Learning faithful graph representations as sets of vertex embeddings has
become a fundamental intermediary step in a wide range of machine learning
applications. We propose the systematic use of symmetric spaces in
representation learning, a class encompassing many of the previously used
embedding targets. This enables us to introduce a new method, the use of
Finsler metrics integrated in a Riemannian optimization scheme, that better
adapts to dissimilar structures in the graph. We develop a tool to analyze the
embeddings and infer structural properties of the data sets. For
implementation, we choose Siegel spaces, a versatile family of symmetric
spaces. Our approach outperforms competitive baselines for graph reconstruction
tasks on various synthetic and real-world datasets. We further demonstrate its
applicability on two downstream tasks, recommender systems and node
classification.

使用对称空间和 Finsler 度量结合的 Riemannian 优化方案来生成图形表示，并在实验中表现出优越性。

图嵌入的对称空间：一种 Finsler-Riemannian 方法

Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach

We present a new family of min-max optimization algorithms that automatically
exploit the geometry of the gradient data observed at earlier iterations to
perform more informative extra-gradient steps in later ones. Thanks to this
adaptation mechanism, the proposed method automatically detects whether the
problem is smooth or not, without requiring any prior tuning by the optimizer.
As a result, the algorithm simultaneously achieves order-optimal convergence
rates, i.e., it converges to an $\varepsilon$-optimal solution within
$\mathcal{O}(1/\varepsilon)$ iterations in smooth problems, and within
$\mathcal{O}(1/\varepsilon^2)$ iterations in non-smooth ones. Importantly,
these guarantees do not require any of the standard boundedness or Lipschitz
continuity conditions that are typically assumed in the literature; in
particular, they apply even to problems with singularities (such as resource
allocation problems and the like). This adaptation is achieved through the use
of a geometric apparatus based on Finsler metrics and a suitably chosen
mirror-prox template that allows us to derive sharp convergence rates for the
methods at hand.

提出了一种新的 MinMax 优化算法家族，它利用早期迭代所观察到的梯度数据的几何信息，以在后期执行更具信息性的额外梯度步骤，从而自适应地检测问题是否光滑。