  Contrastive learning has emerged as a prominent branch of self-supervised
learning for several years. Especially, CLIP, which applies contrastive
learning to large sets of captioned images, has garnered significant attention.
Recently, SigLIP, a variant of CLIP, has been proposed, which uses the sigmoid
loss instead of the standard InfoNCE loss. SigLIP achieves the performance
comparable to CLIP in a more efficient manner by eliminating the need for a
global view. However, theoretical understanding of using the sigmoid loss in
contrastive learning is underexplored. In this paper, we provide a theoretical
analysis of using the sigmoid loss in contrastive learning, in the perspective
of the geometric structure of learned embeddings. First, we propose the
double-Constant Embedding Model (CCEM), a framework for parameterizing various
well-known embedding structures by a single variable. Interestingly, the
proposed CCEM is proven to contain the optimal embedding with respect to the
sigmoid loss. Second, we mathematically analyze the optimal embedding
minimizing the sigmoid loss for contrastive learning. The optimal embedding
ranges from simplex equiangular-tight-frame to antipodal structure, depending
on the temperature parameter used in the sigmoid loss. Third, our experimental
results on synthetic datasets coincide with the theoretical results on the
optimal embedding structures.
