Deterministic replacement for the randomized subspace selection

Construct a deterministic procedure that, given a function $f$ and the dominant eigenspace of ${K}_{\varepsilon}(f\otimes\overline{f})$, finds a function $h$ satisfying the spectral separation and cardinality conditions required to recover quadratic characters via Theorem 7.1.

Background

The quadratic-character decomposition algorithm uses a random unit vector in the dominant eigenspace when the leading eigenvalues are not sufficiently separated. The authors prove that this randomized choice succeeds with high probability, but indicate that a deterministic choice should exist and leave its construction unresolved.