# Partial Information Decomposition via Normalizing Flows in Latent Gaussian Distributions

arXiv: https://arxiv.org/abs/2510.04417 · HTML: https://mlsanity.com/paper/2510.04417/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 63 references that are in the map.
- First posted on arXiv 6 Oct 2025.

## Builds on (most cited first, 30 of 37)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Scaling Up Visual and Vision-Language Representation Learning With Noisy Text Supervision](https://mlsanity.com/paper/2102.05918/) (2021)
- [Align before Fuse: Vision and Language Representation Learning with Momentum Distillation](https://mlsanity.com/paper/2107.07651/) (2021)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Multimodal Machine Learning: A Survey and Taxonomy](https://mlsanity.com/paper/1705.09406/) (2017)
- [What Makes for Good Views for Contrastive Learning?](https://mlsanity.com/paper/2005.10243/) (2020)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Multimodal Transformer for Unaligned Multimodal Language Sequences](https://mlsanity.com/paper/1906.00295/) (2019)
- [Mind the Gap: Understanding the Modality Gap in Multi-modal Contrastive Representation Learning](https://mlsanity.com/paper/2203.02053/) (2022)
- [On Variational Bounds of Mutual Information](https://mlsanity.com/paper/1905.06922/) (2019)
- [Tensor Fusion Network for Multimodal Sentiment Analysis](https://mlsanity.com/paper/1707.07250/) (2017)
- [Multimodal datasets: misogyny, pornography, and malignant stereotypes](https://mlsanity.com/paper/2110.01963/) (2021)
- [Efficient Low-rank Multimodal Fusion with Modality-Specific Factors](https://mlsanity.com/paper/1806.00064/) (2018)
- [Multimodal Deep Learning](https://mlsanity.com/paper/2301.04856/) (2023)
- [Multimodal Compact Bilinear Pooling for Visual Question Answering and Visual Grounding](https://mlsanity.com/paper/1606.01847/) (2016)
- [Multimodal Generative Models for Scalable Weakly-Supervised Learning](https://mlsanity.com/paper/1802.05335/) (2018)
- [Towards Multimodal Sarcasm Detection (An _Obviously_ Perfect Paper)](https://mlsanity.com/paper/1906.01815/) (2019)
- [Learning Factorized Multimodal Representations](https://mlsanity.com/paper/1806.06176/) (2018)
- [Contrastive learning, multi-view redundancy, and linear models](https://mlsanity.com/paper/2008.10150/) (2020)
- [MultiBench: Multiscale Benchmarks for Multimodal Representation Learning](https://mlsanity.com/paper/2107.07502/) (2021)
- [Understanding the Limitations of Variational Mutual Information Estimators](https://mlsanity.com/paper/1910.06222/) (2019)
- [UR-FUNNY: A Multimodal Language Dataset for Understanding Humor](https://mlsanity.com/paper/1904.06618/) (2019)
- [Foundations and Trends in Multimodal Machine Learning: Principles, Challenges, and Open Questions](https://mlsanity.com/paper/2209.03430/) (2022)
- [Multimodal Intelligence: Representation Learning, Information Fusion, and Applications](https://mlsanity.com/paper/1911.03977/) (2019)
- [Joint Multimodal Learning with Deep Generative Models](https://mlsanity.com/paper/1611.01891/) (2016)
- [CentralNet: a Multilayer Approach for Multimodal Fusion](https://mlsanity.com/paper/1808.07275/) (2018)
- [Quantifying & Modeling Multimodal Interactions: An Information Decomposition Framework](https://mlsanity.com/paper/2302.12247/) (2023)
- [Factorized Contrastive Learning: Going Beyond Multi-view Redundancy](https://mlsanity.com/paper/2306.05268/) (2023)

## Built on by (most cited first, 5 of 5)

- [How Vision Becomes Language: A Layer-wise Information-Theoretic Analysis of Multimodal Reasoning](https://mlsanity.com/paper/2602.15580/) (2026)
- [DIPHINE: Diffusion-based Phi-ID Neural Estimator](https://mlsanity.com/paper/2606.18997/) (2026)
- [CST: Collaborative Selective Transmission for Communication-Efficient Multimodal Edge Inference](https://mlsanity.com/paper/2608.22115/) (2026)
- [Structured Latent Modeling for Supervised Multimodal Information Decomposition](https://mlsanity.com/paper/2609.35502/) (2026)
- [Gaussian BROJA Partial Information Decomposition in Closed Form](https://mlsanity.com/paper/2610.13464/) (2026)

## Abstract (arXiv)

The study of multimodality has garnered significant interest in fields where the analysis of interactions among multiple information sources can enhance predictive modeling, data fusion, and interpretability. Partial information decomposition (PID) has emerged as a useful information-theoretic framework to quantify the degree to which individual modalities independently, redundantly, or synergistically convey information about a target variable. However, existing PID methods depend on optimizing over a joint distribution constrained by estimated pairwise probability distributions, which are costly and inaccurate for continuous and high-dimensional modalities. Our first key insight is that the problem can be solved efficiently when the pairwise distributions are multivariate Gaussians, and we refer to this problem as Gaussian PID (GPID). We propose a new gradient-based algorithm that substantially improves the computational efficiency of GPID based on an alternative formulation of the underlying optimization problem. To generalize the applicability to non-Gaussian data, we learn information-preserving encoders to transform random variables of arbitrary input distributions into pairwise Gaussian random variables. Along the way, we resolved an open problem regarding the optimality of joint Gaussian solutions for GPID. Empirical validation in diverse synthetic examples demonstrates that our proposed method provides more accurate and efficient PID estimates than existing baselines. We further evaluate a series of large-scale multimodal benchmarks to show its utility in real-world applications of quantifying PID in multimodal datasets and selecting high-performing models.