Remove condition-number dependence in RBKI bounds via the relative gap

Establish whether Theorem LRA (randomized block Krylov iteration with any block size, gap-independent bounds) and Theorem LRA_gap (gap-dependent bounds) can be proved with dependence only on the minimum relative singular value gap Δ_k = min_{i=1,...,k-1} (σ_i^2 − σ_{i+1}^2)/σ_i^2, replacing the ratio Δ_k/κ_k that involves the rank‑k condition number κ_k = σ_1^2/σ_{k-1}^2; specifically, show that randomized block Krylov iteration achieves the stated matrix‑vector complexity guarantees when any dependence on κ_k is eliminated in favor of Δ_k alone.

Background

The paper proves that randomized block Krylov iteration (RBKI) achieves a (1+ε)-factor low-rank approximation using O~(k/√ε) matrix‑vector products for any block size 1 ≤ b ≤ k, and provides gap-dependent refinements. In the current bounds, the dependence on spectral gaps appears as the ratio Δ_k/κ_k, where Δ_k is the minimum relative singular value gap and κ_k is a rank‑k condition number, primarily inside logarithmic factors.

The authors note that their proofs also work with the minimum additive gap, but for consistency with prior work they stated results in terms of relative gaps and condition numbers. They explicitly conjecture that their RBKI bounds can be strengthened to rely on Δ_k alone, i.e., removing κ_k from the dependence, which would yield sharper theoretical guarantees while preserving the established runtime complexity.