Furthermore, this method can be applied to any network by replacing an existing layer.

In neural network literature, angular similarity between feature vectors is frequently used for interpreting or re-using learned representations. However, the inner product in neural networks partially disperses information over the scales and angles of the involved input vectors and weight vectors. Therefore, when using only angular similarity on representations trained with the inner product, information loss occurs in downstream methods, which limits their performance. In this paper, we proposed the $\textit{spherization layer}$ to represent all information on angular similarity. The layer 1) maps the pre-activations of input vectors into the specific range of angles, 2) converts the angular coordinates of the vectors to Cartesian coordinates with an additional dimension, and 3) trains decision boundaries from hyperplanes, without bias parameters, passing through the origin. This approach guarantees that representation learning always occurs on the hyperspherical surface without the loss of any information unlike other projection-based methods. Furthermore, this method can be applied to any network by replacing an existing layer. We validate the functional correctness of the proposed method in a toy task, retention ability in well-known image classification tasks, and effectiveness in word analogy test and few-shot learning.

Knowledge graph embedding (KGE) has become a fundamental technique for representation learning on multi-relational data. Many seminal models, such as TransE, operate in an unbounded Euclidean space, which presents inherent limitations in modeling complex relations and can lead to inefficient training. In this paper, we propose Spherical Knowledge Graph Embedding (SKGE), a model that challenges this paradigm by constraining entity representations to a compact manifold: a hypersphere. SKGE employs a learnable, non-linear Spherization Layer to map entities onto the sphere and interprets relations as a hybrid translate-then-project transformation. Through extensive experiments on three benchmark datasets, FB15k-237, CoDEx-S, and CoDEx-M, we demonstrate that SKGE consistently and significantly outperforms its strong Euclidean counterpart, TransE, particularly on large-scale benchmarks such as FB15k-237 and CoDEx-M, demonstrating the efficacy of the spherical geometric prior. We provide an in-depth analysis to reveal the sources of this advantage, showing that this geometric constraint acts as a powerful regularizer, leading to comprehensive performance gains across all relation types. More fundamentally, we prove that the spherical geometry creates an "inherently hard negative sampling" environment, naturally eliminating trivial negatives and forcing the model to learn more robust and semantically coherent representations. Our findings compellingly demonstrate that the choice of manifold is not merely an implementation detail but a fundamental design principle, advocating for geometric priors as a cornerstone for designing the next generation of powerful and stable KGE models.

Furthermore, this method can be applied to any network by replacing an existing layer.

In neural network literature, angular similarity between feature vectors is frequently used for interpreting or re-using learned representations. However, the inner product in neural networks partially disperses information over the scales and angles of the involved input vectors and weight vectors. Therefore, when using only angular similarity on representations trained with the inner product, information loss occurs in downstream methods, which limits their performance. In this paper, we proposed the \textit{spherization layer} to represent all information on angular similarity. The layer 1) maps the pre-activations of input vectors into the specific range of angles, 2) converts the angular coordinates of the vectors to Cartesian coordinates with an additional dimension, and 3) trains decision boundaries from hyperplanes, without bias parameters, passing through the origin.