  We consider a certifiable object pose estimation problem, where -- given a
partial point cloud of an object -- the goal is to not only estimate the object
pose, but also to provide a certificate of correctness for the resulting
estimate. Our first contribution is a general theory of certification for
end-to-end perception models. In particular, we introduce the notion of
$\zeta$-correctness, which bounds the distance between an estimate and the
ground truth. We show that $\zeta$-correctness can be assessed by implementing
two certificates: (i) a certificate of observable correctness, that asserts if
the model output is consistent with the input data and prior information, (ii)
a certificate of non-degeneracy, that asserts whether the input data is
sufficient to compute a unique estimate. Our second contribution is to apply
this theory and design a new learning-based certifiable pose estimator. We
propose C-3PO, a semantic-keypoint-based pose estimation model, augmented with
the two certificates, to solve the certifiable pose estimation problem. C-3PO
also includes a keypoint corrector, implemented as a differentiable
optimization layer, that can correct large detection errors (e.g. due to the
sim-to-real gap). Our third contribution is a novel self-supervised training
approach that uses our certificate of observable correctness to provide the
supervisory signal to C-3PO during training. In it, the model trains only on
the observably correct input-output pairs, in each training iteration. As
training progresses, we see that the observably correct input-output pairs
grow, eventually reaching near 100% in many cases. Our experiments show that
(i) standard semantic-keypoint-based methods outperform more recent
alternatives, (ii) C-3PO further improves performance and significantly
outperforms all the baselines, and (iii) C-3PO's certificates are able to
discern correct pose estimates.
