Cross-Geometry Generalization

• Cross-Geometry Generalization is a framework that abstracts geometric structures, ensuring invariant performance across diverse settings through combinatorial and algebraic methods.
• It leverages techniques like combinatorial lifting, projective mappings, and algorithmic abstraction to preserve incidence, symmetry, and metric properties.
• Applications span computational geometry, digital processing, and machine learning, enabling robust adaptations in environments with varying topology and curvature.

Cross-geometry generalization refers to the capacity of mathematical, algorithmic, or machine learning frameworks to operate robustly, with consistent performance and behavior, across a range of distinct geometric domains. These domains may differ in underlying incidence structure, metrics, topology, dimensionality, curvature, combinatorial structure, or algebraic invariants. The concept underlies major recent advances in algebraic geometry, discrete mathematics, computational geometry, digital geometry processing, and modern machine learning, where the goal is to devise structures, representations, or algorithms that transcend a specific geometric context and maintain provable or empirical invariances when ported to new geometries.

1. Abstract Incidence Configurations and Combinatorial Lifting

In combinatorial geometry, cross-geometry generalization is systematically realized through the abstraction and generalization of classical geometric configurations. The Cremona–Richmond configuration, for example, is lifted from the specific case of 15 points and 15 lines in $PG(3, F)$ to a family of configurations $\CRSpace(X,k,s)$ parameterized by integers $k,s$ and an underlying finite set $X$. This construction yields abstract incidence structures whose points and blocks correspond to $k$- and $m$-element subsets, and which can then be realized geometrically in projective spaces of appropriately high dimension. The philosophy is general: the combinatorial structure (incidence pattern, adjacency, or higher-level relations) is defined independently of the geometric model, then “realized” in diverse settings such as vector spaces or fields of arbitrary characteristic. Parameter counts (number of points, blocks, incidences, and intersection properties) are maintained by combinatorial enumeration, and geometric realizability is tracked via embeddings that respect the original incidence logic. Symmetry groups (e.g., $S_{ks}$) and adjacency graphs (e.g., Kneser graphs) are defined at the combinatorial level and thus persist under translation to new geometric contexts, provided compatibility conditions (such as characteristic restrictions) are met (Prażmowska et al., 2014).

2. Projective, Metric, and Topological Generalization of Invariants

Many classical geometric invariants extend across distinct geometries via intrinsic projective or algebraic structure. The cross-ratio, originally a ratio of signed Euclidean distances between quadruples of collinear points, generalizes harmonically to spherical ($K=+1$) and hyperbolic ($K=-1$) planes via canonical substitutions for segment length by $\gsin_K(\cdot)$, the unified “curvature-sine” function. Central (gnomic) projection provides a functorial passage from the sphere or hyperboloid model to the affine plane, guaranteeing the invariance of the cross-ratio under isometries and projective maps. The extension is driven by the uniform algebraic formula for the cross-ratio:

$(ABCD)_K = \frac{\gsin_K(d(A,C))}{\gsin_K(d(A,D))} \cdot \frac{\gsin_K(d(B,D))}{\gsin_K(d(B,C))}$

This foundational approach allows classical theorems (such as Carnot’s Theorem for conics or higher-degree curves) to be transported, verbatim, between Euclidean, spherical, and hyperbolic geometries simply by translating algebraic ratios into their curvature-adapted analogues. Projective invariance underlies the persistence of these properties under model change and demonstrates the principle of cross-geometry generalization for symmetry, incidence, and measurement invariants (Palapa et al., 2 Mar 2024).

A parallel development appears in higher-dimensional projective spaces via the generalization of the cross-ratio using (N+1)-minors of representative homogeneous coordinate tuples, leading to multilinear invariants in $CP^N$ that remain invariant under the full linear-fractional group, provided nondegeneracy conditions are ensured. This establishes $(N+2)$-transitivity for linear-fractional maps in higher dimension, so that key group-theoretic transitivity and invariant properties persist in the more general context (Pilla, 2021).

