@inproceedings{shi-etal-2026-manifold,
title = "Manifold Embedding: A Point-to-Hyperplane Approach",
author = "Shi, Xianglong and
Jiang, Yunhan and
Sebe, Nicu and
Chen, Ziheng",
editor = "Lee, Hung-Yi and
Dev, Sunipa and
Li, Jessy and
Elliott, Desmond",
booktitle = "Proceedings of the 2026 Conference on Empirical Methods in Natural Language Processing",
month = oct,
year = "2026",
address = "Budapest, Hungary",
publisher = "Association for Computational Linguistics",
url = "https://aclanthology.org/2026.emnlp-main.1532/",
pages = "34982--35001",
ISBN = "979-8-89176-459-0",
abstract = "Neural networks over non-Euclidean manifolds have attracted increasing attention across various applications. A fundamental step in these networks is to map Euclidean features onto a target manifold. Existing methods mainly use exponential maps and projections, operate through tangent or ambient Euclidean spaces, and may therefore distort intrinsic manifold geometry. On the other hand, point-to-hyperplane distances have recently shown great success in building manifold-valued networks. Based on this, we propose the Point-to-Hyperplane Distance Map (PHDM), a parameter-free mapping that extends Euclidean fully connected layers to Euclidean-to-manifold transformations. Specially, the exponential map is recovered as a pseudo-distance-based special case. We manifest PHDM on the Lorentz, Poincar{\'e}, spherical, and projected-spherical models, covering both negative and positive curvature. Experiments on arithmetic reasoning across four LLMs, image classification, graph node classification, and link prediction demonstrate consistent improvements over the corresponding baselines. The code is available at https://github.com/Longchentong/PHDM."
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%0 Conference Proceedings
%T Manifold Embedding: A Point-to-Hyperplane Approach
%A Shi, Xianglong
%A Jiang, Yunhan
%A Sebe, Nicu
%A Chen, Ziheng
%Y Lee, Hung-Yi
%Y Dev, Sunipa
%Y Li, Jessy
%Y Elliott, Desmond
%S Proceedings of the 2026 Conference on Empirical Methods in Natural Language Processing
%D 2026
%8 October
%I Association for Computational Linguistics
%C Budapest, Hungary
%@ 979-8-89176-459-0
%F shi-etal-2026-manifold
%X Neural networks over non-Euclidean manifolds have attracted increasing attention across various applications. A fundamental step in these networks is to map Euclidean features onto a target manifold. Existing methods mainly use exponential maps and projections, operate through tangent or ambient Euclidean spaces, and may therefore distort intrinsic manifold geometry. On the other hand, point-to-hyperplane distances have recently shown great success in building manifold-valued networks. Based on this, we propose the Point-to-Hyperplane Distance Map (PHDM), a parameter-free mapping that extends Euclidean fully connected layers to Euclidean-to-manifold transformations. Specially, the exponential map is recovered as a pseudo-distance-based special case. We manifest PHDM on the Lorentz, Poincaré, spherical, and projected-spherical models, covering both negative and positive curvature. Experiments on arithmetic reasoning across four LLMs, image classification, graph node classification, and link prediction demonstrate consistent improvements over the corresponding baselines. The code is available at https://github.com/Longchentong/PHDM.
%U https://aclanthology.org/2026.emnlp-main.1532/
%P 34982-35001
Markdown (Informal)
[Manifold Embedding: A Point-to-Hyperplane Approach](https://aclanthology.org/2026.emnlp-main.1532/) (Shi et al., EMNLP 2026)
ACL
- Xianglong Shi, Yunhan Jiang, Nicu Sebe, and Ziheng Chen. 2026. Manifold Embedding: A Point-to-Hyperplane Approach. In Proceedings of the 2026 Conference on Empirical Methods in Natural Language Processing, pages 34982–35001, Budapest, Hungary. Association for Computational Linguistics.