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REVIEW 3 major objections 4 minor 13 references

Fairness Through Matching

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Matching individuals across groups is enough to guarantee group fairness.

desk verdict MDP/FTM is a solid new idea with a correct core theorem, but Theorem 4.2's subset-fairness bound is false as stated and needs a fix. read the letter →

arxiv 2501.02793 v1 pith:PZNWCU3X submitted 2025-01-06 cs.AI cs.LGstat.ML

Kunwoong Kim , Insung Kong , Jongjin Lee , Minwoo Chae , Sangchul Park , Yongdai Kim This is my paper · ORCID
classification cs.AIcs.LGstat.ML
keywords algorithmicfairnessgroupdemographicparitymatchedoptimaltransportmapsubsetfairclassification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper attempts to establish that group fairness can be reframed as a matching problem: every group-fair predictor secretly pairs individuals from different protected groups, and conversely, imposing that a model gives similar scores to matched pairs is enough to make it group-fair. The authors introduce Matched Demographic Parity (MDP), the average absolute difference between predictions for two individuals connected by a transport map, and prove that any transport map yields a sufficient MDP constraint: if MDP is at most $\delta$ then both Wasserstein demographic parity and demographic parity are at most $\delta$. They then build the Fairness Through Matching (FTM) algorithm around that constraint, with two recommended transport maps—the optimal transport map on inputs, which empirically improves fairness on subsets, and the joint optimal transport map on inputs and labels, which improves accuracy and equalized odds. If the claims hold, fairness becomes a matter of choosing who is matched with whom, not merely of balancing group-level statistics.

What carries the argument

The key object is the transport map between the input spaces of the two protected groups: a deterministic or stochastic pairing that moves the distribution of one group to the other. The measure that carries the argument is MDP, defined as $\Delta_{\mathrm{MDP}}(f,T_s)=E_s|f(X,s)-f(T_s(X),s')|$, the average prediction gap over matched pairs. The load-bearing theorem is Theorem 3.7, which converts a small MDP value into bounded Wasserstein and demographic parity via the Lipschitz duality of the Wasserstein-1 distance; this is what licenses training under an MDP constraint with any transport map. The two proposed OT-based maps—the marginal OT map minimizing input-space transport cost and the joint OT map on $X\times Y$—are what give the algorithm its tunable behavior, steering the resulting fair model toward subset fairness or toward accuracy and equalized odds.

What would settle it

Compute the true subset demographic parity gap for a simple example—say $X$ uniform on $[0,1]$, subset $A=[0,0.5]$, a 1-Lipschitz model $f(x)=x$, and the transport map equal to the fair matching function—and compare it with the right-hand side of inequality (3); if the missing normalization factor $1/P_0(A)=2$ changes the comparison, the stated bound is violated.

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Extended reading notes

Core claim

The central claim is a duality between group-fair models and transport maps on the input space. For a perfectly fair model, there exists a transport map $T_s$ between the conditional input distributions $P_0$ and $P_1$ such that the model gives identical predictions to $x$ and $T_s(x)$ almost everywhere; for a model whose total-variation demographic parity is at most $\delta$, the corresponding transport map keeps MDP within $2\delta$. The converse is the paper's main theorem: for any transport map $T_s$, whenever $\Delta_{\mathrm{MDP}}(f,T_s) \leq \delta$, both the Wasserstein demographic parity and the demographic parity gap of $f$ are at most $\delta$. This 'matching implies fairness' direction is the theoretical engine behind FTM, which trains a classifier to minimize loss subject to an MDP constraint built from a user-selected transport map. The paper further proposes two concrete transport maps from optimal transport theory and reports experiments in which FTM models show higher subset fairness, higher accuracy, and better equalized odds than existing group-fair baselines, depending on the chosen map.

Load-bearing premise

The subset-fairness bound in Theorem 4.2 treats the conditional expectation of the model over a subset as if it were the unnormalized integral over that subset, which only coincides when the subset has probability one.

