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
Two-stage neuro-symbolic architectures provide an elegant paradigm for visual problem solving by cleanly separating connectionist perception of predefined symbols from possibly later defined relational reasoning thereon. However, anchoring high-level predicates into visual frames typically necessitates annotations that are expensive to acquire. In this work, we introduce the Dynamic Orthogonal Concept Bottleneck (D-OCB), an object-centric slot- VAE framework designed to extract human-aligned symbolic predicates under extremely weak supervision. D-OCB eliminates the arduous manual tuning of loss-balancing coef- ficients by dynamically learning optimal hyperparameter allocations during training. To infuse prior knowledge on independence of concept categories, in addition to standard re- construction self-supervision we penalize correlation across concept subspaces. Crucially, to combat the instability of very low supervision regimes, D-OCB incorporates a dynamic di- mensionality allocation mechanism; this adaptive formulation allows well-represented con- cepts to yield latent dimensions to underperforming concepts that are lagging behind, effectively preventing representation collapse and significantly improving overall concept accuracy. Through an extensive empirical evaluation, we demonstrate that our framework achieves high concept alignment and downstream visual reasoning accuracy using minimal label budgets, matching or outperforming end-to-end paradigms.