One classical example is that the Riemannian manifoldEm = (Rm,h,iRm) is nothing but the m-dimensionalEuclideanspace. Assume that the heat kernel is Lipschitz continous. We first construct a sequence of probability measures{ρt}t N, such that ρ2t = µtφφφ, ρ2t+1 = µtM, t N. Proof For a given manifoldM, its Riemannian volume element doesn't vary at different time t, thus thelowerbound ofintegral R

Knowledge graph embedding (KGE) relies on the geometry of the embedding space to encode semantic and structural relations. Existing methods place all entities on one homogeneous manifold, Euclidean, spherical, hyperbolic, or their product/multi-curvature variants, to model linear, symmetric, or hierarchical patterns. Yet a predefined, homogeneous manifold cannot accommodate the sharply varying curvature that real-world graphs exhibit across local regions. Since this geometry is imposed a priori, any mismatch with the knowledge graph's local curvatures will distort distances between entities and hurt the expressiveness of the resulting KGE. To rectify this, we propose RicciKGE to have the KGE loss gradient coupled with local curvatures in an extended Ricci flow such that entity embeddings co-evolve dynamically with the underlying manifold geometry towards mutual adaptation. Theoretically, when the coupling coefficient is bounded and properly selected, we rigorously prove that i) all the edge-wise curvatures decay exponentially, meaning that the manifold is driven toward the Euclidean flatness; and ii) the KGE distances strictly converge to a global optimum, which indicates that geometric flattening and embedding optimization are promoting each other. Experimental improvements on link prediction and node classification benchmarks demonstrate RicciKGE's effectiveness in adapting to heterogeneous knowledge graph structures.

Traditional metrics of interbrain synchrony in social neuroscience typically depend on fixed, correlation-based approaches, restricting their explanatory capacity to descriptive observations. Inspired by the successful integration of geometric insights in network science, we propose leveraging discrete geometry to examine the dynamic reconfigurations in neural interactions during social exchanges. Unlike conventional synchrony approaches, our method interprets inter-brain connectivity changes through the evolving geometric structures of neural networks. This geometric framework is realized through a pipeline that identifies critical transitions in network connectivity using entropy metrics derived from curvature distributions. By doing so, we significantly enhance the capacity of hyperscanning methodologies to uncover underlying neural mechanisms in interactive social behavior.

Geometric data analysis and learning has emerged as a distinct and rapidly developing research area, increasingly recognized for its effectiveness across diverse applications. At the heart of this field lies curvature, a powerful and interpretable concept that captures intrinsic geometric structure and underpins numerous tasks, from community detection to geometric deep learning. A wide range of discrete curvature models have been proposed for various data representations, including graphs, simplicial complexes, cubical complexes, and point clouds sampled from manifolds. These models not only provide efficient characterizations of data geometry but also constitute essential components in geometric learning frameworks. In this paper, we present the first comprehensive review of existing discrete curvature models, covering their mathematical foundations, computational formulations, and practical applications in data analysis and learning. In particular, we discuss discrete curvature from both Riemannian and metric geometry perspectives and propose a systematic pipeline for curvature-driven data analysis. We further examine the corresponding computational algorithms across different data representations, offering detailed comparisons and insights. Finally, we review state-of-the-art applications of curvature in both supervised and unsupervised learning. This survey provides a conceptual and practical roadmap for researchers to gain a better understanding of discrete curvature as a fundamental tool for geometric understanding and learning.

Two recent papers, [Wang et al., 2020, Li et al., 2022] also look into a similar problem of bounding

Assume that the heat kernel is Lipschitz continous. Proof We start by introducing the following lemma, which is Proposition 4.4 in [20]. Following some previous work, we first define the projection operator. We then introduce the following lemma which utilize the projection operator. Readers who are interested may also refer to Chapter 5.3 in [8] and proof C (null) 0 as null 0, and Γ is the gamma function.

Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations between triples of points and other points. More significantly, based on this curvature profile, we introduce a quantitative measure to evaluate the effectiveness of data representations, such as those produced by dimensionality reduction techniques. Furthermore, Our experiments demonstrate that this curvature-based analysis can be employed to estimate the intrinsic dimensionality of datasets. We use this to explore the large-scale geometry of empirical networks and to evaluate the effectiveness of dimensionality reduction techniques.

Randomly Wired Neural Networks (RWNNs) serve as a valuable testbed for investigating the impact of network topology in deep learning by capturing how different connectivity patterns impact both learning efficiency and model performance. At the same time, they provide a natural framework for exploring edge-centric network measures as tools for pruning and optimization. In this study, we investigate three edge-centric network measures: Forman-Ricci curvature (FRC), Ollivier-Ricci curvature (ORC), and edge betweenness centrality (EBC), to compress RWNNs by selectively retaining important synapses (or edges) while pruning the rest. As a baseline, RWNNs are trained for COVID-19 chest x-ray image classification, aiming to reduce network complexity while preserving performance in terms of accuracy, specificity, and sensitivity. We extend prior work on pruning RWNN using ORC by incorporating two additional edge-centric measures, FRC and EBC, across three network generators: Erdös-Rényi (ER) model, Watts-Strogatz (WS) model, and Barabási-Albert (BA) model. We provide a comparative analysis of the pruning performance of the three measures in terms of compression ratio and theoretical speedup. A central focus of our study is to evaluate whether FRC, which is computationally more efficient than ORC, can achieve comparable pruning effectiveness. Along with performance evaluation, we further investigate the structural properties of the pruned networks through modularity and global efficiency, offering insights into the trade-off between modular segregation and network efficiency in compressed RWNNs. Our results provide initial evidence that FRC-based pruning can effectively simplify RWNNs, offering significant computational advantages while maintaining performance comparable to ORC.

Graph Neural Networks (GNNs) have recently shown promise as solvers for Boolean Satisfiability Problems (SATs) by operating on graph representations of logical formulas. However, their performance degrades sharply on harder instances, raising the question of whether this reflects fundamental architectural limitations. In this work, we provide a geometric explanation through the lens of graph Ricci Curvature (RC), which quantifies local connectivity bottlenecks. We prove that bipartite graphs derived from random k-SAT formulas are inherently negatively curved, and that this curvature decreases with instance difficulty. Building on this, we show that GNN-based SAT solvers are affected by oversquashing, a phenomenon where long-range dependencies become impossible to compress into fixed-length representations. We validate our claims empirically across different SAT benchmarks and confirm that curvature is both a strong indicator of problem complexity and can be used to predict performance. Finally, we connect our findings to design principles of existing solvers and outline promising directions for future work.