Clustered-gap bounds for randomized block Krylov approximation

For randomized block Krylov iteration with block size $b$, a Gaussian starting
block and output $Z[Z^{\mathsf T}A]_k$, the manuscript proves that
$q = \lceil C\varepsilon^{-1/2}(t\log(2/\Delta) + \log(n/(\delta\varepsilon)))\rceil$
iterations, with $t = \lceil k/b \rceil$, an absolute constant $C$ and $\Delta$
the relative $b$-step gap of the leading $bt$ squared singular values
(assumed positive, with $bt \le \operatorname{rank}(A)$ and
$0 < \varepsilon, \delta < 1/2$), give
relative error $1 + \varepsilon$ in the spectral and Frobenius norms and
right singular-vector energies within $\varepsilon\sigma_{k+1}^2$, with
probability at least $1 - \delta$. The new estimate concerns a positive head
spectrum of size $m = bt$ with relative $b$-step gap at least $\Delta$ and a
Gaussian $m \times b$ block: with probability at least $1 - \eta$, the
tangential interpolation evaluation matrix has spectral norm at most
$2^{24}m^8\eta^{-4}(128e/\Delta)^t$ on $[0, \sigma_m^2]$. The proof combines a
logarithmic root covering, an annular resolvent identity and a variational
descent estimate with a real-algebraic tube theorem of Bürgisser, Cucker and
Lotz. A standard good-start convergence theorem then gives the iteration
count, without an additional $t\log m$ term.