CMMD Is a Mean: The Deployed CLIP Maximum Mean Discrepancy Measures Little Beyond the Average Embedding

CMMD, a maximum mean discrepancy between CLIP image embeddings with a Gaussian kernel, was proposed to replace FID and is used to rank text-to-image models. At large bandwidth such discrepancies are known to behave like comparisons of means; we work out what this means for CMMD as deployed, with bandwidth σ = 10 on unit-norm embeddings and a factor of 1000. The score equals A∥Δμ∥^2, with A = 10e^−0.01 ≈ 9.9005 and Δμ the difference of mean embeddings, plus a remainder that all differences beyond the mean together cannot raise above 0.0997; at least 80% of every published model score (0.512 to 1.14) is therefore the mean term. A set that repeats k real images has expected score B(1/k + 1/n) against an n-image reference, with B exceeding A times the total variance by at most 1.01%, so on the COCO-30k reference of the public PyTorch port the published scores equal the expected scores of sets repeating 4.7 to 10.4 real images. Greedy mean matching lets a selector with access to the reference embeddings make 400 selections of real images (about 392 distinct) that score below an independent set of 5,000 real images in 20 of 20 draws; a selector that only knows the mean of 5,000 other real images does not beat it on average (8 of 20 draws at 800 selections). Reporting the mean term and the remainder separately, and quoting scores in units of repeated real images, make deployed CMMD readable as what it is: a distance between mean embeddings.