Exponential Convex Calibration Dimension for the Multi-Label Jaccard Measure

E XPONENTIAL C ONVEX C ALIBRATION D IMENSION FOR THE M ULTI -L ABEL JACCARD M EASURE P REPRINT Mingyuan Zhang Independent Researcher myz@alumni.upenn.edu

arXiv:2608.13549v1 [cs.LG] 13 Aug 2026

A BSTRACT 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(∅, ∅) = 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 Möbius 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 Bayesoptimal 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 squareloss 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 + s log(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. Keywords multi-label classification · Jaccard loss · intersection over union · convex calibration dimension · loss-matrix rank · approximate consistency · MinHash random features · F-measure

1

Introduction

In multi-label classification, an outcome and a prediction are subsets of a ground set of s labels. The Jaccard score compares them by the size of their intersection divided by the size of their union. The same quantity is widely called intersection over union (IoU) in image segmentation (Berman et al., 2018). Its dependence on the entire predicted and true sets makes the instance-wise loss nondecomposable across labels. We study the decision-theoretic setting in which this per-instance ratio is averaged under a conditional label distribution, rather than a population-utility setting in which confusion counts are averaged before a ratio is formed. The output space contains 2s sets, but an exponential output space alone does not determine the dimension required by a statistically consistent convex surrogate. Some structured losses admit low-dimensional affine representations or calibrated surrogates even when their report spaces are exponential. Convex calibration dimension formalizes the smallest Euclidean dimension in which an exactly calibrated convex surrogate can exist (Ramaswamy and Agarwal, 2012, 2016). Our contributions are threefold. First, we determine the exact ranks of the Jaccard score, shifted-loss, and ordinary loss matrices, together with the affine dimension of the loss columns. The nonempty score matrix is known to be

Exponential Convex Calibration Dimension for Jaccard

P REPRINT

strictly positive definite (Bouchard et al., 2013); we give a self-contained finite proof tailored to the power set using MinHash and Boolean Möbius inversion. Under our empty-set convention, all three matrix ranks equal 2s , and the affine dimension is 2s − 1. The resulting affine-dimension bound gives an exactly calibrated surrogate in 2s − 1 dimensions, although matrix rank alone cannot lower-bound arbitrary nonlinear convex surrogates. Second, we prove the exponential bounds 2s−1 ≤ CCdim(LJac ) ≤ 2s − 1. For the lower bound, fix one core label and assign each outcome containing d optional labels weight proportional to 1/d!. A combinatorial identity makes all reports containing the core label tie, while adding the core label strictly improves every nonempty report that omits it. Mixing with the empty outcome makes the empty report tie as well. The supported outcomes and Bayes-optimal reports then form the same (2s−1 + 1)-element family. Its principal score submatrix is nonsingular, making the relevant two-sided feasible subspace trivial and yielding the lower bound. Hence the convex calibration dimension for exact calibration is Θ(2s ). Third, we give two complementary polynomial-dimensional approximation guarantees. Pointwise, Jac = F1 /(2 − F1 ), but expectation does not commute with this nonlinear transformation, so F1 - and Jaccard-optimal reports need not agree. Under the same empty-set convention, the multi-label F1 loss has convex calibration dimension Θ(s2 ) (Zhang, 2026), and quadratic-dimensional convex calibrated surrogates for F1 are known (Nowak et al., 2019; Zhang et √ al., 2020). Our regret transfer turns one such surrogate into a polynomial-time Jaccard rule whose regret is at most 3 − 2 2 plus a term controlled by its F1 regret. Our second approximation route uses MinHash directly. For every α > 0 and 0 < ρ < 1, a finite MinHash feature map uniformly approximates the entire Jaccard score matrix with probability at least 1 − ρ. Regressing the conditional feature mean with a convex square loss gives an α-approximately consistent surrogate: vanishing surrogate regret guarantees asymptotic Jaccard regret at most α. A signed MinHash variant has dimension O((s + log(1/ρ))/α2 ). These are prediction-dimension guarantees, not efficient-decoding results: the exact link may still maximize over all 2s reports, while a τ -approximate decoder increases the regret floor by at most τ . There is no conflict with the exact-calibration lower bound, because these constructions allow a positive tolerance and their dimension diverges as α ↓ 0.

2

Setup and calibration background

Fix an integer s ≥ 1. Let [s] = {1, . . . , s}, Y = 2[s] , and N = |Y| = 2s . Outcomes and reports are denoted by A, B ∈ Y. For any finite index set I, let 1I ∈ RI denote the all-ones vector, and let δA denote the point mass at A ∈ Y. Define the Jaccard score by   |A ∩ B| , A ∪ B ̸= ∅, Jac(A, B) = |A ∪ B| (1)  1, A = B = ∅. Let S ∈ RN ×N be the score matrix, SA,B = Jac(A, B), let U = 1N 1⊤ N , and let L := LJac = U − S

(2)

be the Jaccard loss matrix. Thus L − U = −S. For a vector x indexed by a finite set I and T ⊆ I, xT denotes its restriction to T . For a matrix M , MT,R denotes the corresponding submatrix and M·,j its column indexed by j. For a finite family V ⊆ Rr , let affdim(V ) = dim aff(V ). For a matrix M = [m1 · · · mk ], define its column-affine dimension by affdim(M ) = affdim{m1 , . . . , mk }. ⊤ Let ∆N = {p ∈ RN + : 1N p = 1}, and write supp(p) = {A ∈ Y : pA > 0} for p ∈ ∆N . The conditional risk of report ⊤ B is p L·,B , and

optL (p) = arg min p⊤ L·,B B∈Y

is the set of Bayes-optimal reports. Its trigger probability set is  ⊤ ⊤ ′ QL B = p ∈ ∆N : p L·,B ≤ p L·,B ′ for every B ∈ Y . 2

(3)

Exponential Convex Calibration Dimension for Jaccard

P REPRINT

For completeness, let C ⊆ Rd be convex, let ψ : C → RN + have convex coordinate functions, and let pred : C → Y be a link. The pair (ψ, pred) is L-calibrated if, for every p ∈ ∆N , inf

u∈C: pred(u)∈opt / L (p)

p⊤ ψ(u) > inf p⊤ ψ(u). u∈C

The convex calibration dimension CCdim(L) is the smallest such d (Ramaswamy and Agarwal, 2016, Definitions 1 and 10). We use two general results. First, Second, for p ∈ QL B,

CCdim(L) ≤ affdim(L).

