Operational Abstractions of Neural Network Concepts via Topological Representations

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Concept-based methods provide a semantic level for interpreting and manipulating learned representations, but existing editing approaches are typically specialized to particular interventions and do not provide a common and editable representation of concept organization. To achieve this, we introduce Topological Concept Representations (TCR), a post-hoc operational abstraction that jointly characterizes the concepts encoded in a learned representation and their relationships. TCR constructs an intermediate concept space from concept recoverability and interaction scores, and compactly encodes its organization through topology. Interventions are expressed through modifications of this abstraction that are propagated back to the underlying learned representation. This allows different concept-level operations to share the same optimization framework and separates the desired concept organization from the mechanism to achieve it. We establish stability and reparameterization-invariance properties of TCR and its connections to existing concept-editing formulations. We use TCR to disentangle concepts as a preprocessing step for existing erasure methods, improving worst-group accuracy by 21.89 on average at comparable concept leakage. We further use TCR to transfer concepts from teacher to student models, improving concept recoverability by up to 5.54 while also improving or maintaining competitive test top-1 accuracy.