Hyperbolic Image Embeddings (1904.02239v2)

Abstract: Computer vision tasks such as image classification, image retrieval and few-shot learning are currently dominated by Euclidean and spherical embeddings, so that the final decisions about class belongings or the degree of similarity are made using linear hyperplanes, Euclidean distances, or spherical geodesic distances (cosine similarity). In this work, we demonstrate that in many practical scenarios hyperbolic embeddings provide a better alternative.

• The paper demonstrates that hyperbolic embeddings outperform Euclidean and spherical methods, notably with Hyperbolic ProtoNet on MiniImageNet and CUB datasets.
• It introduces hyperbolic network layers applying Möbius addition and distance computations to effectively capture hierarchical structures in visual data.
• The study shows that the distance from the origin in the Poincaré model serves as a measure of uncertainty, enhancing confidence assessment in image models.

Hyperbolic Image Embeddings: An Overview

The paper "Hyperbolic Image Embeddings" explores the application of hyperbolic geometry to image embedding tasks in computer vision, such as image classification, image retrieval, and few-shot learning. Traditionally, these tasks have utilized Euclidean and spherical embedding spaces. The authors propose that hyperbolic spaces, with their negative curvature, can provide a more effective alternative in certain scenarios.

Introduction

In conventional image embedding approaches, Euclidean and spherical geometries are leveraged, which involve methods like linear hyperplanes, Euclidean distances, or spherical geodesic distances. By contrast, this research advocates for hyperbolic embeddings. Hyperbolic spaces, characterized by negative curvature, are argued to be more suited for representing hierarchical and complex relational structures often found in visual data, analogous to their success in natural language processing tasks.

Motivation and Methodology

The motivation for utilizing hyperbolic embeddings in vision tasks stems from their capacity to represent hierarchies efficiently, akin to their use in NLP. The researchers incorporate hyperbolic network layers into existing vision architectures, demonstrating through experiments the improved performance in various tasks such as image classification and one-shot learning.

The paper relies on the Poincaré ball model of hyperbolic space, outlining operations like M\"obius addition, distance computation, and hyperbolic averaging. These operations enable the transformation of deep network outputs into hyperbolic space, effectively capturing hierarchical structures inherent in image data.

Key Results

The paper reveals several significant findings:

1. Performance Improvement: Hyperbolic embeddings often outperform their Euclidean and spherical counterparts, particularly noticeable in few-shot learning scenarios. For example, Hyperbolic ProtoNet demonstrated superior performance on datasets like MiniImageNet and CUB by effectively utilizing the hierarchical relationships between classes.
2. Measure of Uncertainty: The distance of embeddings from the origin in Poincaré space serves as a reliable measure of uncertainty. This property was validated by experimenting with models trained on MNIST and tested on Omniglot, where more ambiguous images gravitated towards the center of the hyperbolic space.
3. Dataset Hyperbolicity: Through the concept of $\delta$-hyperbolicity, the research quantifies the suitability of datasets for hyperbolic embedding, suggesting that many image datasets naturally align well with hyperbolic geometry.

Implications and Future Directions

The implications of this work are both practical and theoretical. On the practical side, employing hyperbolic embeddings can lead to more precise and confidence-aware models in image-related tasks. Theoretically, this work prompts a reevaluation of how intrinsic data geometries are represented and used in machine learning models.

Future developments could focus on refining these models to better utilize hyperbolic embeddings across different tasks and datasets. Additionally, addressing the numeric precision issues arising from different models of hyperbolic geometry could pave the way for more robust application scenarios.

This paper marks a meaningful step in exploring the potential of hyperbolic geometry in computer vision, suggesting that it could become an integral part of future developments in embedding techniques.