Combined Hyperbolic and Euclidean Soft Triple Loss Beyond the Single Space Deep Metric Learning

TL;DR

CHEST addresses the challenge of proxy-based deep metric learning in hyperbolic space by jointly optimizing a proxy-based loss in hyperbolic space and in Euclidean space, with proxies mapped via exponential mapping. It adds a hierarchical regularization (HypHC) to encourage a tree-like proxy organization in hyperbolic space. Through experiments on four benchmarks, CHEST achieves new state-of-the-art results and demonstrates improved learning stability and generalization bounds compared to single-space or pair-based methods. The approach is scalable to large datasets due to lower training complexity of proxy-based losses and the cross-space regularization.

Abstract

Deep metric learning (DML) aims to learn a neural network mapping data to an embedding space, which can represent semantic similarity between data points. Hyperbolic space is attractive for DML since it can represent richer structures, such as tree structures. DML in hyperbolic space is based on pair-based loss or unsupervised regularization loss. On the other hand, supervised proxy-based losses in hyperbolic space have not been reported yet due to some issues in applying proxy-based losses in a hyperbolic space. However, proxy-based losses are attractive for large-scale datasets since they have less training complexity. To address these, this paper proposes the Combined Hyperbolic and Euclidean Soft Triple (CHEST) loss. CHEST loss is composed of the proxy-based losses in hyperbolic and Euclidean spaces and the regularization loss based on hyperbolic hierarchical clustering. We find that the combination of hyperbolic and Euclidean spaces improves DML accuracy and learning stability for both spaces. Finally, we evaluate the CHEST loss on four benchmark datasets, achieving a new state-of-the-art performance.

Figures (3)

• Figure 1: Network architecture for the CHEST loss. $D_{E}$ means the dimension of Euclidean space, and $D_{H}$ means the dimension of hyperbolic space. Each point shape means a class. Red symbols are feature vectors and black symbols are proxies. The CHEST loss encourages red lines to be close and blue lines to be far away.
• Figure 2: Distribution of embeddings on CUB200 train and test datasets and examples of retrieval results on the CUB200 dataset. The red circles in the distribution represent the retrieval query of examples. Four examples of retrieval show. Each retrieval result has two rows, including one query and 19 retrieval results. The top left images are query images, and the other images are arranged in order of similarity from top left to top right and bottom left to bottom right. A red frame means the retrieved image is a different class image.
• Figure 3: The histogram of similarities between batch data and classes. The left histogram result is without Euclidean space loss, and the right histogram result is of the combined hyperbolic and Euclidean loss (Hyperbolic + Euclidean). The blue area represents the similarities to negative classes, and the red area represents the similarities to positive class.