Editorial extensions

If this is right

  • If Theorem 3.7 holds, any practitioner can enforce demographic parity by pairing individuals and penalizing score differences within pairs, with the pairings chosen to reflect the application's priorities.
  • The marginal OT map should produce models with fewer subset-fairness outliers than standard group-fair baselines, as observed on Adult, German, Dutch, and Bank datasets.
  • Tuning the joint OT map's label-weight $\alpha$ trades accuracy against transport cost; at fairness levels where an accurate fair model exists, larger $\alpha$ improves accuracy and equalized odds.
  • The transport cost of the fair matching function can serve as a model-selection diagnostic: among models with similar fairness and accuracy, the one with lower transport cost is preferable because it avoids extreme subset targeting and self-fulfilling prophecies.
  • Under a linear Gaussian structural causal model, the marginal OT map coincides with the counterfactual input, so FTM with this map also implements a form of counterfactual fairness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One could plausibly extend the same matching template to other fairness notions by conditioning the MDP constraint on the label, yielding an equalized-odds analogue of FTM; the paper notes this direction for equal opportunity but does not develop it.
  • A natural testable extension is to learn the transport map jointly with the model instead of fixing it beforehand, so that the map tracks the model's evolving level sets; the paper fixes the map in advance, and alternating optimization would probe whether the sufficiency theorem still holds with coupled maps.
  • The matching view suggests an audit recipe that goes beyond the paper's experiments: compute the fair matching function of any deployed fair model and inspect the matched pairs' distance to surface latent subset biases that group-level metrics miss.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces Matched Demographic Parity (MDP), a group fairness measure that uses a transport map between the input spaces of two protected groups to match individuals and then averages the absolute prediction differences over the matched pairs. It proves (Theorem 3.7) that any transport map yields a sufficient condition for demographic parity, and it develops the FTM algorithm, which trains a model under an MDP constraint with a user-chosen transport map. Two transport maps are proposed: the marginal optimal transport map on X, claimed to improve subset fairness (Theorem 4.2), and the joint optimal transport map on X x Y, claimed to improve prediction accuracy and equalized odds. Experiments on Adult, German, Dutch, and Bank datasets compare FTM with Reduction, Reg, Adv, and fair representation learning baselines.

Significance. If all claims were valid, this would be a valuable bridge between optimal transport and algorithmic fairness, offering a flexible family of in-processing constraints. The core implication of Theorem 3.7 is correctly proven via an IPM/Wasserstein argument and is a genuine sufficient condition for demographic parity; this is the main theoretical strength of the paper. The paper also provides public source code and falsifiable empirical comparisons. However, the theoretical justification for the advertised subset-fairness benefit of the marginal OT map is invalid: Theorem 4.2 is false as stated, and even a corrected version would not directly apply to the transport map actually used in FTM. The equalized-odds theory for the joint OT map also has a proof gap. The empirical findings may still be useful, but the paper's stated theoretical guarantees for these two transport maps require substantial correction before the claims can be accepted as stated.