(4)

CCdim(L) ≥ ∥p∥0 − µQLB (p) − 1,

(5)

where ∥p∥0 = | supp(p)|. To define µ, for Q ⊆ ∆N and p ∈ Q, let  dirQ (p) = v ∈ RN : p + εv ∈ Q for all sufficiently small ε > 0 , and set
linQ (p) = dirQ (p) ∩ − dirQ (p) , µQ (p) = dim linQ (p). Equations (4) and (5) are Theorems 12 and 16 of Ramaswamy and Agarwal (2016).

3

Related work

Jaccard matrices and MinHash. The collision probability underlying MinHash is the Jaccard similarity (Broder, 1997). Following an earlier positive-semidefiniteness result of Gower (1971), Bouchard et al. (2013) proved strict positive definiteness of the complete nonempty Jaccard index matrix by an infinite positive-semidefinite decomposition. We give an alternative finite proof based on MinHash and Boolean Möbius inversion, then use the result to analyze the associated loss matrix. We also subsample the same Gram representation to obtain a uniformly accurate approximate factorization and elicit its conditional mean with a convex square loss. Instance-wise Jaccard prediction. Dembczyński et al. (2012) studied decision-theoretic risk minimization for several multi-label losses and highlighted the difficulty of exact Jaccard-risk minimization from an arbitrary conditional joint distribution. For an empirical distribution, Bayes-optimal Jaccard prediction is the Jaccard-median problem; Chierichetti et al. (2010) established its computational hardness and gave approximation algorithms. Although Jaccard is a pointwise monotone transform of the F1 score, the corresponding expected-score maximizers need not agree (Waegeman et al., 2014). If δ(P ) denotes the Bayes-optimal expected F1 score, Waegeman et al. (2014, Theorem 3) bound the Jaccard regret of an F1 -optimal report by 1 − δ(P )/2; this is an approximation guarantee, not exact Jaccard consistency. They take F1 (∅, ∅) = 1 but Jac(∅, ∅) = 0, whereas our main convention aligns the two empty-set values. Under the aligned convention, we derive a convention-matched uniform transfer and combine it with the quadratic-dimensional F1 surrogate of Zhang et al. (2020). This is the setting considered here: the Jaccard loss is evaluated separately on each outcome–report pair and then averaged under a conditional distribution. It differs from population-confusion or expected-utility formulations of micro- and instance-averaged linear-fractional metrics, for which threshold and calibrated-utility methods are available (Koyejo et al., 2015; Bao and Sugiyama, 2020). Convex surrogates for IoU. The Lovász hinge and Lovász–Softmax losses provide tractable objectives motivated by submodular losses and IoU (Yu and Blaschko, 2015; Berman et al., 2018). These objectives have been influential in segmentation, but empirical usefulness does not imply calibration for the finite instance-wise loss studied here. Indeed, Finocchiaro et al. (2022) showed that the Lovász hinge is inconsistent for its intended structured target unless the underlying set function is modular. Dai and Li (2023) derived an IoU-calibrated ranking-based plug-in rule under conditional independence of the component labels. Their exact guarantee relies on conditional independence. Our exact lower bound allows arbitrary conditional dependence and concerns all convex calibrated surrogates, independently of smoothness or polyhedrality. Our MinHash construction gives a distribution-free approximate guarantee without an independence assumption. Prediction dimension. Convex calibration dimension was introduced by Ramaswamy and Agarwal (2012) and developed using trigger-set geometry by Ramaswamy and Agarwal (2016). For comparison, under the aligned emptyset convention, Zhang (2026) proved that the F1 score, shifted-loss, and ordinary loss matrices have rank s2 − s + 2, that the loss columns have affine dimension s2 − s + 1, and that CCdim(LF1 ) = Θ(s2 ). Thus exact F1 prediction 3

Exponential Convex Calibration Dimension for Jaccard

P REPRINT

dimension is quadratic, whereas the Jaccard bounds proved here are exponential. The present exact lower bound applies the feasible-subspace theorem of Ramaswamy and Agarwal (2016) to a Jaccard-specific family of tied Bayes reports. We are not aware of a previous exponential prediction-dimension lower bound for the instance-wise Jaccard loss. Convex calibration dimension is an exact notion and does not itself lower-bound the dimension needed for a fixed additive target-regret tolerance.

4

Exact rank and affine dimension

Let Y+ = Y \ {∅} and let K = SY+ ,Y+ be the score matrix restricted to nonempty sets. Lemma 4.1 (Strict positive definiteness). For every s ≥ 1, the matrix K is positive definite. Consequently, rank(K) = 2s − 1. The proof is given in Appendix A.1. It writes K as a MinHash Gram matrix and uses Boolean Möbius inversion to show that the Gram features have no nontrivial common null vector. Theorem 4.2 (Exact Jaccard ranks). For every s ≥ 1, under the convention in (1), rank(S) = rank(L − U ) = rank(L) = 2s . Moreover, affdim(L) = 2s − 1. Proof sketch. Ordering the empty set first gives S = diag(1, K), so Lemma 4.1 makes S and L − U = −S nonsingular. s Let e ∈ R2 −1 indicate the singleton sets. For every nonempty A, (Ke)A =

s X

Jac(A, {j}) = 1,

j=1

so Ke = 1N −1 . The block kernel equations for L then force every null vector to vanish, proving that L is nonsingular. Finally, the 2s score columns are linearly independent and hence have affine dimension 2s − 1; the invertible affine map x 7→ 1N − x transfers this dimension to the loss columns. The full proof is in Appendix A.2. Corollary 4.3 (Calibration-dimension upper bound). For every s ≥ 1, CCdim(LJac ) ≤ 2s − 1. This follows immediately from Theorem 4.2 and (4). Equivalently, one may estimate an arbitrary 2s -class conditional distribution in 2s − 1 coordinates and decode by minimizing the estimated Jaccard risk. Remark 4.4 (Alternative empty-set convention). If instead Jac(∅, ∅) = 0, then rank(S) = rank(L − U ) = 2s − 1,

rank(L) = 2s ,

affdim(L) = 2s − 1.

