Design an algorithm matching truncated generalized Nyström guarantees with standard generalized Nyström accuracy

Develop a randomized low-rank approximation algorithm that achieves the theoretical embedding-dimension guarantees of the truncated generalized Nyström estimator while attaining the empirical accuracy of the standard generalized Nyström estimator for approximating a matrix A.

Background

The generalized Nyström method provides strong empirical accuracy for single-pass low-rank approximation but, when the oversampling factor γ is large, requires an embedding dimension p ≈ γ^2 r for the left sketch. A theoretically appealing variant—truncated generalized Nyström—reduces this to p ≈ γ r, but in practice its truncation step can significantly degrade accuracy.

The paper states that, although one can adapt the theory to justify truncated generalized Nyström with improved embedding dimension, the resulting method underperforms empirically. Bridging this gap—retaining the reduced embedding dimension while matching the standard generalized Nyström’s accuracy—remains unresolved.