Abstract
Empowering visual algorithms/models quickly adapt to new environments/tasks is a key research direction in computer vision and machine learning. In the adaptation process, the matching degree between the geometry of space and the geometric structure of data plays an important role. Real-world data exhibits various froms of non-Euclidean geometric structures, such as hierarchical structures in natural language and cyclical structures in facial images. Previous research has shown that the non-Euclidean structure of real-world data are consistent with Riemannian manifold structures, providing theoretical feasibility for modeling data using Riemannian manifolds. Here, we study to create suitable Riemannian geometry of the underlying data space to match the given data from the data-level, model-level, and optimizer-level perspectives, as shown in Figure 1. We further provide an illustration of geometry adaptation in Figure.2
Key words: Riemannian Manifolds, Geometry Adaptation, Riemannian Optimization, Riemannian Neural Networks
Data-Level Geometry Adaptation
Model-Level Geometry Adaptation
MODALITY ALIGNMENT ACROSS TREES ON HETEROGENEOUS
HYPERBOLIC MANIFOLDS
Wei Wu*, Xiaomeng Fan*, Yuwei Wu, Zhi Gao, Pengxiang Li, Yunde Jia,
Mehrtash Harandi
ICLR, 2026.
[PDF] [Project] [Code]
Metric learning on heterogeneous hyperbolic manifolds.
Curvature Learning for the Generalization of Hyperbolic Neural Networks
Xiaomeng Fan, Yuwei Wu, Zhi Gao, Mehrtash Harandi, Yunde Jia
IJCV, 2025.
[PDF] [Project] [Code]
Searching optimal curvatures for hyperbolic neural networks.
Exploring Data Geometry for Continual Learning
Zhi Gao, Chen Xu, Feng Li, Yunde Jia, Mehrtash Harandi, Yuwei Wu
CVPR, 2023.
[PDF] [Project] [Code]
A exapndable geometry of mixed-curvature space for dynamic data stream.
Curvature-Adaptive Meta-Learning for Fast Adaptation to Manifold Data
Zhi Gao, Yuwei Wu, Mehrtash Harandi, Yunde Jia
T-PAMI, 2022.
[PDF] [Project] [Code]
Curvature adaptation for backbone in mixed-curvature space.
Optimizer-Level Geometry Adaptation
Riemannian Implicit Differentiation via a Fixed-Point Equation
for Riemannian Bi-Level Optimization
Xiaomeng Fan, Yuwei Wu, Zhi Gao, Zhipeng Lu, Feng Li, Mehrtash Harandi, Yunde Jia
T-NNLS, 2025.
[PDF] [Project] [Code]
A unified Riemannian bi-level optimization framework.
Large-Scale Riemannian Meta-Optimization via Subspace Adaptation
Peilin Yu, Yuwei Wu, Zhi Gao, Xiaomeng Fan, Yunde Jia
CVIU, 2025.
[PDF] [Project] [Code]
Riemannian Meta-Optimization for Deep Neural Networks.
Learning to Optimize on Riemannian Manifolds
Zhi Gao, Yuwei Wu, Xiaomeng Fan, Mehrtash Harandi, Yunde Jia
T-PAMI, 2023.
[PDF] [Project] [Code]
Learning Riemannian optimizers for vairous manifolds.
Efficient Riemannian Meta-Optimization by Implicit Differentiation
Xiaomeng Fan, Yuwei Wu, Zhi Gao, Yunde Jia, Mehrtash Harandi
AAAI, 2022.
[PDF] [Project] [Code]
Efficiently learning Riemannian optimizers via implicit differentiation.
Learning a Gradient-free Riemannian Optimizer on Tangent Spaces
Xiaomeng Fan, Zhi Gao, Yuwei Wu, Yunde Jia, Mehrtash Harandi
AAAI, 2021.
[PDF] [Project] [Code]
Learning gredient-free Riemannian optimizers.
Tutorial on Hyperbolic Representation Learning (updated to 2023)
Hyperbolic Learning: Theory and Applications
Pengxiang Li, Peilin Yu, Yangkai Xue, Yuwei Wu , Zhi Gao
[PDF]
A tutorial explores hyperbolic learning's theoretical underpinnings and applications,
highlighting its advantages in modeling hierarchical data in diverse downstream felds.
Resource
[THESIS 2023] Geometry-Adaptive Meta-Learning in Riemannian Manifolds. Zhi Gao. (in Chinese) [pdf]