# On the Consistency of Maximum Likelihood Estimation of Probabilistic Principal Component Analysis

arXiv: https://arxiv.org/abs/2311.05046 · HTML: https://mlsanity.com/paper/2311.05046/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 29 references that are in the map.
- First posted on arXiv 8 Nov 2023.

## Builds on (most cited first, 3 of 3)

- [A Unifying Review of Deep and Shallow Anomaly Detection](https://mlsanity.com/paper/2009.11732/) (2020)
- [Instance Credibility Inference for Few-Shot Learning](https://mlsanity.com/paper/2003.11853/) (2020)
- [APPLE Picker: Automatic Particle Picking, a Low-Effort Cryo-EM Framework](https://mlsanity.com/paper/1802.00469/) (2018)

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

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- [MM-PoE: Multiple Choice Reasoning via. Process of Elimination using Multi-Modal Models](https://mlsanity.com/paper/2412.07148/) (2024)
- [CLT and Edgeworth Expansion for m-out-of-n Bootstrap Estimators of The Studentized Median](https://mlsanity.com/paper/2505.11725/) (2025)
- [Exponential Convergence of CAVI for Bayesian PCA](https://mlsanity.com/paper/2505.16145/) (2025)
- [Is Sliding Window All You Need? An Open Framework for Long-Sequence Recommendation](https://mlsanity.com/paper/2604.12372/) (2026)

## Abstract (arXiv)

Probabilistic principal component analysis (PPCA) is currently one of the most used statistical tools to reduce the ambient dimension of the data. From multidimensional scaling to the imputation of missing data, PPCA has a broad spectrum of applications ranging from science and engineering to quantitative finance. Despite this wide applicability in various fields, hardly any theoretical guarantees exist to justify the soundness of the maximal likelihood (ML) solution for this model. In fact, it is well known that the maximum likelihood estimation (MLE) can only recover the true model parameters up to a rotation. The main obstruction is posed by the inherent identifiability nature of the PPCA model resulting from the rotational symmetry of the parameterization. To resolve this ambiguity, we propose a novel approach using quotient topological spaces and in particular, we show that the maximum likelihood solution is consistent in an appropriate quotient Euclidean space. Furthermore, our consistency results encompass a more general class of estimators beyond the MLE. Strong consistency of the ML estimate and consequently strong covariance estimation of the PPCA model have also been established under a compactness assumption.