major comments (3)
  1. [Theorem 4.2 / Appendix A, Eqs. (17)–(20)] The proof of Theorem 4.2 contains normalization errors that invalidate the stated bound. In the first term of Eq. (17), the conditional expectation E(f(X,0)|S=1,X in A) is bounded by L E1||X-T1(X)||, but the correct bound after conditioning is at most L (E1||X-T1(X)||^2)^{1/2} / sqrt(P1(A)), which is unbounded for rare subsets A. In the third term, Eq. (19) replaces the conditional expectations with unnormalized integrals, dropping the factors 1/P0(A) and 1/P1(A). The claimed bound fails even when the fair-matching cost is zero: let P0=P1=Unif[0,1], set f(x,0)=psi((x-(0.5-eps))/eps) and f(x,1)=psi((x-0.5)/eps) with psi the triangular bump; then delta=0, L(E||X-T^f(X)||^2)^{1/2} = O(sqrt(eps)) -> 0, TV=0, but for A={x: f(x,0) >= 1/2} of measure eps/2, DeltaDP_A = 3/4, contradicting Eq. (3).
  2. [Section 4.1 / Definition 3.4 / Eq. (2)] Even if Theorem 4.2 were corrected to include the proper normalizers, it would not support FTM's use of the marginal OT map. The theorem bounds subset unfairness in terms of the fair matching function T^f_s (Definition 3.4), while the FTM objective (Eq. (2) and loss (7)) imposes DeltaMDP(f,T_s) <= delta for a user-chosen transport map T_s. Although DeltaMDP(f,T^f_s) <= DeltaMDP(f,T_s) holds by definition, the transport-cost term E||X - T^f_s(X)||^2 in Eq. (3) is not controlled by the cost of T_s. The manuscript provides no transfer result between T_s and T^f_s, so the statement in Section 4.1 that using the marginal OT map improves subset fairness is not theoretically justified.
  3. [Section 4.2] In the theoretical discussion of the joint OT map, the assertion that for alpha->infinity one has P(Y=y|X=x,S=s)=P(Y=y|X=T_s(x),S=s') for almost all x is not a consequence of T_s being an optimal transport map between the joint distributions. A transport map that pushes the joint distribution forward to the target joint distribution does not generally preserve the conditional label distributions pointwise; the subsequent inequality bounding DeltaTPR and DeltaFPR in terms of DeltaMDP therefore lacks a valid proof. The empirical equalized-odds improvements may still stand, but the theoretical support stated in the text is not established.
minor comments (4)
  1. [Abstract] The phrase 'an user-specified transport map' should be 'a user-specified transport map'.
  2. [Section 2.1] The definition s' = 1-s is stated twice in the notation paragraph; one occurrence can be removed.
  3. [Table 1] The checkmark symbol next to FTM in Table 1 and subsequent tables is not defined in the caption; please explain what it indicates.
  4. [Section 5.3.2] The text refers to the 'right panel of Figure 6' and then to the 'left panel'; since Figure 6 contains two panels, please label them or refer to them by their positions consistently.

Circularity Check

0 steps flagged · score 2.0 of 10

Central derivation is self-contained: MDP is a genuine sufficient condition for DP/WDP by Theorem 3.7, not a definitional tautology. Self-citations are only baseline references, not load-bearing. Theorem 4.2 has a proof gap, but that is a correctness issue, not circularity.

full rationale

The paper's main theoretical chain is self-contained. Theorem 3.7 proves that if ΔMDP(f,Ts)≤δ for any transport map Ts, then ΔWDP(f)≤δ and ΔDP(f)≤δ. This is a real implication: the proof uses the IPM representation of the Wasserstein-1 distance and the push-forward property Ts#Ps=Ps', so the conclusion is not merely a restatement of the MDP definition. Proposition 3.1 and Theorem 3.3 construct a transport map from a fair model via quantile/Brenier arguments; the map is derived, not assumed. FTM trains under an MDP constraint with a user-chosen marginal or joint OT map; the paper does not fit a parameter and then rename it as a prediction. The hyperparameters λ and α are user-chosen and do not appear in the theoretical implication. Self-citations (Kim et al. 2022a,b) appear only as baselines and related-work references; no uniqueness theorem or modeling ansatz is imported from the authors' prior work, so there is no load-bearing self-citation. The apparent flaw in Theorem 4.2's proof (dropping conditional normalizers in Appendix A) is a correctness/rigor concern, not circularity: the stated bound is not equivalent to its assumptions by construction. Moreover, since Tf_s is defined as the minimizer of ΔMDP over transport maps, any f satisfying ΔMDP(f,Ts)≤δ also satisfies the theorem's premise, so the transfer from the user-chosen map to the fair matching function is not a circular step. Overall, the paper's central claims are independent mathematical statements with proofs, and the only circularity-adjacent issue is the presence of minor self-citations, which are not load-bearing.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No fitted parameters enter the theoretical claims. The algorithm has two user-chosen hyperparameters (λ and α). The theory rests on standard optimal transport regularity conditions plus a Lipschitz assumption. No new physical or statistical entities are postulated.

