Strengthen RBKI and Krylov conditioning results using b‑th order gaps without condition number

Establish whether the main results—Theorem LRA (RBKI with any block size, gap-independent bounds), Theorem LRA_gap (gap-dependent bounds), and Theorem krylov_sigmamin_bound (lower bound on the minimum singular value of a random block Krylov matrix)—can be proved to hold when the b‑th order relative singular value gap Δ_k^{(b)} = min_{i=1,...,k−b} (σ_i^2 − σ_{i+b}^2)/σ_i^2 replaces the standard minimum relative gap Δ_k, and any dependence on the rank‑k condition number κ_k is removed entirely.

Background

Beyond establishing O~(k/√ε) matrix‑vector complexity for RBKI across all block sizes, the paper’s technical core proves lower bounds on the minimum singular value of random block Krylov matrices. These results currently depend on the standard minimum relative gap Δ_k and include condition-number κ_k terms (typically inside logarithms) in the bounds.

To achieve a “best of both worlds” dependence—reflecting both block size b and sharper spectral gap structure—the authors conjecture strengthening the gap parameter to the b‑th order relative gap Δ_k^{(b)} and eliminating κ_k entirely across their principal theorems. Such a result would refine the theoretical understanding of RBKI’s convergence for intermediate block sizes and tighten bounds on the conditioning of block Krylov matrices.