Sufficiency of O(k/ε) squared-norm column samples

Determine whether sampling only O(k/ε) columns according to squared column norms suffices for additive-error low-rank approximation of a positive-semidefinite matrix, thereby yielding an O(nk/ε)-time algorithm.

Background

The paper develops a simple sublinear-time algorithm for additive-error rank-k approximation of a positive-semidefinite matrix. Its procedure samples O(k/ε²) columns according to the squared column-norm distribution, uses these columns to approximate the leading k-dimensional left singular subspace, and then applies approximate matrix multiplication to construct the right factor. This yields an additive Frobenius-norm guarantee with O(nk/ε²) entry queries.

The authors note that the best previously known algorithm achieves a stronger relative-error guarantee with approximately Õ(nk/ε) queries, while the method presented in the paper incurs an additional 1/ε factor. The unresolved question is whether the paper’s sampling-based approach can reduce its sample count from O(k/ε²) to O(k/ε), which would correspondingly improve the query complexity to O(nk/ε).

References

We pose as an open question whether it is enough to sample only O(k/\epsilon) columns by squared column norms, thus leading to an O(nk/\epsilon) algorithm.

— Fast Length-Squared Sampling for Positive-Semidefinite Matrices  (2608.12503 - Bhattacharjee et al., 12 Aug 2026) in Section 4, Subsection “Additive-Error Low-Rank Approximation” (immediately before Table 1)