Geometry

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

Figure 1. Framework of geometry adaptation for computer vision.

Figure 2. One illustration of geometry adaptation.

Data-Level Geometry Adaptation

Hyperbolic Feature Augmentation via Distribution Estimation and
Infinite Sampling on Manifolds
Zhi Gao, Yuwei Wu, Yunde Jia, Mehrtash Harandi
NeurIPS, 2022.
[PDF] [Project] [Code]
Infinite feature augmentation on hyperbolic spaces.

Model-Level Geometry Adaptation

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.

Curvature Generation in Curved Spaces for Few-Shot Learning
Zhi Gao, Yuwei Wu, Yunde Jia, Mehrtash Harandi
ICCV, 2021.
[PDF] [Project] [Code]
Generate suitable curvature for hyperbolic classifier.

Optimizer-Level Geometry Adaptation

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.

Learning to Optimize on SPD Manifolds
Zhi Gao, Yuwei Wu, Yunde Jia, Mehrtash Harandi
CVPR, 2020.
[PDF] [Project] [Code]
Learning a optimizers on SPD manifold.

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]

Geometry Adaptation for Computer Vision