AI 中文总结
该研究针对多标签Jaccard测度，证明精确校准的凸代理需指数级预测坐标，同时给出多项式维度的近似校准保证及显式遗憾传递结果。
AI 中文摘要
实例级Jaccard得分（即交并比IoU）是多标签分类和二值分割中的标准指标。对于s个标签，其损失矩阵包含2^s种结果与报告。在约定Jac(∅,∅)=1下，我们证明Jaccard得分、移位损失及普通损失矩阵均非奇异，且损失列的仿射维度为2^s−1，该证明结合了有限MinHash Gram表示与布尔Möbius反演。对于精确校准，我们证明2^{s−1}≤CCdim(L^{Jac})≤2^s−1；下界采用具有2^{s−1}+1个支撑结果及贝叶斯最优报告的因子加权分布，因此每个精确校准的凸代理都需要指数级数量的预测坐标。我们还给出两个多项式维度的近似保证及显式遗憾传递：新的F₁到Jaccard传递将现有(s²+1)维F₁代理转化为多项式时间规则，其渐近Jaccard遗憾最多为3−2√2；对于任意α>0和0<ρ<1，MinHash平方损失代理在任意条件标签分布上可均匀达到Jaccard遗憾下限α，直接构造的维度以至少1−ρ的概率为O((s²+s log(1/ρ))/α²)，带符号变体的维度为O((s+log(1/ρ))/α²)。综上，零遗憾校准需要指数维度，而每个固定加性遗憾容差都允许多项式预测维度。
英文摘要
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 $\mathrm{Jac}(\varnothing,\varnothing)=1$, 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 $2^{s-1} \leq \mathrm{CCdim}(L^{\mathrm{Jac}}) \leq 2^s-1$. 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.
