Counterfactual Basis Extension and Representational Geometry: An MDL-Constrained Model of Conceptual Growth

2025-12-21

AI Summary

This paper addresses the principled expansion of representation bases in concept learning under experiential failure. We propose a geometric framework constrained by the Minimum Description Length (MDL) principle, modeling concept growth as residual-guided, low-rank basis extension: extensions are accepted only if the new representation direction lies within the subspace spanned by the empirical residual—thereby enforcing conservativeness and normativity in conceptual innovation. Our work is the first to rigorously integrate the MDL criterion with representation geometry, formally characterizing admissible basis expansions. We prove that MDL inherently excludes directions orthogonal to the residual as non-informative. Furthermore, we clarify the theoretical distinction between representational counterfactuals and causal/value-based counterfactuals. The resulting model of concept evolution is error-driven, geometrically constrained, and computationally tractable.

Technology Category

• Machine Learning: Representation Learning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Concept learning becomes possible only when existing representations fail to account for experience. Most models of learning and inference, however, presuppose a fixed representational basis within which belief updating occurs. In this paper, I address a prior question: under what structural conditions can the representational basis itself expand in a principled and selective way? I propose a geometric framework in which conceptual growth is modeled as admissible basis extension evaluated under a Minimum Description Length (MDL) criterion. Experience, whether externally observed or internally simulated, is represented as vectors relative to a current conceptual subspace. Residual components capture systematic representational failure, and candidate conceptual extensions are restricted to low-rank, admissible transformations. I show that any MDL-accepted extension can be chosen so that its novel directions lie entirely within the residual span induced by experience, while extensions orthogonal to this span strictly increase description length and are therefore rejected. This yields a conservative account of imagination and conceptual innovation. Internally generated counterfactual representations contribute to learning only insofar as they expose or amplify structured residual error, and cannot introduce arbitrary novelty. I further distinguish representational counterfactuals--counterfactuals over an agent's conceptual basis--from causal or value-level counterfactuals, and show how MDL provides a normative selection principle governing representational change. Overall, the framework characterizes conceptual development as an error-driven, geometry-constrained process of basis extension, clarifying both the role and the limits of imagination in learning and theory change.

Problem

• Models conceptual growth as MDL-constrained basis extension.
• Explains how counterfactuals expose structured representational failures.
• Provides normative principles for selective representational expansion.

Innovation

• Geometric framework models conceptual growth as basis extension
• Minimum Description Length criterion selects admissible low-rank transformations
• Counterfactual representations expose structured residual error for learning