Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, firstly because of the absence of corresponding hyperbolic neural network layers.

Non-Euclidean foundation models increasingly place representations in curved spaces such as hyperbolic geometry. We show that this geometry creates a boundary-driven asymmetry that backdoor triggers can exploit. Near the boundary, small input changes appear subtle to standard input-space detectors but produce disproportionately large shifts in the model's representation space. Our analysis formalizes this effect and also reveals a limitation for defenses: methods that act by pulling points inward along the radius can suppress such triggers, but only by sacrificing useful model sensitivity in that same direction. Building on these insights, we propose a simple geometry-adaptive trigger and evaluate it across tasks and architectures. Empirically, attack success increases toward the boundary, whereas conventional detectors weaken, mirroring the theoretical trends. Together, these results surface a geometry-specific vulnerability in non-Euclidean models and offer analysis-backed guidance for designing and understanding the limits of defenses.

Thanks to the authors for the detailed response. The new results presented in the rebuttal are indeed convincing, hence I am updating my score to an 8 now. This is with the understanding that these would be incorporated in the revised version of the paper. Several works in the last year have explored using hyperbolic representations for data which exhibits hierarchical latent structure. Some promising results on the efficiency of these representations at capturing hierarchical relationships have been shown, most notably by Nickel & Kiela (Nips, 2017). However one big hindrance for utilizing them so far is the lack of deep neural network models which can consume these representations as input for some other downstream task.

Hyperbolic representations have shown remarkable efficacy in modeling inherent hierarchies and complexities within data structures. Hyperbolic neural networks have been commonly applied for learning such representations from data, but they often fall short in preserving the geometric structures of the original feature spaces. In response to this challenge, our work applies the Gromov-Wasserstein (GW) distance as a novel regularization mechanism within hyperbolic neural networks. The GW distance quantifies how well the original data structure is maintained after embedding the data in a hyperbolic space. Specifically, we explicitly treat the layers of the hyperbolic neural networks as a transport map and calculate the GW distance accordingly. We validate that the GW distance computed based on a training set well approximates the GW distance of the underlying data distribution. Our approach demonstrates consistent enhancements over current state-of-the-art methods across various tasks, including few-shot image classification, as well as semi-supervised graph link prediction and node classification.

Learning representations according to the underlying geometry is of vital importance for non-Euclidean data. Studies have revealed that the hyperbolic space can effectively embed hierarchical or tree-like data. In particular, the few past years have witnessed a rapid development of hyperbolic neural networks. However, it is challenging to learn good hyperbolic representations since common Euclidean neural operations, such as convolution, do not extend to the hyperbolic space. Most hyperbolic neural networks do not embrace the convolution operation and ignore local patterns. Others either only use non-hyperbolic convolution, or miss essential properties such as equivariance to permutation. We propose HKConv, a novel trainable hyperbolic convolution which first correlates trainable local hyperbolic features with fixed kernel points placed in the hyperbolic space, then aggregates the output features within a local neighborhood. HKConv not only expressively learns local features according to the hyperbolic geometry, but also enjoys equivariance to permutation of hyperbolic points and invariance to parallel transport of a local neighborhood. We show that neural networks with HKConv layers advance state-of-the-art in various tasks.

With the recent advance of deep learning, neural networks have been extensively used for the task of molecular generation. Many deep generators extract atomic relations from molecular graphs and ignore hierarchical information at both atom and molecule levels. In order to extract such hierarchical information, we propose a novel hyperbolic generative model. Our model contains three parts: first, a fully hyperbolic junction-tree encoder-decoder that embeds the hierarchical information of the molecules in the latent hyperbolic space; second, a latent generative adversarial network for generating the latent embeddings; third, a molecular generator that inherits the decoders from the first part and the latent generator from the second part. We evaluate our model on the ZINC dataset using the MOSES benchmarking platform and achieve competitive results, especially in metrics about structural similarity.

Hyperbolic neural networks have been popular in the recent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in developing these networks lies in the nonlinearity of the embedding space namely, the Hyperbolic space. Hyperbolic space is a homogeneous Riemannian manifold of the Lorentz group. Most existing methods (with some exceptions) use local linearization to define a variety of operations paralleling those used in traditional deep neural networks in Euclidean spaces. In this paper, we present a novel fully hyperbolic neural network which uses the concept of projections (embeddings) followed by an intrinsic aggregation and a nonlinearity all within the hyperbolic space. The novelty here lies in the projection which is designed to project data on to a lower-dimensional embedded hyperbolic space and hence leads to a nested hyperbolic space representation independently useful for dimensionality reduction. The main theoretical contribution is that the proposed embedding is proved to be isometric and equivariant under the Lorentz transformations. This projection is computationally efficient since it can be expressed by simple linear operations, and, due to the aforementioned equivariance property, it allows for weight sharing. The nested hyperbolic space representation is the core component of our network and therefore, we first compare this ensuing nested hyperbolic space representation with other dimensionality reduction methods such as tangent PCA, principal geodesic analysis (PGA) and HoroPCA. Based on this equivariant embedding, we develop a novel fully hyperbolic graph convolutional neural network architecture to learn the parameters of the projection. Finally, we present experiments demonstrating comparative performance of our network on several publicly available data sets.

Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, firstly because of the absence of corresponding hyperbolic neural network layers. As a result, we derive hyperbolic versions of important deep learning tools: multinomial logistic regression, feed-forward and recurrent neural networks. This allows to embed sequential data and perform classification in the hyperbolic space. Empirically, we show that, even if hyperbolic optimization tools are limited, hyperbolic sentence embeddings either outperform or are on par with their Euclidean variants on textual entailment and noisy-prefix recognition tasks. Papers published at the Neural Information Processing Systems Conference.