# Foundations of Independent Component Analysis

arXiv: https://arxiv.org/abs/2608.13229 · HTML: https://mlsanity.com/paper/2608.13229/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 50 references that are in the map.
- First posted on arXiv 13 Aug 2026.

## Builds on (most cited first, 5 of 5)

- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICA](https://mlsanity.com/paper/1605.06336/) (2016)
- [Independent mechanism analysis, a new concept?](https://mlsanity.com/paper/2106.05200/) (2021)
- [Multi-View Independent Component Analysis with Shared and Individual Sources](https://mlsanity.com/paper/2210.02083/) (2022)
- [Robust Multi-view Co-expression Network Inference](https://mlsanity.com/paper/2409.19991/) (2024)

## Abstract (arXiv)

We present the mathematical foundations of linear independent component analysis (ICA) models based on standard literature in a self-contained note. It is aimed at readers with a background in measure-theoretic probability theory. We first develop the theory of the characteristic functions of probability measures on ℝd, including their analyticity and the way in which they determine and characterise the distributions. We then focus on several identifiability results of ICA models with successively strengthened assumptions on the sources: from merely non-constant, to non-Gaussian, to Gaussian-free independent sources. Under the strictest assumptions, we show that the independent sources are identifiable up to translation, permutation, scales and signs, and this even in the presence of additive Gaussian noise. Furthermore, we present the online equivariant gradient descent ICA algorithm for recovering the independent sources from data, in the standard complete noiseless non-Gaussian ICA setting.