  We show how spectra of Type Ia supernovae (SNe Ia) at maximum light can be
used to improve cosmological distance estimates. In a companion article, we
used manifold learning to build a three-dimensional parameterization of the
intrinsic diversity of SNe Ia at maximum light that we call the "Twins
Embedding". In this article, we discuss how the Twins Embedding can be used to
improve the standardization of SNe Ia. With a single
spectrophotometrically-calibrated spectrum near maximum light, we can
standardize our sample of SNe Ia with an RMS of $0.101 \pm 0.007$ mag, which
corresponds to $0.084 \pm 0.009$ mag if peculiar velocity contributions are
removed and $0.073 \pm 0.008$ mag if a larger reference sample were obtained.
Our techniques can standardize the full range of SNe Ia, including those
typically labeled as peculiar and often rejected from other analyses. We find
that traditional light curve width + color standardization such as SALT2 is not
sufficient. The Twins Embedding identifies a subset of SNe Ia including but not
limited to 91T-like SNe Ia whose SALT2 distance estimates are biased by $0.229
\pm 0.045$ mag. Standardization using the Twins Embedding also significantly
decreases host-galaxy correlations. We recover a host mass step of $0.040 \pm
0.020$ mag compared to $0.092 \pm 0.024$ mag for SALT2 standardization on the
same sample of SNe Ia. These biases in traditional standardization methods
could significantly impact future cosmology analyses if not properly taken into
account.
