@inproceedings{
zhang2024enriching,
title={Enriching Disentanglement: From Logical Definitions to Quantitative Metrics},
author={Yivan Zhang and Masashi Sugiyama},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=tvQ3XCKWbB}
}
Disentangling the explanatory factors in complex data is a promising approach for generalizable and data-efficient representation learning. While a variety of quantitative metrics for learning and evaluating disentangled representations have been proposed, it remains unclear what properties these metrics truly quantify. In this work, we establish algebraic relationships between logical definitions and quantitative metrics to derive theoretically grounded disentanglement metrics. Concretely, we introduce a compositional approach for converting a higher-order predicate into a real-valued quantity by replacing (i) equality with a strict premetric, (ii) the Heyting algebra of binary truth values with a quantale of continuous values, and (iii) quantifiers with aggregators. The metrics induced by logical definitions have strong theoretical guarantees, and some of them are easily differentiable and can be used as learning objectives directly. Finally, we empirically demonstrate the effectiveness of the proposed metrics by isolating different aspects of disentangled representations.
Enriching Disentanglement: From Logical Definitions to Quantitative Metrics
Yivan Zhang Masashi Sugiyama