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 2s 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 2s-1. The proof combines a finite MinHash Gram representation with Boolean M"obius inversion. For exact calibration, we prove 2s-1 ≤ CCdim(LJac) ≤ 2s-1. The lower bound uses a factorially weighted distribution with 2s-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 F1-to-Jaccard transfer turns an existing (s2+1)-dimensional F1 surrogate into a polynomial-time rule with asymptotic Jaccard regret at most 3-2√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((s2+slog(1/ρ))/α2), while a signed variant has dimension O((s+log(1/ρ))/α2). Thus zero-regret calibration requires exponential dimension, whereas every fixed additive regret tolerance admits polynomial prediction dimension.