Frequency of large singular-value gaps in practical matrices

Determine how frequently practical matrices satisfy the large consecutive-singular-value-gap condition required by the Approximate-and-Round algorithm of Bhardwaj and Vu.

Background

The Approximate-and-Round algorithm removes dependence on the condition number but assumes that consecutive singular values are sufficiently separated. The paper describes this gap requirement as mild from a technical perspective, yet notes that its practical prevalence is unclear.

This question is included because the authors explicitly state that they do not know how often the assumption is satisfied by matrices arising in applications. The discussion contrasts large spectral gaps with the small-condition-number assumption and motivates the search for guarantees that require neither condition.

References

While the required bound for the gaps is mild (much weaker than what one requires for the application of Davis-Kahan theorem; see for more discussion), we do not know how often matrices in practice satisfy it.

— Fast exact recovery of noisy matrix from few entries: the infinity norm approach  (2501.19224 - Tran et al., 31 Jan 2025) in Section 1.4.4, "Low rank approximation with rounding-off"