# Enriching Disentanglement: From Logical Definitions to Quantitative Metrics

arXiv: https://arxiv.org/abs/2305.11512 · HTML: https://mlsanity.com/paper/2305.11512/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 62 of its 125 references that are in the map.
- First posted on arXiv 19 May 2023.

## Builds on (most cited first, 30 of 62)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [A Survey on Contrastive Self-supervised Learning](https://mlsanity.com/paper/2011.00362/) (2020)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [A Metric Learning Reality Check](https://mlsanity.com/paper/2003.08505/) (2020)
- [Robust Aggregation for Federated Learning](https://mlsanity.com/paper/1912.13445/) (2019)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Weakly-Supervised Disentanglement Without Compromises](https://mlsanity.com/paper/2002.02886/) (2020)
- [On the Fairness of Disentangled Representations](https://mlsanity.com/paper/1905.13662/) (2019)
- [Weakly supervised causal representation learning](https://mlsanity.com/paper/2203.16437/) (2022)
- [Robustly Disentangled Causal Mechanisms: Validating Deep Representations for Interventional Robustness](https://mlsanity.com/paper/1811.00007/) (2018)
- [Equivariant Architectures for Learning in Deep Weight Spaces](https://mlsanity.com/paper/2301.12780/) (2023)
- [Learning Deep Disentangled Embeddings with the F-Statistic Loss](https://mlsanity.com/paper/1802.05312/) (2018)
- [On the Transfer of Inductive Bias from Simulation to the Real World: a New Disentanglement Dataset](https://mlsanity.com/paper/1906.03292/) (2019)
- [On Disentangled Representations Learned From Correlated Data](https://mlsanity.com/paper/2006.07886/) (2020)
- [Weakly Supervised Disentanglement with Guarantees](https://mlsanity.com/paper/1910.09772/) (2019)
- [Measuring Disentanglement: A Review of Metrics](https://mlsanity.com/paper/2012.09276/) (2020)
- [On the Transfer of Disentangled Representations in Realistic Settings](https://mlsanity.com/paper/2010.14407/) (2020)
- [Theory and Evaluation Metrics for Learning Disentangled Representations](https://mlsanity.com/paper/1908.09961/) (2019)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Graph Neural Networks are Dynamic Programmers](https://mlsanity.com/paper/2203.15544/) (2022)
- [Categorical Foundations of Gradient-Based Learning](https://mlsanity.com/paper/2103.01931/) (2021)
- [Compositional Generalization in Unsupervised Compositional Representation Learning: A Study on Disentanglement and Emergent Language](https://mlsanity.com/paper/2210.00482/) (2022)

## Built on by (most cited first, 2 of 2)

- [Disentangled Representations for Causal Cognition](https://mlsanity.com/paper/2407.00744/) (2024)
- [Compositional Behavioral Semantics for State Abstraction in Reinforcement Learning](https://mlsanity.com/paper/2606.25357/) (2026)

## Abstract (arXiv)

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.