free parameters (2)
  • alpha (joint OT map weight) = 100
    User-specified weight on the label distance in the X×Y cost. Set to 100 in experiments so the label term fully dominates the input distance.
  • lambda (Lagrange multiplier) = Swept over {0.1, 0.2, ..., 10.0}
    Trades off prediction loss against the MDP constraint. A list of values is used to produce different fairness levels, standard for in-processing fairness methods.
assumptions (4)
  • domain assumption Condition (C): P_s are absolutely continuous with respect to the Lebesgue measure.
    Used to guarantee the existence of deterministic transport maps in Theorems 3.1 and 3.3. The paper provides a discrete analogue via stochastic maps in Proposition A.1.
  • ad hoc to paper The cost function |f(x,s)-f(y,s')| is lower semi-continuous and bounded below, ensuring existence and uniqueness of the fair matching function via Brenier's theorem.
    Boundedness holds since f∈[0,1], but lower semi-continuity is asserted without proof and requires continuity of f, which is not assumed. This is invoked in Definition 3.4.
  • domain assumption In Theorem 4.2, the hypothesis class F consists of L-Lipschitz functions.
    Needed to relate transport cost to subset fairness. MLPs with bounded weights are Lipschitz, but the empirical models are not verified to satisfy this.
  • domain assumption The linear Gaussian structural causal model in equation (4) for Proposition 4.3.
    Used to show the marginal OT map matches counterfactual inputs. This is an illustrative example, not a general claim.

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Cite this review

Pith. "Pith review of Fairness Through Matching." pith.science (2026). https://pith.science/paper/PZNWCU3X

@misc{pith2026250102793,
  author       = {Pith},
  title        = {Pith review of: Fairness Through Matching},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PZNWCU3X}},
  note         = {Machine review of arXiv:2501.02793}
}
read the original abstract

Group fairness requires that different protected groups, characterized by a given sensitive attribute, receive equal outcomes overall. Typically, the level of group fairness is measured by the statistical gap between predictions from different protected groups. In this study, we reveal an implicit property of existing group fairness measures, which provides an insight into how the group-fair models behave. Then, we develop a new group-fair constraint based on this implicit property to learn group-fair models. To do so, we first introduce a notable theoretical observation: every group-fair model has an implicitly corresponding transport map between the input spaces of each protected group. Based on this observation, we introduce a new group fairness measure termed Matched Demographic Parity (MDP), which quantifies the averaged gap between predictions of two individuals (from different protected groups) matched by a given transport map. Then, we prove that any transport map can be used in MDP to learn group-fair models, and develop a novel algorithm called Fairness Through Matching (FTM), which learns a group-fair model using MDP constraint with an user-specified transport map. We specifically propose two favorable types of transport maps for MDP, based on the optimal transport theory, and discuss their advantages. Experiments reveal that FTM successfully trains group-fair models with certain desirable properties by choosing the transport map accordingly.

Figures

Figures reproduced from arXiv: 2501.02793 by the authors.

Figure 1
Figure 1. Simplified illustration of MDP. Once two individuals [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. An example DAG of the SCM in equation (4). Let xs be a realization of Xs and assume (Id − A) has its inverse B ∶= (Id − A) −1 , where Id ∈ R d×d is the identity matrix. Then, its counterfactual becomes x˜ CF s = Bµs ′ + σs ′σ −1 s Id(xs − Bµs) (proved with Proposition 4.3 together in Section A of Appendix). On the other hand, by Lemma A.3 in Section A of Appendix, the image of xs by the marginal OT map is given as x… view at source ↗
Figure 3
Figure 3. Fairness-prediction trade-offs: ∆WDP vs. Acc on (Left to right) Adult, German, Dutch, Bank datasets. Similar results are observed for ∆DP and ∆DP in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Fairness on random subsets: Boxplots of the levels of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Fairness-prediction trade-offs: ∆DP vs. Acc on (Left to right) Adult, German, Dutch, Bank datasets. FTM (joint) = FTM with the joint OT map. FTM (marginal) = FTM with the marginal OT map. In addition, we compare FTM using the marginal OT map, FTM using the joint OT map…
Figure 6
Figure 6. Figure 6: Transport cost-prediction trade-offs: Trans [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: (Top) A group-fair model with the risk of discrim [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]
Figure 8
Figure 8. Figure 8: shows the trade-offs between the fairness levels with respect to ∆DP, ∆DP and classification accuracy [PITH_FULL_IMAGE:figures/full_fig_p034_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 11 canonical work pages

