1 research outputs found
The Lanczos Algorithm Under Few Iterations: Concentration and Location of the Output
We study the Lanczos algorithm where the initial vector is sampled uniformly
from . Let be an Hermitian matrix. We show
that when run for few iterations, the output of Lanczos on is almost
deterministic. More precisely, we show that for any
there exists depending only on and a certain global
property of the spectrum of (in particular, not depending on ) such that
when Lanczos is run for at most iterations, the output Jacobi
coefficients deviate from their medians by with probability at most
for . We directly obtain a similar
result for the Ritz values and vectors. Our techniques also yield asymptotic
results: Suppose one runs Lanczos on a sequence of Hermitian matrices whose spectral distributions converge in Kolmogorov distance
with rate to a density for some .
Then we show that for large enough , and for , the
Jacobi coefficients output after iterations concentrate around those for
. The asymptotic setting is relevant since Lanczos is often used to
approximate the spectral density of an infinite-dimensional operator by way of
the Jacobi coefficients; our result provides some theoretical justification for
this approach.
In a different direction, we show that Lanczos fails with high probability to
identify outliers of the spectrum when run for at most iterations,
where again depends only on the same global property of the spectrum of
. Classical results imply that the bound is tight up to a
constant factor.Comment: v2: A detailed discussion of the motivation and relevance of the main
results and definitions has been added. Minor corrections have been made. 38
pages, 3 figure