# DIPHINE: Diffusion-based Phi-ID Neural Estimator

arXiv: https://arxiv.org/abs/2606.18997 · HTML: https://mlsanity.com/paper/2606.18997/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 43 references that are in the map.
- First posted on arXiv 17 Jun 2026.

## Builds on (most cited first, 15 of 15)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Formal Limitations on the Measurement of Mutual Information](https://mlsanity.com/paper/1811.04251/) (2018)
- [Information-Theoretic Diffusion](https://mlsanity.com/paper/2302.03792/) (2023)
- [MINDE: Mutual Information Neural Diffusion Estimation](https://mlsanity.com/paper/2310.09031/) (2023)
- [Mutual Information Estimation via Normalizing Flows](https://mlsanity.com/paper/2403.02187/) (2024)
- [Measuring Dependence with Matrix-based Entropy Functional](https://mlsanity.com/paper/2101.10160/) (2021)
- [Exploration of synergistic and redundant information sharing in static and dynamical Gaussian systems](https://mlsanity.com/paper/1411.2832/) (2014)
- [Measuring multivariate redundant information with pointwise common change in surprisal](https://mlsanity.com/paper/1602.05063/) (2016)
- [InfoBridge: Mutual Information estimation via Bridge Matching](https://mlsanity.com/paper/2502.01383/) (2025)
- [Partial Information Decomposition via Normalizing Flows in Latent Gaussian Distributions](https://mlsanity.com/paper/2510.04417/) (2025)
- [Information-Theoretic Measures on Lattices for High-Order Interactions](https://mlsanity.com/paper/2408.07533/) (2024)
- [A unified framework for information integration based on information geometry](https://mlsanity.com/paper/1510.04455/) (2015)
- [TENDE: Transfer Entropy Neural Diffusion Estimation](https://mlsanity.com/paper/2510.14096/) (2025)

## Abstract (arXiv)

Uncovering the true informational architecture of real-world complex systems requires disentangling how their components uniquely store, redundantly share, and synergistically integrate information over time. Integrated Information Decomposition (PhiID) is a framework for decomposing the information dynamics of multivariate systems into sixteen non-overlapping atoms that characterize redundant, unique, and synergistic modes of information storage, transfer, and integration. Existing methods to compute PhiID are restricted to Gaussian or discrete systems, preventing its application to continuous non-Gaussian dynamical systems. We address this limitation by proposing DIPHINE (Diffusion-based Phi-ID Neural Estimator), the first neural estimator that leverages score-based diffusion models to jointly estimate all the mutual information terms required by PhiID from a single amortized network, recovering the sixteen atoms through M"obius inversion. We provide a theoretical analysis of error propagation through the inversion, showing that the Jacobian of the mapping from mutual informations to atoms is integer-valued and that the synergy-to-synergy atom is provably the hardest to estimate. We demonstrate accurate recovery of ground-truth atoms on synthetic benchmarks, superior performance compared to established mutual information estimators, and the ability to extract physiologically interpretable information-dynamic structure on an application involving real data without any distributional assumptions.