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    We partition the interval[0,z] into subintervals of length at mostδ

    (Constructing subsets based on Fcont s ) Since Fcont 1 (1) ≤ Fcont 0 (1), there exists z ≤ 1 such that Fcont 1 (1)=Fcont 0 (z). We partition the interval[0,z] into subintervals of length at mostδ. Let v0 = 0 and definevk ∶= min{vk−1+δ,z} for k = 1,...,m. Here, m ∈ N is the number that satisfiesvm =z. Note thatFcont 1 (1)=∑m k=1Fcont 1 ((vk−1,vk]). For eac...

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    By Brenier’s Theorem (Villani, 2008; Hütter & Rigollet, 2021), there exists a transport mapT(1) 0,k from P0,k(⋅) to P1,k(⋅), under (C)

    (Defining probability measures and transport maps on subsets) For eachk ∈ {1,...,m }, define probability measures Ps,k,s ∈ {0, 1} such that Ps,k(A)∶= Ps(A∩Xs,k) Ps(Xs,k) for measurable subsetsA⊆ X. By Brenier’s Theorem (Villani, 2008; Hütter & Rigollet, 2021), there exists a transport mapT(1) 0,k from P0,k(⋅) to P1,k(⋅), under (C). Sincevk−vk−1 ≤δ,∀k, we ...

    work page 2008

  3. [3]

    Fix d ∈D0,1

    (Handling discontinuity points) LetD0,1∶=D0∩D1 be the intersection ofD0 and D1, which is the set of common discontinuity points. Fix d ∈D0,1. Suppose that P1(f−1 1 ({d})) ≤ P0(f−1 0 ({d})). Then, there existsf−1 0 ({d})′ ⊂f−1 0 ({d}) such that P0(f−1 0 ({d})′) = P1(f−1 1 ({d})). Define ˜X0,d ∶= f−1 0 ({d})′ and ˜X1,d ∶= f−1 1 ({d}). We can define ˜X0,d an...

    work page 2024

  4. [4]

    Because P0(⋃m k=1 X0,k)= P1(⋃m k=1 X1,k) and P0(⋃d∈D0,1 ˜X0,d)= P1(⋃d∈D0,1 ˜X1,d), we haveP0(X′ 0)= 1− P0(⋃k∈{1,...,m} X0,k)− P0(⋃d∈D0,1 ˜X0,d)= P1(X′ 1)≤δ

    (Constructing the complement parts) We collect the complements as X′ 0∶= X0∖⎛ ⎝ m ⋃ k=1 X0,k∪ ⋃ d∈D0,1 ˜X0,d ⎞ ⎠ and X′ 1∶= X1∖⎛ ⎝ m ⋃ k=1 X1,k∪ ⋃ d∈D0,1 ˜X1,d ⎞ ⎠. Because P0(⋃m k=1 X0,k)= P1(⋃m k=1 X1,k) and P0(⋃d∈D0,1 ˜X0,d)= P1(⋃d∈D0,1 ˜X1,d), we haveP0(X′ 0)= 1− P0(⋃k∈{1,...,m} X0,k)− P0(⋃d∈D0,1 ˜X0,d)= P1(X′ 1)≤δ. Define probability measuresP′ s,s ∈...

    work page

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    (11) We note that{{X0,k}m k=1,{ ˜X0,d}d∈D0,1, X′ 0} and {{X1,k}m k=1,{ ˜X1,d}d∈D0,1, X′ 1} are partitions ofX0 and X1, respectively

    (Overall transport map) Finally, combining 2 to 4 above, we define the (overall) transport mapT0 as T0(⋅)∶= m ∑ k=1 T(1) 0,k(⋅)1 (⋅∈ X0,k)+ ∑ d∈D0,1 T(2) 0,d(⋅)1 (⋅∈ ˜X0,d)+ T(3) 0 (⋅)1 (⋅∈ X′ 0). (11) We note that{{X0,k}m k=1,{ ˜X0,d}d∈D0,1, X′ 0} and {{X1,k}m k=1,{ ˜X1,d}d∈D0,1, X′ 1} are partitions ofX0 and X1, respectively. Moreover,P0(X0,k)= P1(X1,k)...