The verification is given in Appendix E.

5

An exponential calibration-dimension lower bound

The lower-bound witness uses a factorial balancing identity. For an integer n ≥ 0, write [n] = {1, . . . , n}, with [0] = ∅. Lemma 5.1 (Factorial balancing). For every C ⊆ [n], n   X X 1 + |C ∩ D| n 1 = . (6) (1 + |C ∪ D|) |D|! d (d + 1)! d=0

D⊆[n]

In particular, the left-hand side is independent of C. The proof is a finite binomial calculation and is given in Appendix B. Theorem 5.2 (Exponential CC-dimension bounds). For every s ≥ 1, 2s−1 ≤ CCdim(LJac ) ≤ 2s − 1. Consequently, CCdim(LJac ) = Θ(2s ). 4

(7)

Exponential Convex Calibration Dimension for Jaccard

P REPRINT

Proof sketch. Fix a core label 1 and put O = [s] \ {1}, so |O| = s − 1. Let U = {{1} ∪ D : D ⊆ O}. First define a full-support distribution q on U by assigning {1} ∪ D probability proportional to 1/|D|!. For a report {1} ∪ C, its expected score under q is the normalized left-hand side of (6); hence all 2s−1 reports in U tie. Every nonempty report missing label 1 is pointwise improved by adding it, because all outcomes in U contain label 1. Let κ > 0 be the common score of the reports in U. Mixing q with the empty outcome using weights 1/(1 + κ) and κ/(1 + κ), respectively, makes the empty report tie with every report in U. These are exactly the Bayes-optimal reports, and the support is A = {∅} ∪ U, |A| = 2s−1 + 1. The active score submatrix indexed by A is diag(1, SU ,U ) and is nonsingular by Lemma 4.1. Its columns therefore have affine dimension |A| − 1 = 2s−1 . At the witness distribution, their difference span is the entire hyperplane orthogonal to the positive probability vector. Adding the normalization constraint leaves no nonzero two-sided feasible direction, so µ = 0. Substitution into (5) gives the lower bound |A| − 1 = 2s−1 . Corollary 4.3 gives the upper bound. The detailed feasible-subspace argument is in Appendix C. Remark 5.3 (Size of the remaining gap). The bounds in (7) differ by less than a factor of two. Determining the exact convex calibration dimension, or improving either constant, remains open. The theorem rules out every polynomialdimensional convex surrogate that is exactly calibrated uniformly over all conditional label distributions. Section 6 shows that the qualifier “exactly” is essential. Remark 5.4 (Alternative empty-set convention). Under Jac(∅, ∅) = 0, the same factorial witness without the empty-outcome mixture yields 2s−1 − 1 ≤ CCdim(LJac ) ≤ 2s − 1. Thus the exponential conclusion is unchanged. See Appendix E.

6

Polynomial-dimensional approximate surrogates

The exponential lower bound in Theorem 5.2 concerns exact calibration. We now allow a nonzero, distribution-free target-regret floor, combine a prior F1 surrogate with a new regret transfer, and construct a MinHash surrogate of polynomial prediction dimension. We use both conditional and population regrets. For a score G : Y × Y → [0, 1], a distribution p ∈ ∆N , and a report B ∈ Y, define X X rG (p, B) = max pA G(A, C) − pA G(A, B). (8) C∈Y

A∈Y

A∈Y

For a surrogate Ψ : Y × Rd → R+ , define its conditional regret by X X rΨ (p, u) = pA Ψ(A, u) − inf pA Ψ(A, v). v∈Rd

A∈Y

(9)

A∈Y

For a distribution D on X × Y, let px be the conditional distribution of the outcome given X = x, and set   RegG (h) = EX rG (pX , h(X)) , RegΨ (f ) = E(X,A)∼D [Ψ(A, f (X))] − inf E(X,A)∼D [Ψ(A, g(X))], g

(10) (11)

where the infimum is over all measurable g : X → Rd . For α ≥ 0, we call a surrogate–link pair α-approximately consistent for Jaccard if, for every D and every sequence (fn ), RegΨ (fn ) −→ 0

=⇒

lim sup RegJac (pred ◦fn ) ≤ α.

(12)

n→∞

For both surrogates below, the infimum in (11) decomposes pointwise, and hence   RegΨ (f ) = EX rΨ (pX , f (X)) . 5

(13)

Exponential Convex Calibration Dimension for Jaccard

6.1

P REPRINT

An F1 -proxy surrogate

Define the instance-wise F1 score using the same empty-set convention as in (1):   2|A ∩ B| , |A| + |B| > 0, F (A, B) = |A| + |B|  1, A = B = ∅.

(14)

Pointwise,

t . 2−t √ Proposition 6.1 (F1 -to-Jaccard regret transfer). Let c⋆ = 3 − 2 2 and define H : [0, 1] → [0, 1] by √  c⋆ + r, 0 ≤ r ≤ 2 − 1, H(r) = √  2r , 2 − 1 ≤ r ≤ 1. 1+r Then, for every distribution D and every classifier h : X → Y,
RegJac (h) ≤ H RegF (h) ≤ c⋆ + RegF (h). Jac(A, B) = g(F (A, B)),

g(t) =

(15)

(16)

(17)

In particular, an F1 -Bayes classifier has Jaccard regret at most c⋆ ≈ 0.1716. The proof is given in Appendix D.1. Let (ΨF , predF ) denote the (s2 + 1)-dimensional convex surrogate and polynomial-time link for F1 of Zhang et al. (2020). Their calibration and regret-transfer results imply that, for every distribution D and every sequence (fn ), RegΨF (fn ) −→ 0

RegF (predF ◦fn ) −→ 0;

=⇒

see Zhang et al. (2020, Theorems 2 and 5). Proposition 6.1 therefore immediately gives lim sup RegJac (predF ◦fn ) ≤ c⋆ .