Generalized metric spaces (arbitrary nondegenerate symmetric bilinear forms) permit the extension of inner products, cross products, and all classical vector-calculus identities to settings beyond Euclidean geometry (signatures, finite fields, relativistic geometries), maintaining the form and algebraic structure of trigonometric and spread laws (Notowidigdo et al., 2019).

3. Cross-Geometry Generalization in Discrete and Digital Geometry

In computational and digital geometry, cross-geometry generalization is operationalized through abstraction of algorithms and data structures at the conceptual level rather than at the implementation level. Within frameworks such as Milena, image operators, topological thinning, and neighborhood systems are formulated using “concepts” and “traits” that express only primitive requirements: domain interfaces, adjacency relations, and predicate logic. The core algorithm (e.g., breadth-first thinning) is agnostic to the underlying geometry—be it a regular grid, a graph, a cell complex, or a higher-dimensional simplicial or cubical complex. All geometry-specific logic and site-specific behavior are encapsulated in functor parameters or domain objects.

As such, a single codebase (not merely an algorithmic template) is instantiated identically for 2D image grids, 3D volume data, or arbitrary mesh geometries simply by supplying the relevant domain functors and adjacency objects. All topology-preserving operators (detach, test-simple, constraint checks) follow the same invocation model, resulting in cross-geometry generalization without loss of performance or correctness. This supports direct cross-domain experimentation and transfer, strictly by design abstraction, and demonstrates that geometric generalization may be achieved algorithmically at the software level when mathematical structure is faithfully mirrored in the programming model (Levillain et al., 2012).

4. Domain Generalization and Geometry-Invariant Learning

In machine learning, especially geometric deep learning and generative modeling, cross-geometry generalization refers to the ability of algorithms (notably, neural networks) to maintain predictive accuracy and fidelity across differing geometric domains, such as graphs with varying topology, surfaces of distinct genus or local metric, or data manifolds with divergent structure.

For graph neural networks (GNNs), domain generalization is realized by constructing augmentations that remove and add edges in a manner designed to strip away domain- or instance-specific artifacts and reinforce only those interactions or neighborhoods that are invariant under changes of graph geometry. The explicit graph augmentation pipeline applies statistically motivated low-weight edge dropping (removing fragile, likely non-invariant or spurious edges) and clustering-based edge adding (inducing invariant structure from feature-space similarity, presumed stable across environments). These combined augmentations force the model to learn representations predictive under all geometry perturbations, implementing an implicit form of invariant risk minimization over the space of possible graph topologies. Empirically, this yields significant gains in cross-graph node classification tasks and is directly driven by the principle of capturing structure that persists across geometric variation (Chen et al., 25 Feb 2025).

In generative modeling of geometric data (e.g., surfaces for quad mesh generation), cross-geometry generalization is achieved by constructing joint latent representations for the geometric object (e.g., signed distance field) and its relevant geometric field (e.g., cross field for meshing) that are explicitly architecture-agnostic. In the CrossGen framework, all surface input types are normalized to point clouds, embedded via sparse CNN encoders, and decoded into geometry-plus-field jointly. The network, trained on a diverse dataset, builds local patch descriptors that are generic enough to permit inference on surfaces with wildly different geometry, topological connectivity, and local feature complexity. Augmentation with a latent diffusion model further extends generative capacity to novel, incomplete, or partial geometry inputs, again relying on the transferability of the joint latent space and the universal character of the auto-encoder architecture. Evaluation demonstrates fast, robust generalization to out-of-domain and degraded inputs, evidence of true cross-geometry generalization (Dong et al., 8 Jun 2025).

In diffusion models for image synthesis, generalization across different data geometries stems from the inductive bias of convolutional architectures: denoising functions are realized as soft-thresholders in geometry-adaptive harmonic bases (GAHB), which efficiently represent structure in natural images and degrade gracefully to the tangent spaces of low-dimensional manifolds. The learned bias—favoring locally harmonic, shift-invariant feature atoms—explains rapid generalization and near-optimal performance on photorealistic and synthetic geometric data alike, with empirical alignment between theory and implementation (Kadkhodaie et al., 2023).

