Exponential Convex Calibration Dimension for the Multi-Label Jaccard Measure

The per-instance Jaccard score, or intersection over union (IoU), is standard in multi-label classification and binary segmentation. With s labels, its loss matrix has 2^s outcomes and reports. Under the convention Jac(\varnothing,\varnothing)=1, we prove that the Jaccard score, shifted-loss, and ordinary loss matrices are nonsingular and that the loss columns have affine dimension 2^s-1. The proof combines a finite MinHash Gram representation with Boolean Möbius inversion. For exact calibration, we prove 2^{s-1} \leq CCdim(L^{Jac}) \leq 2^s-1. The lower bound uses a factorially weighted distribution with 2^{s-1}+1 supported outcomes and Bayes-optimal reports. Consequently, every exactly calibrated convex surrogate requires exponentially many prediction coordinates. We also give two polynomial-dimensional approximation guarantees with explicit regret transfers. A new F_1-to-Jaccard transfer turns an existing (s^2+1)-dimensional F_1 surrogate into a polynomial-time rule with asymptotic Jaccard regret at most 3-2\sqrt{2}. For any α>0 and 0<ρ<1, a MinHash square-loss surrogate attains Jaccard-regret floor α uniformly over arbitrary conditional label distributions. With probability at least 1-ρ, the direct construction has dimension O((s^2+s\log(1/ρ))/α^2), while a signed variant has dimension O((s+\log(1/ρ))/α^2). Thus zero-regret calibration requires exponential dimension, whereas every fixed additive regret tolerance admits polynomial prediction dimension.