constant, linear, quintic, and step interpolation schedules, as well as an increasing warm-up variant. Experiments on the ABC CAD dataset demonstrate that time-varying schedules consistently outperform fixed weights. Our method achieves up to a 35% improvement in Chamfer Distance over the FlatCAD baseline, establishing scheduling as a simple yet effective extension of curvature regularization for robust CAD reconstruction.

Additional Feedback: * One philosophical question that comes to mind when reading the three observations in the Introduction and while going over the example in Figure 1 is the following: Could it be that the exact reason that the representation methods learn interesting patterns is the fact that they are allowed to twist and curve the space as required, in order to bring nodes that are far apart in terms of graph distance close together in terms of Euclidean distance? It is not obvious to me that constraining the optimization algorithm of this ability can only have positive outcome. There's a parallel to be drawn here with the kernel trick in Support Vector Machines, where we are allowed to embed the data in a higher dimension, where the classes become linearly separable. This way, the sum could be defined over (q', q'') \in \Gamma_{i, j} * The sectional curvature could have been more thoroughly introduced. Unfortunately, for a paper that is heavily based on geometric notions, there's a clear shortage of pictures.

Recent research on graph embedding has achieved success in various applications. Most graph embedding methods preserve the proximity in a graph into a manifold in an embedding space. We argue an important but neglected problem about this proximity-preserving strategy: Graph topology patterns, while preserved well into an embedding manifold by preserving proximity, may distort in the ambient embedding Euclidean space, and hence to detect them becomes difficult for machine learning models. To address the problem, we propose curvature regularization, to enforce flatness for embedding manifolds, thereby preventing the distortion. We present a novel angle-based sectional curvature, termed ABS curvature, and accordingly three kinds of curvature regularization to induce flat embedding manifolds during graph embedding.

To address the challenges of long-tailed classification, researchers have proposed several approaches to reduce model bias, most of which assume that classes with few samples are weak classes. However, recent studies have shown that tail classes are not always hard to learn, and model bias has been observed on sample-balanced datasets, suggesting the existence of other factors that affect model bias. In this work, we systematically propose a series of geometric measurements for perceptual manifolds in deep neural networks, and then explore the effect of the geometric characteristics of perceptual manifolds on classification difficulty and how learning shapes the geometric characteristics of perceptual manifolds. An unanticipated finding is that the correlation between the class accuracy and the separation degree of perceptual manifolds gradually decreases during training, while the negative correlation with the curvature gradually increases, implying that curvature imbalance leads to model bias. Therefore, we propose curvature regularization to facilitate the model to learn curvature-balanced and flatter perceptual manifolds. Evaluations on multiple long-tailed and non-long-tailed datasets show the excellent performance and exciting generality of our approach, especially in achieving significant performance improvements based on current state-of-the-art techniques. Our work opens up a geometric analysis perspective on model bias and reminds researchers to pay attention to model bias on non-long-tailed and even sample-balanced datasets. The code and model will be made public.

Recent research on graph embedding has achieved success in various applications. Most graph embedding methods preserve the proximity in a graph into a manifold in an embedding space. We argue an important but neglected problem about this proximity-preserving strategy: Graph topology patterns, while preserved well into an embedding manifold by preserving proximity, may distort in the ambient embedding Euclidean space, and hence to detect them becomes difficult for machine learning models. To address the problem, we propose curvature regularization, to enforce flatness for embedding manifolds, thereby preventing the distortion. We present a novel angle-based sectional curvature, termed ABS curvature, and accordingly three kinds of curvature regularization to induce flat embedding manifolds during graph embedding. We integrate curvature regularization into five popular proximity-preserving embedding methods, and empirical results in two applications show significant improvements on a wide range of open graph datasets.