Explain the comparable finite-sample performance of low-rank imputation and local PCA

Explain why low-rank imputation and local principal-components analysis achieve nearly identical imputation errors in the simulations when the underlying graphon is nonlinear and high-rank, including whether approximate low-rank structure and sample loss in local principal-components analysis account for the observed pattern.

Background

The paper compares the low-rank imputation methods of Bai et al. and Li et al. with the local PCA method of Feng et al. in simulations based on a nonlinear graphon that is theoretically high-rank. Although local PCA should have an asymptotic advantage in this setting, the reported Frobenius-norm RMSEs of the two methods are nearly identical across sampling rates and network sparsity levels.

The authors conjecture that the similarity may result from the graphon being approximately low-rank and from local PCA losing observations when it restricts estimation to neighborhood submatrices. The conjecture is unresolved because the paper does not formally establish which mechanism, or combination of mechanisms, generates the observed finite-sample equivalence.

References

We conjecture that this pattern is driven by two reasons: (1) Although the graphon is not exactly low-rank, it is approximately low-rank, so the low-rank approach still provides a good approximation to the graphon matrix, especially when performance is measured using Frobenius norm. (2) Local PCA relies only on the subnetwork formed by neighbors, which leads to substantial sample loss and limits its performance in finite samples.

— Flexible Imputation of Incomplete Network Data  (2604.03171 - Sun et al., 3 Apr 2026) in Section 5.2, Simulation for Imputation Accuracy, discussion following Table 1