What Does Fréchet Distance Measure? A Directional Decomposition

The Fréchet distance is a *de facto* standard for evaluating generative models across domains, appearing as FID for images and FVD for videos. It summarizes the discrepancy between generated and reference distributions in a single scalar, with lower values typically interpreted as better generation quality. However, this scalar view can obscure what drives the comparison. For example, in COCO dataset, increasing the number of diffusion sampling steps improves ImageReward scores yet worsens (increases) FID. Motivated by this mismatch, we seek to make the Fréchet distance more interpretable by uncovering *where* the discrepancy lies. To this end, we introduce *directional Fréchet distance*, the expected squared projection of the optimal transport displacement onto a given direction. Across our image, video, and protein case studies, we find that a small number of interpretable directions account for much of the distance. We use these directions to explain the FID increase in terms of semantic concepts represented by CLIP embeddings, quantify FVD's bias toward per-frame appearance, and revisit the interpretation of Protein FID.