  The obstruction to constructing localized degrees of freedom is a signature
of several interesting condensed matter phases. We introduce a localization
renormalization procedure that harnesses this property, and apply our method to
distinguish between topological and trivial phases in quantum Hall and Chern
insulators. By iteratively removing a fraction of maximally-localized
orthogonal basis states, we find that the localization length in the residual
Hilbert space exhibits a power-law divergence as the fraction of remaining
states approaches zero, with an exponent of $\nu=0.5$. In sharp contrast, the
localization length converges to a system-size-independent constant in the
trivial phase. We verify this scaling using a variety of algorithms to truncate
the Hilbert space, and show that it corresponds to a statistically self-similar
expansion of the real-space projector. This result accords with a
renormalization group picture and motivates the use of localization
renormalization as a versatile numerical diagnostic for quantum Hall systems.