(18)

n→∞

Thus their surrogate–link pair is c⋆ -approximately consistent for Jaccard. We refer to Zhang et al. (2020) for the construction, decoder, and proof. Its quadratic order is necessary for exact F1 calibration (Zhang, 2026); after the transfer above, however, it provides only a constant-floor approximate guarantee for Jaccard. Proposition 6.1 is a convention-matched refinement of Waegeman et al. (2014, Theorem 3). For δ(p) = maxC Ep [F (A, C)], their coarse comparison gives an F1 -optimal report BF the bound rJac (p, BF ) ≤ 1 − δ(p)/2. Under the aligned empty-set convention, the exact identity Jac = g(F ) and Jensen’s inequality instead give, for any report B with F1 regret r, rJac (p, B) ≤ δ(p) − g(δ(p) − r) ≤ H(r). In particular, √ δ(p)(1 − δ(p)) rJac (p, BF ) ≤ ≤ 3 − 2 2, 2 − δ(p) whereas the earlier bound is at least 1/2. Thus the proposition both sharpens the exact-optimizer guarantee and covers approximate F1 optimization; concavity of H supplies the population transfer used above. Because F1 - and Jaccard-optimal reports need not agree, the resulting guarantee remains approximate rather than exact. 6.2

A MinHash random-feature surrogate

Write a permutation of [s] as π = (π(1), . . . , π(s)). For every nonempty A ⊆ [s], its MinHash value is the first element of A in this order: mπ (A) = π(kA ), kA = min{k ∈ [s] : π(k) ∈ A}. Introduce an additional symbol ⊥ and extend this map to all A ∈ Y by setting  mπ (A), A ̸= ∅, mπ (A) = ⊥, A = ∅, Under the convention Jac(∅, ∅) = 1, a uniformly random permutation then satisfies the MinHash identity for every A, B ∈ Y: (19) Jac(A, B) = Pr(mπ (A) = mπ (B)) . π

6

Exponential Convex Calibration Dimension for Jaccard

P REPRINT

Lemma 6.2 (Uniform finite-sample MinHash approximation). Let π1 , . . . , πM be independent uniformly random permutations of [s], and let {ej : j ∈ [s] ∪ {⊥}} be the standard basis of Rs+1 . Define
1 Φ(A) = √ emπ1 (A) , . . . , emπM (A) ∈ RM (s+1) M

(20)

M 1 X e S(A, B) = ⟨Φ(A), Φ(B)⟩ = 1{mπr (A) = mπr (B)} . M r=1

(21)

  e Pr max |S(A, B) − Jac(A, B)| > η ≤ 2 4s exp(−2M η 2 ).

(22)

and

For every η > 0, A,B∈Y

Consequently, for 0 < ρ < 1, the uniform error is at most η with probability at least 1 − ρ whenever M≥

2s log 2 + log(2/ρ) . 2η 2

(23)

The proof is given in Appendix D.2. Fix a realization of the feature map and define the convex square-loss surrogate and link by ΨM (A, u) = ∥u − Φ(A)∥22 , u ∈ RM (s+1) , predM (u) ∈ arg max⟨u, Φ(B)⟩, B∈Y

(24) (25)

using any fixed rule to break ties. Theorem 6.3 (MinHash approximate consistency). On the uniform-approximation event in Lemma 6.2, for every p ∈ ∆N and every u ∈ RM (s+1) , p rJac (p, predM (u)) ≤ 2η + 2rΨM (p, u). (26) The corresponding population guarantee is RegJac (predM ◦f ) ≤ 2η +

q

2 RegΨM (f ).

(27)

In particular, for every α > 0, taking η = α/2 makes the pair α-approximately consistent. With probability at least 1 − ρ, this is achieved in dimension &
' 2 2s log 2 + log(2/ρ) dMH = M (s + 1), M= , (28) α2 and hence

 2  s + s log(1/ρ) dMH = O . α2 The probabilistic construction also proves the existence of a fixed deterministic feature map with this guarantee. The proof is given in Appendix D.3. For the compressed construction, fix an integer d ≥ 1, draw independent uniform permutations π1 , . . . , πd , and, independently of the permutations and of one another, draw Rademacher variables ξr,j ∈ {−1, 1} for r ∈ [d] and j ∈ [s] ∪ {⊥}. Define
1 Φ± (A) = √ ξ1,mπ1 (A) , . . . , ξd,mπd (A) ∈ Rd , (29) d and set Ψ± (A, u) = ∥u − Φ± (A)∥22 , pred± (u) ∈ arg max⟨u, Φ± (B)⟩, B∈Y

using any fixed rule to break ties. 7

(30) (31)

Exponential Convex Calibration Dimension for Jaccard

P REPRINT

Corollary 6.4 (Rademacher-compressed approximate consistency). For every α > 0 and 0 < ρ < 1, take the preceding random surrogate–link pair with d = d± , where &
' 8 2s log 2 + log(2/ρ) d± = (32) α2 Then, with probability at least 1 − ρ over its feature construction, simultaneously for every distribution D and every measurable f , q (33) RegJac (pred± ◦f ) ≤ α + 2 RegΨ± (f ). Thus the pair is α-approximately consistent in dimension O((s + log(1/ρ))/α2 ). Fixing ρ = 1/2 and any successful realization proves the deterministic existence of such a pair in dimension O(s/α2 ). The proof is given in Appendix D.4. Remark 6.5 (Prediction dimension versus decoding). The MinHash construction has polynomial prediction dimension, but the exact link in (25) still maximizes over all 2s reports. No polynomial-time decoding claim is implicit in b satisfying Theorem 6.3. If a decoder returns B b ≥ max⟨u, Φ(B)⟩ − τ, ⟨u, Φ(B)⟩ B∈Y

the argument in Appendix D.3 adds only τ to the regret bounds. The same argument applies to the signed construction. b satisfying If a signed-feature decoder returns B b ≥ max⟨u, Φ± (B)⟩ − τ, ⟨u, Φ± (B)⟩ B∈Y

then, on the uniform-approximation event used in Corollary 6.4, q b ≤ 2η + τ + 2 rΨ (p, u). rJac (p, B) ± Consequently, with the choice η = α/2 in that corollary, the population bound for the corresponding approximate link d becomes pred ± q d ◦ f ) ≤ α + τ + 2 Reg (f ). Reg (pred Jac

±

Ψ±

Thus a fixed decoding error gives (α + τ )-approximate consistency; to retain an overall floor α, choose the kernelapproximation and decoding budgets so that 2η + τ ≤ α. In contrast, the F1 proxy of Zhang et al. (2020) has polynomial-time decoding but, after the transfer in Proposition 6.1, retains the nonzero worst-case floor c⋆ . Remark 6.6 (Alternative empty-set convention). If Jac(∅, ∅) = 0, define a convention-matched F1 score with F (∅, ∅) = 0. Then (15) and Proposition 6.1 remain valid; any surrogate calibrated for this convention-matched score inherits the same transfer. For the MinHash construction, set Φ(∅) = 0 and use the ordinary nonempty-set features in (20). All feature norms are then at most one, and the concentration and regret bounds remain unchanged; the direct feature dimension becomes M s. The signed construction likewise remains valid after setting Φ± (∅) = 0. For every fixed α > 0, the MinHash constructions have polynomial dimension. There is no contradiction with Theorem 5.2: that theorem concerns the zero-floor case α = 0, whereas the dimensions above grow as the requested approximation floor tends to zero.

