# Phase transition in PCA with missing data: Reduced signal-to-noise ratio, not sample size!

arXiv: https://arxiv.org/abs/1905.00709 · HTML: https://mlsanity.com/paper/1905.00709/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 May 2019.

## Built on by (most cited first, 4 of 4)

- [The Shape of Learning Curves: a Review](https://mlsanity.com/paper/2103.10948/) (2021)
- [Learning Dynamics in Linear VAE: Posterior Collapse Threshold, Superfluous Latent Space Pitfalls, and Speedup with KL Annealing](https://mlsanity.com/paper/2310.15440/) (2023)
- [High-dimensional Asymptotics of VAEs: Threshold of Posterior Collapse and Dataset-Size Dependence of Rate-Distortion Curve](https://mlsanity.com/paper/2309.07663/) (2023)
- [Statistical Mechanics of Min-Max Problems](https://mlsanity.com/paper/2409.06053/) (2024)

## Abstract (arXiv)

How does missing data affect our ability to learn signal structures? It has been shown that learning signal structure in terms of principal components is dependent on the ratio of sample size and dimensionality and that a critical number of observations is needed before learning starts (Biehl and Mietzner, 1993). Here we generalize this analysis to include missing data. Probabilistic principal component analysis is regularly used for estimating signal structures in datasets with missing data. Our analytic result suggests that the effect of missing data is to effectively reduce signal-to-noise ratio rather than - as generally believed - to reduce sample size. The theory predicts a phase transition in the learning curves and this is indeed found both in simulation data and in real datasets.