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.
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