# On the Estimation Performance of Generalized Power Method for Heteroscedastic Probabilistic PCA

arXiv: https://arxiv.org/abs/2312.03438 · HTML: https://mlsanity.com/paper/2312.03438/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 47 references that are in the map.
- First posted on arXiv 6 Dec 2023.

## Builds on (most cited first, 7 of 7)

- [Optimal Non-Convex Exact Recovery in Stochastic Block Model via Projected Power Method](https://mlsanity.com/paper/2106.05644/) (2021)
- [On the Local Linear Rate of Consensus on the Stiefel Manifold](https://mlsanity.com/paper/2101.09346/) (2021)
- [Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCA](https://mlsanity.com/paper/2303.06198/) (2023)
- [Orthogonal Group Synchronization with Incomplete Measurements: Error Bounds and Linear Convergence of the Generalized Power Method](https://mlsanity.com/paper/2112.06556/) (2021)
- [Seeded graph matching for the correlated Gaussian Wigner model via the projected power method](https://mlsanity.com/paper/2204.04099/) (2022)
- [A Semidefinite Relaxation for Sums of Heterogeneous Quadratic Forms on the Stiefel Manifold](https://mlsanity.com/paper/2205.13653/) (2022)
- [Projected Tensor Power Method for Hypergraph Community Recovery](https://mlsanity.com/paper/2307.00210/) (2023)

## Abstract (arXiv)

The heteroscedastic probabilistic principal component analysis (PCA) technique, a variant of the classic PCA that considers data heterogeneity, is receiving more and more attention in the data science and signal processing communities. In this paper, to estimate the underlying low-dimensional linear subspace (simply called ground truth) from available heterogeneous data samples, we consider the associated non-convex maximum-likelihood estimation problem, which involves maximizing a sum of heterogeneous quadratic forms over an orthogonality constraint (HQPOC). We propose a first-order method -- generalized power method (GPM) -- to tackle the problem and establish its estimation performance guarantee. Specifically, we show that, given a suitable initialization, the distances between the iterates generated by GPM and the ground truth decrease at least geometrically to some threshold associated with the residual part of certain "population-residual decomposition". In establishing the estimation performance result, we prove a novel local error bound property of another closely related optimization problem, namely quadratic optimization with orthogonality constraint (QPOC), which is new and can be of independent interest. Numerical experiments are conducted to demonstrate the superior performance of GPM in both Gaussian noise and sub-Gaussian noise settings.