Neuro-Symbolic Geometric Abstraction (NeuSOGA): Fro...

Neuro-Symbolic Geometric Abstraction (NeuSOGA): Fro... | AI Research

Key Takeaways

Neuro-Symbolic Geometric Abstraction (NeuSOGA): From Observations to Symbolic Mathematical Representations Modern artificial intelligence excels at recognizi...
A fundamental challenge in artificial intelligence is the transformation of observations into explicit symbolic representations suitable for abstraction, interpretation, and reasoning.
Inspired by Neuro-Symbolic AI and theories of human abstraction, this paper investigates the formation of symbolic mathematical representations from geometric observations.
The resulting representation is an analytical implicit model supporting arbitrary-order smoothness, additive composition, and closed-form evaluation.
Unlike neural latent encodings, the generated representation remains interpretable, editable, and mathematically explicit.

Paper AbstractExpand

A fundamental challenge in artificial intelligence is the transformation of observations into explicit symbolic representations suitable for abstraction, interpretation, and reasoning. While modern AI systems achieve remarkable perceptual capabilities through large-scale statistical learning, the resulting knowledge is typically encoded within latent parameters that are difficult to inspect or manipulate analytically. Inspired by Neuro-Symbolic AI and theories of human abstraction, this paper investigates the formation of symbolic mathematical representations from geometric observations. We propose NeuSOGA (Neuro-Symbolic Geometric Abstraction), a framework that progressively transforms observations into topological abstractions, geometric abstractions, and ultimately symbolic mathematical representations. The architecture combines topology-guided structural discovery using Euclidean Distance Transforms, foundation-model perception using Segment Anything, adaptive multi-scale geometric abstraction, and symbolic synthesis through Implicit Area Splines. The resulting representation is an analytical implicit model supporting arbitrary-order smoothness, additive composition, and closed-form evaluation. Unlike neural latent encodings, the generated representation remains interpretable, editable, and mathematically explicit. Experiments on ModelNet40 point clouds, arbitrary-view projections, and segmented optical observations demonstrate that NeuSOGA transforms diverse observations into compact symbolic representations while preserving essential geometric and topological structure across sensing modalities and viewing directions. NeuSOGA provides an interpretable and explainable pathway from observation to symbol and establishes

Neuro-Symbolic Geometric Abstraction (NeuSOGA): From Observations to Symbolic Mathematical Representations
Modern artificial intelligence excels at recognizing patterns, but it often struggles to explain its reasoning or provide results that humans can easily edit. When a neural network processes a 3D object, it typically stores that information in "latent parameters"—a complex web of numbers that are difficult for humans to interpret or manipulate. This paper introduces NeuSOGA, a framework designed to bridge the gap between raw perceptual data and clear, symbolic mathematical models. By mimicking how humans abstract geometric concepts, NeuSOGA transforms messy observations into structured, editable, and mathematically explicit representations.

From Observations to Symbols

The core of NeuSOGA is a progressive transformation pipeline: Observations (𝒪) → Topological Abstractions (𝒯) → Geometric Abstractions (𝒢) → Symbolic Mathematical Representations (𝒮). Instead of relying solely on statistical learning, the framework uses deterministic geometric operators to identify the "skeleton" of an object. By using tools like Euclidean Distance Transforms, the system identifies the essential structure of an object before it even attempts to define its surface, ensuring that the final model is built on a solid, logical foundation.

How the Framework Works

NeuSOGA combines several advanced techniques to achieve its results:

Topology-Guided Perception: It uses foundation models like Segment Anything to isolate geometric structures, guided by the topological "core" of the object.
Multi-Scale Abstraction: Using scale-space theory, the system analyzes the object at different levels of detail. It captures the "big picture" structure first and then adds finer details, similar to how a human might sketch a shape.
Implicit Area Splines: The final output is an analytical model based on Implicit Area Splines. Unlike standard neural representations, these splines are mathematically explicit, meaning they support smooth, continuous surfaces that can be easily edited or analyzed using standard mathematical tools.

Why This Matters

The primary advantage of NeuSOGA is that it provides "explainability by construction." Because the final representation is a set of explicit mathematical formulas rather than a "black box" of neural weights, the resulting models are transparent and editable. This approach allows for a more reliable way to perform geometric reasoning and machine-assisted mathematical discovery.

Results and Versatility

Experiments show that NeuSOGA is highly effective across different types of data, including 3D point clouds, 2D projections, and segmented optical images. Regardless of the viewing angle or the sensing method, the framework consistently produces compact, accurate symbolic representations. By separating the process of perception from the process of symbolic synthesis, NeuSOGA offers a robust alternative to purely data-driven methods, providing a pathway for AI systems to perform higher-level reasoning about the physical world.