Self-supervised learning by predicting transformations has demonstrated
outstanding performances in both unsupervised and (semi-)supervised tasks.
Among the state-of-the-art methods is the AutoEncoding Transformations (AET) by
decoding transformations from the learned representations of original and
transformed images. Both deterministic and probabilistic AETs rely on the
Euclidean distance to measure the deviation of estimated transformations from
their groundtruth counterparts. However, this assumption is questionable as a
group of transformations often reside on a curved manifold rather staying in a
flat Euclidean space. For this reason, we should use the geodesic to
characterize how an image transform along the manifold of a transformation
group, and adopt its length to measure the deviation between transformations.
Particularly, we present to autoencode a Lie group of homography
transformations PG(2) to learn image representations. For this, we make an
estimate of the intractable Riemannian logarithm by projecting PG(2) to a
subgroup of rotation transformations SO(3) that allows the closed-form
expression of geodesic distances. Experiments demonstrate the proposed AETv2
model outperforms the previous version as well as the other state-of-the-art
self-supervised models in multiple tasks.

提出了一种使用提取自动编码转换的 Lie 群 PG（2）对图像进行表示学习，并通过在旋转群 SO（3）中估计无法计算的黎曼对数的方法来度量变换之间的测地线距离的自监督学习方法（AETv2），在多个任务中表现优于其他最先进的自监督模型。