Link prediction is a fundamental problem for network-structured data. However, the prevalent research paradigm tends to assume abundant observed links, overlooking more challenging scenarios with a scarcity of observed links, which results in insufficient sample sizes. In real-world networks, hierarchical modularity, characterized by structured and nested connections, remains robust even with sparsely observed links. To address the challenge of limited link samples, we propose leveraging hierarchical modularity as a prior structure. We introduce complete-tree (CT) space, a discrete metric space with latent complete-tree structures, to formalize hierarchical modularity with an emphasis on its hierarchical permutation symmetry. Utilizing the group theory to quantize and compare permutation symmetries of different spaces, we prove that the CT space provides a significantly lower bound on sample complexity than the commonly used Euclidean space. We develop leaf matching, a data-efficient network embedding that maps nodes onto the CT space and conducts discrete optimization by reducing it to decentralized search. Experiments verify the data efficiency of CT space over other spaces. Moreover, leaf matching outperforms the state-of-the-art graph transformer in data-scarce scenarios while exhibiting excellent scalability. The code is available at: **https://github.com/KevinGao7/LeafMatching**.

Abstract The image‐to‐image translation (I2IT) task aims to transform images from the source domain into the specified target domain. State‐of‐the‐art CycleGAN‐based translation algorithms typically use cycle consistency loss and latent regression loss to constrain translation. In this work, it is demonstrated that the model parameters constrained by the cycle consistency loss and the latent regression loss are equivalent to optimizing the medians of the data distribution and the generative distribution. In addition, there is a style bias in the translation. This bias interacts between the generator and the style encoder and visually exhibits translation errors, e.g. the style of the generated image is not equal to the style of the reference image. To address these issues, a new I2IT model termed high‐quality‐I2IT (HQ‐I2IT) is proposed. The optimization scheme is redesigned to prevent the model from optimizing the median of the data distribution. In addition, by separating the optimization of the generator and the latent code estimator, the redesigned model avoids error interactions and gradually corrects errors during training, thereby avoiding learning the median of the generated distribution. The experimental results demonstrate that the visual quality of the images produced by HQ‐I2IT is significantly improved without changing the generator structure, especially when guided by the reference images. Specifically, the Fréchet inception distance on the AFHQ and CelebA‐HQ datasets are reduced from 19.8 to 10.2 and from 23.8 to 17.0, respectively.

Learning to detect and encode temporal regularities (TRs) in events is a prerequisite for human-like intelligence. These regularities should be formed from limited event samples and stored as easily retrievable representations. Existing event embeddings, however, cannot effectively decode TR validity with well-trained vectors, let alone satisfy the efficiency requirements. We develop Noether Embedding (NE) as the first efficient TR learner with event embeddings. Specifically, NE possesses the intrinsic time-translation symmetries of TRs indicated as conserved local energies in the embedding space. This structural bias reduces the calculation of each TR validity to embedding each event sample, enabling NE to achieve data-efficient TR formation insensitive to sample size and time-efficient TR retrieval in constant time complexity. To comprehensively evaluate the TR learning capability of embedding models, we define complementary tasks of TR detection and TR query, formulate their evaluation metrics, and assess embeddings on classic ICEWS14, ICEWS18, and GDELT datasets. Our experiments demonstrate that NE consistently achieves about double the F1 scores for detecting valid TRs compared to classic embeddings, and it provides over ten times higher confidence scores for querying TR intervals. Additionally, we showcase NE's potential applications in social event prediction, personal decision-making, and memory-constrained scenarios.