7

Conclusion

The instance-wise multi-label Jaccard loss has maximal matrix rank and affine dimension. A finite MinHash Gram representation and Boolean Möbius inversion establish the rank results, while a factorially weighted family with a trivial two-sided feasible subspace yields 2s−1 ≤ CCdim(LJac ) ≤ 2s − 1. Thus exact calibration requires Θ(2s ) prediction coordinates, although the precise dimension remains open within a factor of two. Approximation exposes an exactness–dimension tradeoff. Our F1 transfer,√combined with the polynomial-time decoder of Zhang et al. (2020), gives asymptotic Jaccard regret at most 3 − 2 2. For any α > 0, MinHash square-loss surrogates attain regret floor α in polynomial dimension; a signed variant achieves O((s + log(1/ρ))/α2 ) dimensions with probability at least 1 − ρ. These prediction-dimension results do not resolve inference complexity: the MinHash link may still search over 2s reports. Natural next steps are efficient links with arbitrarily small floors, matching approximate-dimension lower bounds, and closing the factor-of-two gap in CCdim(LJac ). 8

Exponential Convex Calibration Dimension for Jaccard

A

Proofs for the exact-rank results

A.1

Proof of Lemma 4.1

P REPRINT

Proof of Lemma 4.1. For a permutation π of [s] and a nonempty A ⊆ [s], let mπ (A) be the first element of A under π. The MinHash identity (Broder, 1997) states that, for nonempty A, B,
Jac(A, B) = Pr mπ (A) = mπ (B) . (34) π

Indeed, the first element of A ∪ B under a uniformly random permutation is uniform on A ∪ B, and the two minima agree exactly when that element lies in A ∩ B. For every permutation π and j ∈ [s], define the column vector
hπ,j := 1{mπ (A) = j} A∈Y ∈ RY+ ∼ = RN −1 . +

Thus hπ,j h⊤ π,j is an (N − 1) × (N − 1) matrix indexed by pairs of nonempty sets. For every A, B ∈ Y+ ,  

s 1 XX

s!



s

1 XX 1{mπ (A) = j}1{mπ (B) = j} s! π j=1 A,B
= Pr mπ (A) = mπ (B) = Jac(A, B) = KA,B ,

 hπ,j h⊤ π,j

π j=1

=

π

where the sum over j is the indicator that the two MinHashes agree. Consequently, Equation (34) gives the Gram representation s 1 XX hπ,j h⊤ K= π,j , s! π j=1 Indeed, for every x ∈ RY+ , s

x⊤ Kx =

s

1 XX ⊤ 1 X X ⊤
2 x hπ,j h⊤ h x ≥ 0, π,j x = s! π j=1 s! π j=1 π,j

so K ⪰ 0. If x⊤ Kx = 0, this finite sum of nonnegative squares is zero, and therefore h⊤ π,j x = 0 for every π and j. By the definition of hπ,j , X X h⊤ 1{mπ (A) = j}xA = xA . π,j x = A∈Y+

A:mπ (A)=j

Consequently, X

xA = 0

for every π and j.

(35)

A:mπ (A)=j

Fix j and any T ⊆ [s] \ {j}. Choose a permutation in which every element of [s] \ (T ∪ {j}) precedes j and every element of T follows j. The sets whose first element is j are exactly {j} ∪ R with R ⊆ T . Hence (35) becomes X x{j}∪R = 0 for every T ⊆ [s] \ {j}. R⊆T

Boolean Möbius inversion gives x{j}∪T = 0 for every T . Every nonempty set contains some j, so x = 0. Therefore K ≻ 0. A.2

Proof of Theorem 4.2

Proof of Theorem 4.2. Order the empty set first. Since its score with a nonempty set is zero,   1 0 S= . 0 K Lemma 4.1 makes S nonsingular, so rank(S) = rank(L − U ) = N . 9

Exponential Convex Calibration Dimension for Jaccard

P REPRINT

Let e ∈ RN −1 be the indicator of the singleton subsets. For every nonempty A, (Ke)A =

s X

Jac(A, {j}) =

j=1

X 1 = 1. |A|

j∈A

Thus Ke = 1N −1 and 1⊤ N −1 e = s. The loss has block form

 L=

0

1N −1

 1⊤ N −1 . 1N −1 1⊤ N −1 − K

(36)

Suppose L(α, x)⊤ = 0. The first block equation gives 1⊤ N −1 x = 0. The second then reduces to Kx = α1N −1 . Since K is invertible and Ke = 1N −1 , x = αe. Therefore 0 = 1⊤ N −1 x = αs, so α = 0 and x = 0. Hence rank(L) = N . The N score columns are linearly independent, and therefore affinely independent, so affdim(S) = N − 1. Applying the invertible affine map z 7→ 1N − z to each score column gives the corresponding loss column. Thus affdim(L) = N − 1.

B

Proof of the factorial balancing identity

Proof of Lemma 5.1. Put c = |C|. Every D ⊆ [n] has the unique decomposition D = I ⊔ E,

I = D ∩ C ⊆ C,

E = D \ C ⊆ [n] \ C.

Let i = |I| and j = |E|. For this decomposition, |C ∩ D| = i, |C ∪ D| = |C| + |E| = c + j, |D| = i + j.

c n−c For fixed i and j, there are i choices for I and j choices for E. Consequently, the left-hand side of (6) becomes   n−c c  XX n−c c i+1 . (c + j + 1)(i + j)! j i j=0 i=0 Factoring the terms that do not depend on i gives n−c c   X n − c X 1 c i+1 . j c + j + 1 i=0 i (i + j)! j=0 We claim that

c   c   X X 1 c i+1 c 1 = . c + j + 1 i=0 i (i + j)! r=0 r (j + r + 1)!