    work page

  6. [6]

    (Calculation of the bound for∆MDP(f, T0)) Using the constructed transport mapT0, we have that ∆MDP(f, T0)= E0∣f(X, 0)−f(T0(X), 1)∣=∫ ∣f(X, 0)−f(T0(X), 1)∣dP0(X) = m ∑ k=1 ∫X0,k ∣f(x, 0)−f(T(1) 0,k(x), 1)∣dP0(x) + ∑ d∈D0,1 ∫ ˜X0,d ∣f(X, 0)−f(T(2) 0,d(x), 1)∣⋅dP0(x) +∫X ′ 0 ∣f(x, 0)−f(T(3) 0 (x), 1)∣dP0(x) ≤ (∗) δ m ∑ k=1 P0(X0,k)+∫X ′ 0 ∣f(x, 0)−f(T(3) 0 (...

    work page 2024

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    Furthermore, sincef(⋅)∈ [0, 1], we have that E0(∣f(X, 0)−f(T(3) 0 (X), 1)∣⋅ 1 (X∈ X′ 0))=∫ ∣f(X, 0)−f(T(3) 0 (X), 1)∣⋅ 1 (X∈ X′ 0)dP0(X) ≤∫ 1 (X∈ X′ 0)dP0(X)= P0(X′ 0)≤δ

    = 1− (∑m k=1 P0(X0,k)+∑d∈D0,1 P0( ˜X0,d)) ≤ δ, since ∆TVDP(f) = TV(Pf(X,0)∣S=0, Pf(X,1)∣S=1)≤δ. Furthermore, sincef(⋅)∈ [0, 1], we have that E0(∣f(X, 0)−f(T(3) 0 (X), 1)∣⋅ 1 (X∈ X′ 0))=∫ ∣f(X, 0)−f(T(3) 0 (X), 1)∣⋅ 1 (X∈ X′ 0)dP0(X) ≤∫ 1 (X∈ X′ 0)dP0(X)= P0(X′ 0)≤δ. (10)

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    Build a diagonal matrixΓ11 = 1 n1 Im ∈ Rm×m + , where Im is the identity matrix of sizem×m

    work page

Show all 13 references
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    Build a zero matrixΓ21 = 0(n0−m)×m ∈ R(n0−m)×m + , where 0 denotes the zero matrix

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    Build matricesΓ12 ∈ Rm×(n1−m) + and matrixΓ22 ∈ R(n0−m)×(n1−m) + satisfying Γ121n1−m = ( 1 n0 − 1 n1 ) 1m, Γ221n1−m = 1 n0 1n0−m, and (Γ12 Γ22 ) ⊺ 1n0 = 1 n1 1n1−m

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    Last, note that theΓ is a coupling matrix (i.e., satisfying the constraints∑n0 i=1γi,j = 1/n0,∀j ∈ [n1] and ∑n1 j=1γi,j = 1/n1,∀i∈ [n0])

    Complete Γ∶= (Γ11 Γ12 Γ21 Γ22 )= ( 1 n1 Im Γ12 0(n0−m)×m Γ22 ). Last, note that theΓ is a coupling matrix (i.e., satisfying the constraints∑n0 i=1γi,j = 1/n0,∀j ∈ [n1] and ∑n1 j=1γi,j = 1/n1,∀i∈ [n0]). Hence, we can define the joint distributionQ by the constructedΓ (i.e., Γ i...

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    PMLR. URL http://proceedings.mlr.press/v108/wei20a.html. Yongkai Wu, Lu Zhang, and Xintao Wu. On convexity and bounds of fairness-aware classification. InThe World Wide Web Conference, WWW ’19, pp. 3356–3362, New York, NY, USA, 2019. Association for Computing Machinery. ISBN 9...

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Reviewed August 10, 2026 · model on record in the stance chip above.