# Non-Gaussian Component Analysis using Entropy Methods

arXiv: https://arxiv.org/abs/1807.04936 · HTML: https://mlsanity.com/paper/1807.04936/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jul 2018.

## Built on by (most cited first, 8 of 8)

- [Lattice-Based Methods Surpass Sum-of-Squares in Clustering](https://mlsanity.com/paper/2112.03898/) (2021)
- [Optimal Spectral Recovery of a Planted Vector in a Subspace](https://mlsanity.com/paper/2105.15081/) (2021)
- [Non-Gaussian Component Analysis via Lattice Basis Reduction](https://mlsanity.com/paper/2112.09104/) (2021)
- [SQ Lower Bounds for Non-Gaussian Component Analysis with Weaker Assumptions](https://mlsanity.com/paper/2403.04744/) (2024)
- [Sum-of-squares lower bounds for Non-Gaussian Component Analysis](https://mlsanity.com/paper/2410.21426/) (2024)
- [Optimal SQ Lower Bounds for Robustly Learning Discrete Product Distributions and Ising Models](https://mlsanity.com/paper/2206.04589/) (2022)
- [Learning from higher-order statistics, efficiently: hypothesis tests, random features, and neural networks](https://mlsanity.com/paper/2312.14922/) (2023)
- [PTF Testing Lower Bounds for Non-Gaussian Component Analysis](https://mlsanity.com/paper/2511.19398/) (2025)

## Abstract (arXiv)

Non-Gaussian component analysis (NGCA) is a problem in multidimensional data analysis which, since its formulation in 2006, has attracted considerable attention in statistics and machine learning. In this problem, we have a random variable X in n-dimensional Euclidean space. There is an unknown subspace Gamma of the n-dimensional Euclidean space such that the orthogonal projection of X onto Gamma is standard multidimensional Gaussian and the orthogonal projection of X onto Gammaperp, the orthogonal complement of Gamma, is non-Gaussian, in the sense that all its one-dimensional marginals are different from the Gaussian in a certain metric defined in terms of moments. The NGCA problem is to approximate the non-Gaussian subspace Gammaperp given samples of X. Vectors in Gammaperp correspond to `interesting' directions, whereas vectors in Gamma correspond to the directions where data is very noisy. The most interesting applications of the NGCA model is for the case when the magnitude of the noise is comparable to that of the true signal, a setting in which traditional noise reduction techniques such as PCA don't apply directly. NGCA is also related to dimension reduction and to other data analysis problems such as ICA. NGCA-like problems have been studied in statistics for a long time using techniques such as projection pursuit. We give an algorithm that takes polynomial time in the dimension n and has an inverse polynomial dependence on the error parameter measuring the angle distance between the non-Gaussian subspace and the subspace output by the algorithm. Our algorithm is based on relative entropy as the contrast function and fits under the projection pursuit framework. The techniques we develop for analyzing our algorithm maybe of use for other related problems.