(37)

(38)

To verify the claim, multiply its right-hand side by c + j + 1 and use c + j + 1 = (j + r + 1) + (c − r) in each summand. This gives c   X c 1 (c + j + 1) r (j + r + 1)! r=0   c c   X c j+r+1 X c c−r = + (j + r + 1)! (j + r + 1)! r r r=0 r=0 c   c−1   X X c 1 c c−r = + . r (j + r)! r=0 r (j + r + 1)! r=0 In the second sum, the r = c term is zero. Using (c − r)

    c c = (r + 1) r r+1

and then setting i = r + 1, we obtain c−1   X c r=0

  c X c−r c 1 = i . i (j + i)! r (j + r + 1)! i=1 10

Exponential Convex Calibration Dimension for Jaccard

P REPRINT

Renaming r as i in the first sum and combining the two sums therefore yields   c   c c   X X X c 1 c 1 c i+1 + = . i i (j + i)! i=1 i (j + i)! i (j + i)! i=0 i=0 This is the numerator on the left-hand side of (38); dividing by c + j + 1 proves the claim. Substituting (38) into (37) yields n−c c  XX j=0 r=0

n−c j

  c 1 . r (j + r + 1)!

Set d = j + r. Since 0 ≤ j ≤ n − c and 0 ≤ r ≤ c, the new index runs from 0 to n. For fixed d, the denominator is (d + 1)!, while the sum of the binomial coefficients over all admissible pairs (j, r) is      X c n−c n = r d−r d 0≤r≤c 0≤d−r≤n−c

by Vandermonde’s identity. Hence the preceding double sum equals    X n n   X X n 1 1 c n−c , = (d + 1)! d (d + 1)! r d−r 0≤r≤c 0≤d−r≤n−c

d=0

d=0

which is the right-hand side of (6).

C

Proof of the exponential lower bound

Proof of Theorem 5.2. The upper bound is Corollary 4.3. We prove the lower bound. Step 1: the factorial distribution.

Fix the core label 1, let O = [s] \ {1} and n = s − 1, and set U = {{1} ∪ D : D ⊆ O}.

Define Zn =

n   X n 1 d=0

d d!

,

Hn =

n   X n d=0

1 , d (d + 1)!

κ=

Hn > 0. Zn

Let q be supported on U with q{1}∪D =

1 , Zn |D|!

D ⊆ O.

(39)

Every coordinate in this support is positive. For C ⊆ O, let A ∼ q and write A = {1} ∪ D. Lemma 5.1 gives EA∼q [Jac(A, {1} ∪ C)] =

1 X 1 + |C ∩ D| Zn (1 + |C ∪ D|)|D|! D⊆O

Hn = = κ. Zn Thus all reports in U tie under q. If B ̸= ∅ and 1 ∈ / B, then for every outcome A ∈ U, Jac(A, B ∪ {1}) − Jac(A, B) =

1 > 0. |A ∪ B|

(40)

Hence B is strictly suboptimal under q. The empty report has score zero, whereas κ > 0. Therefore optL (q) = U . 11

(41)

Exponential Convex Calibration Dimension for Jaccard

Step 2: tie the empty report.

P REPRINT

Define

κ 1 δ∅ + q. (42) 1+κ 1+κ The empty report has expected score κ/(1 + κ). Every report in U has the same score because its score on the empty outcome is zero and its score under q is κ. Equations (41) and (40) show that all remaining reports have strictly lower score. Thus optL (p) = A := {∅} ∪ U , supp(p) = A, |A| = 2s−1 + 1. (43) p=

Step 3: active-column dimension.

The score submatrix with rows and columns in A is   1 0 SA,A = . 0 SU ,U

The lower-right block is a principal submatrix of the positive-definite matrix in Lemma 4.1; hence SA,A is nonsingular. Its |A| columns are linearly independent and therefore have affine dimension |A| − 1. Fix B0 = ∅ and define the active loss-difference space E = span {LA,B − LA,B0 : B ∈ A} ⊆ RA . Loss differences are the negatives of score differences, so dim E = |A| − 1 = 2s−1 .

(44)

All reports in A tie at p, so p⊤ A z = 0 for every z ∈ E. Since pA is nonzero and (44) has codimension one, E = p⊥ A.

(45)

Step 4: the two-sided feasible subspace. At p, the reports in A are exactly tied and every other report has a strict risk gap. It follows directly from the simplex constraints and the active trigger inequalities that  linQLB (p) = v ∈ RN : vAc = 0, 1⊤ vA ⊥ E . (46) A vA = 0, 0

Indeed, two-sided feasibility forces zero motion on the zero-probability coordinates, preserves total mass, and turns every active comparison into an equality. Conversely, any vector satisfying these conditions remains in the simplex and preserves the active equalities for sufficiently small motion in either direction; the strict inactive gaps remain strict. By (45), the last condition in (46) gives vA = apA for some scalar a. Normalization gives 0 = 1⊤ A vA = a, because ⊤ 1A pA = 1. Thus the lineality space is {0} and µQLB (p) = 0. 0

Applying (5) and (43) gives CCdim(LJac ) ≥ |A| − 0 − 1 = 2s−1 .

D

Proofs for the polynomial-dimensional approximation results

D.1

Proof of Proposition 6.1

Proof of Proposition 6.1. Fix p ∈ ∆N and B ∈ Y. Let A be a Y-valued random variable with Pr(A = D) = pD , and write X Ep [φ(A)] := pD φ(D) D∈Y

for any function φ : Y → R. Define the optimal conditional F1 score and the conditional F1 regret of B by δ = max Ep [F (A, C)], C∈Y

r = rF (p, B),

respectively. By the definition of rF in (8), Ep [F (A, B)] = δ − r. Moreover, 0 ≤ r ≤ δ ≤ 1, since F takes values in [0, 1]. Because g(t) ≤ t on [0, 1], max Ep [Jac(A, C)] ≤ max Ep [F (A, C)] = δ. C∈Y

C∈Y

12

Exponential Convex Calibration Dimension for Jaccard

P REPRINT

The function g is convex. Jensen’s inequality and (15) therefore give
Ep [Jac(A, B)] = Ep [g(F (A, B))] ≥ g Ep [F (A, B)] = g(δ − r). Consequently, rJac (p, B) ≤ δ − g(δ − r).