5. Cross-Geometry Principles in Incidence, Polytopal, and Combinatorial Geometries

Algebraic and combinatorial generalizations yield cross-geometry results through categorical or group-theoretic constructions. For example, halving operations originally defined on the faces of regular polytopes are abstracted via properties (residual connectivity, thinness, flag transitivity) that can be established in any regular hypertope with a non-degenerate leaf. Partitioned and bipartitioned halving constructions apply to general incidence geometries by satisfying combinatorial compatibility conditions (B₁), (B₂) and diagram constraints, resulting in new families of geometries and infinite chains of derived objects (e.g., toroidal polytopes, Coxeter complexes) (Piedade et al., 29 May 2024).

Obstruction-theoretic approaches further reveal that the existence or nonexistence of certain embedding categories (e.g., contracting endomorphisms or nonbijective embeddings of polygons) simultaneously governs open problems about the nature of local finiteness in incidence geometry (Tits’s problem) and the linearity of translation generalized quadrangles. This reflects a form of cross-geometry “control”: disparate geometric phenomena are unified by categorical relationships that transcend specific ambient constructions (Thas, 2014).

6. Generalization of Crossing Concepts in Discrete Geometry

Recent results show that notions of “crossing” in geometric graphs can be generalized by parameterizing both the crossing pattern (e.g., $H$-crossing families) and the class of subgraphs under consideration (paths, stars, cliques). Main theorems provide universal lower bounds for the existence and count of such crossing families for given graph sizes, utilizing geometric partitioning and combinatorial rearrangements independent of the metric or embedding used, provided the point set is in general position. The existence of $H$-intersecting families and their asymptotic size bounds further generalize crossing concepts over varying geometric domains, supporting robustness with respect to geometry (Lara et al., 2018).

Similarly, the Crossing Tverberg theorem establishes results about partitions of points into vertex-disjoint simplices whose boundaries pairwise cross, not only in the Euclidean plane but in arbitrary dimension and, via combinatorial arguments, potentially in further generalizations such as pseudolinear or topological settings. This demonstrates how crossing properties and bound-optimality transfer from classical to more abstract settings via invariant combinatorial logic and parity/cocycle arguments (Fulek et al., 2018).

7. Limitations, Open Problems, and Theoretical Outlook

While structural invariance and algorithmic abstraction provide robust mechanisms for cross-geometry generalization, several limitations remain:

• Full algebraic characterization of all possible cross-geometry invariants—such as providing a precise functional description of the class of data for which convolutional architectures are minimax-optimal—is unresolved (Kadkhodaie et al., 2023).
• In category-theoretic frameworks, the non-existence or explicit classification of objects capturing generalization failures remains an open question (Thas, 2014), with implications for longstanding open problems.
• Algorithmic realizability (e.g., polynomial-time algorithm for crossing Tverberg partitions) often lags behind existential or combinatorial proofs (Fulek et al., 2018).
• For digital geometry frameworks, extension to highly irregular or degenerate adjacency structures, or to data types lacking canonical neighborhood structure, may pose nontrivial practical barriers (Levillain et al., 2012).
• Invariant augmentation strategies in geometric learning depend on feature-homophily or other domain-invariant prior assumption; if the feature space or its distribution shifts, generalization can degrade (Chen et al., 25 Feb 2025).

These areas motivate further exploration at the interfaces of combinatorics, topology, machine learning, and geometric group theory. The unifying principle is that cross-geometry generalization emerges from the formulation of mathematical structures, algorithmic representations, or learning priors that can be expressed or realized independently of a particular geometric instantiation, and that preserve key incidence, invariance, or optimality properties under appropriate functorial, algebraic, or categorical mappings.