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。

AETv2: 借助李群上的测地距离最小化进行自监督表示学习的自编码转换

AETv2: AutoEncoding Transformations for Self-Supervised Representation  Learning by Minimizing Geodesic Distances in Lie Groups

Transformation Equivariant Representations (TERs) aim to capture the
intrinsic visual structures that equivary to various transformations by
expanding the notion of {\em translation} equivariance underlying the success
of Convolutional Neural Networks (CNNs). For this purpose, we present both
deterministic AutoEncoding Transformations (AET) and probabilistic AutoEncoding
Variational Transformations (AVT) models to learn visual representations from
generic groups of transformations. While the AET is trained by directly
decoding the transformations from the learned representations, the AVT is
trained by maximizing the joint mutual information between the learned
representation and transformations. This results in Generalized TERs (GTERs)
equivariant against transformations in a more general fashion by capturing
complex patterns of visual structures beyond the conventional linear
equivariance under a transformation group. The presented approach can be
extended to (semi-)supervised models by jointly maximizing the mutual
information of the learned representation with both labels and transformations.
Experiments demonstrate the proposed models outperform the state-of-the-art
models in both unsupervised and (semi-)supervised tasks.

该研究介绍了确定性自编码变换和概率自编码变分变换模型，以从通用转换组中学习视觉表示，并通过最大化学习表示和转换之间的联合互信息来训练 AutoEncoding Variational Transformations 模型，该模型能更加一般性地捕获视觉结构的 复杂模式， 这将导致广义的 Transformation Equivariant Representations，可以应用到无监督和半监督的场景中。