(47)

Set z = δ − r = Ep [F (A, B)]. Then z ≥ 0, and z + r = δ ≤ 1, so z ∈ [0, 1 − r]. The right-hand side of (47) becomes r + z − g(z) = r +

z(1 − z) . 2−z

a(z) := z − g(z) =

z(1 − z) . 2−z

Let

√ The function a increases on [0, z⋆ ] and decreases on [z⋆ , 1], where z⋆ = 2 − 2 and a(z⋆ ) = c⋆ . Since the actual value of z lies in [0, 1 − r], we have r + a(z) ≤ r + max a(u) = H(r), 0≤u≤1−r

with H as defined in (16): if r ≤ occurs at u = 1 − r and equals

√

2 − 1, the interval contains z⋆ and the maximum is r + c⋆ ; otherwise the maximum r + a(1 − r) =

2r . 1+r

It follows that
rJac (p, B) ≤ H rF (p, B) . It remains to pass from conditional to population regret. Let (X, A) have distribution D, let pX denote the conditional law of A given X, and let h be the classifier in the proposition. Define the random variable RX := rF (pX , h(X)) ∈ [0, 1]. √ The two pieces of H agree in both value and first derivative at r = 2 − 1, and the second piece is increasing and concave. Hence H is increasing and concave on [0, 1]. Applying the conditional inequality with p = pX and B = h(X), then using Jensen’s inequality and the population-regret definition (10), gives   RegJac (h) = EX rJac (pX , h(X)) ≤ EX [H(RX )] ≤ H(EX [RX ]) = H(RegF (h)). Finally, a(z) ≤ c⋆ for every z ∈ [0, 1], so the representation H(r) = r + max0≤u≤1−r a(u) gives H(r) ≤ r + c⋆ . This proves the second inequality in (17). In particular, if h is F1 -Bayes, then RegF (h) = 0 and the first inequality gives RegJac (h) ≤ H(0) = c⋆ . D.2

Proof of Lemma 6.2

Proof of Lemma 6.2. For fixed A, B, the summands in (21) are independent Bernoulli random variables with mean Jac(A, B) by (19). Hoeffding’s inequality gives   e Pr |S(A, B) − Jac(A, B)| > η ≤ 2 exp(−2M η 2 ). There are |Y|2 = 4s ordered outcome–report pairs. A union bound proves (22), and solving its right-hand side for M gives (23). 13

Exponential Convex Calibration Dimension for Jaccard

D.3

P REPRINT

Proof of Theorem 6.3

Proof of Theorem 6.3. Fix a realization for which the uniform-approximation event in Lemma 6.2 holds. For p ∈ ∆N , set X X µp = pA Φ(A), gp (B) = pA Jac(A, B), gep (B) = ⟨µp , Φ(B)⟩. A∈Y

A∈Y

e By the definitions of µp and S, gep (B) =

X

pA ⟨Φ(A), Φ(B)⟩ =

A∈Y

X

e pA S(A, B).

(48)

A∈Y

On the uniform-approximation event, |gp (B) − gep (B)| =

X

≤

X

e pA Jac(A, B) − S(A, B)

A∈Y

e pA | Jac(A, B) − S(A, B)|

(49)

A∈Y

≤η

X

pA = η

A∈Y

b = predM (u). Adding and subtracting the two approximate for every p and B. Let B ⋆ ∈ arg maxB gp (B) and B scores gives b = gp (B ⋆ ) − gp (B) b rJac (p, B)


b + gep (B) b − gp (B) b = gp (B ⋆ ) − gep (B ⋆ ) + gep (B ⋆ ) − gep (B) b ≤ 2η + gep (B ⋆ ) − gep (B) b = 2η + ⟨µp , Φ(B ⋆ ) − Φ(B)⟩ b + ⟨u, Φ(B ⋆ ) − Φ(B)⟩ b = 2η + ⟨µp − u, Φ(B ⋆ ) − Φ(B)⟩ b 2. ≤ 2η + ∥µp − u∥2 ∥Φ(B ⋆ ) − Φ(B)∥ Here

b ≤0 ⟨u, Φ(B ⋆ ) − Φ(B)⟩

b maximizes B 7→ ⟨u, Φ(B)⟩; the final line then follows from Cauchy–Schwarz. Moreover, each of the M because B blocks in Φ(B) is a scaled standard basis vector, so M

∥Φ(B)∥22 =

1 X ∥emπr (B) ∥22 = 1. M r=1

All feature coordinates are nonnegative, and hence ⟨Φ(B), Φ(B ′ )⟩ ≥ 0. It follows that ∥Φ(B) − Φ(B ′ )∥22 = ∥Φ(B)∥22 + ∥Φ(B ′ )∥22 − 2⟨Φ(B), Φ(B ′ )⟩ = 2 − 2⟨Φ(B), Φ(B ′ )⟩ ≤ 2. Combining the last two displays yields b ≤ 2η + rJac (p, B)

√

2 ∥u − µp ∥2 .

(50)

It remains to identify the distance on the right with square-loss regret. For every A, u − Φ(A) = (u − µp ) + (µp − Φ(A)). After squaring and averaging, the cross term vanishes because X X pA (µp − Φ(A)) = µp − pA Φ(A) = 0. A

A

Therefore the square-loss bias–variance identity is X X pA ∥u − Φ(A)∥22 = ∥u − µp ∥22 + pA ∥Φ(A) − µp ∥22 . A

A

14

(51)

