多标签F1损失的精确秩与凸校准维下界
Exact Rank and Convex Calibration Dimension Lower Bounds for the Multi-Label F1 Loss
AI总结：
本文针对多标签F1损失，确定其相关矩阵的精确秩，通过分析贝叶斯几何给出凸校准维的下界，证明其凸校准维为Θ(s²)。
AI中文摘要：
实例级F1测度是多标签分类的核心性能指标，对于具有s个标签的问题，它定义了一个2^s×2^s的损失矩阵。先前研究给出了s²+1维坐标仿射移位低秩表示，并据此构造了二次维的凸校准代理函数。本文确定了其精确秩：在约定F1(∅,∅)=1的情况下，F1得分矩阵、移位损失矩阵与未移位损失矩阵的秩均为s²-s+2，而损失的列仿射维为s²-s+1。证明过程通过子集关联矩阵与正定柯西矩阵对非空得分矩阵进行因式分解。精确秩本身并不直接给出任意凸校准代理函数的维下界，因此本文直接分析F1的贝叶斯几何：构造一个分布，使得固定核心标签集的所有超集恰好为贝叶斯最优，证明对应活动损失列在见证支撑上的仿射维为hn，其中n=s-⌊s/3⌋，h=⌈(s⌊s/3⌋)^(1/2)⌉-1。应用凸校准维的可行子空间下界可得CCdim(L^{F1})≥(2/(3√3)-o(1))s²，结合二次上界，确立CCdim(L^{F1})=Θ(s²)。
英文摘要：
The instance-wise $F_1$ measure is a central performance measure for multi-label classification. For a problem with $s$ labels, it defines a $2^s\times 2^s$ loss matrix. Previous work exhibited $s^2+1$-coordinate affine and shifted low-rank representations and used them to construct quadratic-dimensional convex calibrated surrogates. We determine the exact rank. Under the convention $F_1(\varnothing,\varnothing)=1$, the $F_1$ score matrix, the shifted loss matrix, and the unshifted loss matrix all have rank $s^2-s+2$, while the column-affine dimension of the loss is $s^2-s+1$. The proof factors the nonempty score matrix through subset-incidence matrices and a positive-definite Cauchy matrix. Exact rank does not, by itself, lower-bound the dimension of an arbitrary convex calibrated surrogate. We therefore analyze the Bayes geometry of $F_1$ directly. We construct a distribution for which precisely all supersets of a fixed core label set are Bayes optimal, and show that the corresponding active loss columns, restricted to the witness support, have affine dimension $hn$, where $n=s-\lfloor s/3\rfloor$ and $h=\lceil(s\lfloor s/3\rfloor)^{1/2}\rceil-1$. Applying the feasible-subspace lower bound for convex calibration dimension gives \[ \operatorname{CCdim}(L^{F_1}) \ge \left(\frac{2}{3\sqrt{3}}-o(1)\right)s^2. \] Together with the quadratic upper bound, this establishes $\operatorname{CCdim}(L^{F_1})=Θ(s^2)$.