Exponential Convex Calibration Dimension for Jaccard

P REPRINT

The second term is independent of u, so the conditional surrogate risk is minimized at u = µp , and rΨM (p, u) = ∥u − µp ∥22 . Substitution into (50) proves (26). Now apply the conditional bound at each x with p = px and u = f (x). Using (10), then Cauchy–Schwarz (equivalently, Jensen’s inequality for the square root), gives RegJac (predM ◦f ) = EX [rJac (pX , predM (f (X)))] hp i √ ≤ 2η + 2 EX rΨM (pX , f (X)) p ≤ 2η + 2 EX [rΨM (pX , f (X))] q = 2η + 2 RegΨM (f ), where the last equality uses (13). This proves (27). Finally, set η = α/2. If RegΨM (fn ) → 0, the population bound gives lim sup RegJac (predM ◦fn ) ≤ 2η = α, n→∞

which is approximate consistency. With this value of η, (23) becomes
2 2s log 2 + log(2/ρ) . α2 Taking the ceiling gives (28); multiplying by the block size s + 1 gives the stated order for dMH . Since the uniformapproximation event has positive probability, at least one fixed realization of the feature map satisfies all these bounds. M≥

D.4

Proof of Corollary 6.4

Proof of Corollary 6.4. Take d = d± in (29). For every A, B, E⟨Φ± (A), Φ± (B)⟩ = Jac(A, B): conditional on πr , the sign product has expectation one when the two hashes agree and zero otherwise. The summands lie in [−1, 1], so Hoeffding’s inequality and the same union bound give uniform error η with probability at least 1 − ρ provided
2 2s log 2 + log(2/ρ) d± ≥ . η2 The proof of Theorem 6.3 applies to (30)–(31), with ∥Φ± (B) − Φ± (B ′ )∥2 ≤ 2. Taking η = α/2 gives (33), while (32) is exactly the preceding sample-size condition with this choice of η.

E

The alternative empty-set convention

Suppose Jac(∅, ∅) = 0. Ordering the empty set first, the score matrix becomes diag(0, K). Lemma 4.1 therefore gives rank(S) = rank(L − U ) = N − 1. The empty score column is zero and the other N − 1 columns are independent, so affdim(S) = N − 1 and hence affdim(L) = N − 1. The loss block is now

 L=

⊤

1

1N −1

 1⊤ N −1 . 1N −1 1⊤ N −1 − K

= 0, the first equation is α + 1⊤ N −1 x = 0. Substitution into the second gives Kx = 0, so x = 0 and then

If L(α, x) α = 0. Thus rank(L) = N .

For the calibration lower bound, use the factorial distribution q from (39) without mixing in the empty outcome. Its support and its exact Bayes-optimal report set are both U, of size 2s−1 . The restricted active score matrix SU ,U is positive definite. Repeating Steps 3–4 of the proof of Theorem 5.2 gives zero lineality and hence the lower bound 2s−1 − 1. The affine-dimension upper bound remains 2s − 1.

15

Exponential Convex Calibration Dimension for Jaccard

P REPRINT

References Han Bao and Masashi Sugiyama. Calibrated surrogate maximization of linear-fractional utility in binary classification. In Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, volume 108 of PMLR, pages 2337–2347, 2020. Maxim Berman, Amal Rannen Triki, and Matthew B. Blaschko. The Lovász–Softmax loss: A tractable surrogate for the optimization of the intersection-over-union measure in neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4413–4421, 2018. Mathieu Bouchard, Anne-Laure Jousselme, and Pierre-Emmanuel Doré. A proof for the positive definiteness of the Jaccard index matrix. International Journal of Approximate Reasoning, 54(5):615–626, 2013. Andrei Z. Broder. On the resemblance and containment of documents. In Compression and Complexity of Sequences 1997, pages 21–29. IEEE, 1997. Flavio Chierichetti, Ravi Kumar, Sandeep Pandey, and Sergei Vassilvitskii. Finding the Jaccard median. In Proceedings of the Twenty-First Annual ACM–SIAM Symposium on Discrete Algorithms, pages 293–311, 2010. Ben Dai and Chunlin Li. RankSEG: A consistent ranking-based framework for segmentation. Journal of Machine Learning Research, 24(224):1–50, 2023. Krzysztof Dembczyński, Willem Waegeman, Weiwei Cheng, and Eyke Hüllermeier. On label dependence and loss minimization in multi-label classification. Machine Learning, 88(1–2):5–45, 2012. Jessica J. Finocchiaro, Rafael Frongillo, and Enrique B. Nueve. The structured abstain problem and the Lovász hinge. In Proceedings of the 35th Conference on Learning Theory, volume 178 of PMLR, pages 3718–3740, 2022. John C. Gower. A general coefficient of similarity and some of its properties. Biometrics, 27(4):857–871, 1971. Oluwasanmi O. Koyejo, Nagarajan Natarajan, Pradeep K. Ravikumar, and Inderjit S. Dhillon. Consistent multilabel classification. In Advances in Neural Information Processing Systems 28, pages 3321–3329, 2015. Alex Nowak, Francis Bach, and Alessandro Rudi. Sharp analysis of learning with discrete losses. In Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, volume 89 of PMLR, pages 1920–1929, 2019. Harish G. Ramaswamy and Shivani Agarwal. Classification calibration dimension for general multiclass losses. In Advances in Neural Information Processing Systems 25, pages 2087–2095, 2012. Harish G. Ramaswamy and Shivani Agarwal. Convex calibration dimension for multiclass loss matrices. Journal of Machine Learning Research, 17(14):1–45, 2016. Willem Waegeman, Krzysztof Dembczyński, Arkadiusz Jachnik, Weiwei Cheng, and Eyke Hüllermeier. On the Bayes-optimality of F-measure maximizers. Journal of Machine Learning Research, 15(103):3513–3568, 2014. Jiaqian Yu and Matthew Blaschko. Learning submodular losses with the Lovász hinge. In Proceedings of the 32nd International Conference on Machine Learning, volume 37 of PMLR, pages 1623–1631, 2015. Mingyuan Zhang, Harish G. Ramaswamy, and Shivani Agarwal. Convex calibrated surrogates for the multi-label F-measure. In Proceedings of the 37th International Conference on Machine Learning, volume 119 of PMLR, pages 11246–11255, 2020. Mingyuan Zhang. Exact rank and convex calibration dimension lower bounds for the multi-label F1 loss. arXiv preprint arXiv:2608.08399, 2026.

16

Related documents

• On dissonance and orthogonal projections of self-conformal measures#1035345
• On sliced Cramér metrics#635623
• On sliced Cramér metrics#633274
• The Teichmüller-Randers metric#147524
• Inductive Volume Measure: A Geometric Alternative to Hausdorff Measure#131354