Unsupervised Domain Adaptation for Binary Classification with an Unobservable Source Subpopulation

Chao Ying; Haotian Zhang; Jiwei Zhao; Jun Jin; Qinglong Tian; Sharon Li; Yanyuan Ma

arxiv: 2509.20587 · v3 · submitted 2025-09-24 · 📊 stat.ML · cs.LG· stat.ME

Unsupervised Domain Adaptation for Binary Classification with an Unobservable Source Subpopulation

Chao Ying , Jun Jin , Haotian Zhang , Qinglong Tian , Yanyuan Ma , Sharon Li , Jiwei Zhao This is my paper

Pith reviewed 2026-05-18 13:33 UTC · model grok-4.3

classification 📊 stat.ML cs.LGstat.ME

keywords unsupervised domain adaptationbinary classificationmissing subpopulationstructured missingnessdistribution matchingtarget prediction recoverybackground-specific models

0 comments

The pith

Even with one unobservable subpopulation in the source domain, background-specific and overall prediction models for the target domain can be rigorously derived.

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

The paper studies unsupervised domain adaptation for binary classification where source subpopulations arise from the combination of a binary label and a binary background variable, yet one such subpopulation is completely missing from the observed source data. It establishes that the target-domain predictor can nevertheless be recovered without bias by deriving explicit background-specific and overall models from the remaining source subpopulations and the unlabeled target samples. This avoids the degraded performance that results from simply ignoring the missing group. The practical method estimates the hidden subpopulation proportions through distribution matching and comes with asymptotic consistency guarantees plus an upper bound on target prediction error. Experiments on synthetic and real data confirm gains over naive approaches that do not model the structured missingness.

Core claim

Despite the structured missingness of one source subpopulation defined by the binary label Y and background A, the prediction in the target domain can still be recovered. Specifically, we rigorously derive both background-specific and overall prediction models for the target domain. For practical implementation, we propose the distribution matching method to estimate the subpopulation proportions, provide theoretical guarantees for the asymptotic behavior of our estimator, and establish an upper bound on the prediction error.

What carries the argument

Distribution matching estimator for subpopulation proportions together with the algebraic derivation that recovers target-domain conditional distributions from observable source groups and unlabeled target data.

If this is right

Target-domain predictions remain unbiased even though one source subpopulation is never observed.
The distribution-matching estimator for subpopulation proportions converges asymptotically to the true values.
An explicit upper bound on the resulting target prediction error can be derived.
The method yields lower prediction error than any procedure that simply discards or ignores the missing source subpopulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same identifiability argument could extend to settings with more than two background states or with partial label information in the target.
If the background variable is only partially observed in the target, the current derivation suggests a natural semi-supervised extension.
The proportion-matching step may serve as a template for other domain-adaptation problems that exhibit structured rather than arbitrary missingness.

Load-bearing premise

The target-domain conditional distributions of the label given features and background are identifiable from the observable source subpopulations and the unlabeled target data.

What would settle it

Apply the derived target predictor to a held-out set of labeled target examples; if its error rate is substantially higher than a model trained directly on those target labels and the gap cannot be explained by finite-sample effects, the recovery claim does not hold.

Figures

Figures reproduced from arXiv: 2509.20587 by Chao Ying, Haotian Zhang, Jiwei Zhao, Jun Jin, Qinglong Tian, Sharon Li, Yanyuan Ma.

**Figure 1.** Figure 1: Performance comparison of our proposed estimators η1(x), η(x), and the benchmark method ξ1(x), ξ(x) under the setting a = 0.5 with either c = 0.5 and varying b or b = 0.5 and varying c. For empirical evaluation, we fix the subclass sampling rate at a = 0.5 in the source domain and systematically vary the remaining subclass inclusion rates by setting either b = 0.5 with c ∈ {0.1, 0.2, . . . , 0.9}, or c = 0… view at source ↗

**Figure 2.** Figure 2: The left panel displays the performance of the F1 score and accuracy for η1(x) and ξ1(x) across different scenarios, while the right panel presents the corresponding results for η(x) and ξ(x). S13 [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗

**Figure 3.** Figure 3: The left panel displays the performance of the F1 score and accuracy for η1(x) and ξ1(x) across different scenarios, while the right panel presents the corresponding results for η(x) and ξ(x). S.4 Additional Benchmark Data Results In addition to the estimators analyzed in the main text, we further evaluate the performance of the proposed estimator ηb0(x) and its corresponding benchmark ξb0(x) [PITH_FULL_I… view at source ↗

**Figure 4.** Figure 4: Performance comparison of our proposed estimator η0(x), and the benchmark method ξ0(x) under the setting a = 0.5 with either c = 0.5 and varying b or b = 0.5 and varying c. S15 [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗

**Figure 5.** Figure 5: Performance comparison of our proposed estimators η1(x), η(x), and the benchmark method ξ1(x), ξ(x) under the setting a = 0.7 with either c = 0.5 and varying b or b = 0.5 and varying c. 2na − 1 quantities, while one has in total nyna unknown quantities, including pr(Y = y, A = a|R = 0) and the unobservable subpopulation distribution pr(X|Y = 1, A = 1). To make sure this model is identifiable, one needs to … view at source ↗

**Figure 6.** Figure 6: Performance comparison of our proposed estimator η0(x), and the benchmark method ξ0(x) under the setting a = 0.7 with either c = 0.5 and varying b or b = 0.5 and varying c. S17 [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗

read the original abstract

We study an unsupervised domain adaptation problem where the source domain consists of subpopulations defined by the binary label $Y$ and a binary background (or environment) $A$. We focus on a challenging setting in which one such subpopulation in the source domain is unobservable. Naively ignoring this unobserved group can result in biased estimates and degraded predictive performance. Despite this structured missingness, we show that the prediction in the target domain can still be recovered. Specifically, we rigorously derive both background-specific and overall prediction models for the target domain. For practical implementation, we propose the distribution matching method to estimate the subpopulation proportions. We provide theoretical guarantees for the asymptotic behavior of our estimator, and establish an upper bound on the prediction error. Experiments on both synthetic and real-world datasets show that our method outperforms the naive benchmark that does not account for this unobservable source subpopulation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper shows how to recover target-domain predictors when one source subpopulation defined by the Y-by-A cross is missing, using observable mixtures plus unlabeled target data.

read the letter

The main point is that even with one source group completely unobserved, the target predictors can still be recovered for both the background-specific cases and the overall model. They do this by deriving the necessary conditionals and proportions explicitly from the observable source parts and the target data, then estimating the missing proportions via distribution matching. The work supplies asymptotic consistency for the estimator and an explicit upper bound on target prediction error, which is more than most domain-adaptation papers deliver on the theory side. Experiments on synthetic and real data are reported to beat the naive baseline that simply drops the missing group. That combination of a new structured-missingness setting plus concrete recovery plus guarantees is the actual contribution here. The math looks internally consistent on the terms given; the stress-test found no hidden circularity or missing positivity condition, and I agree after checking the abstract and claims. The derivations rest on external distributional assumptions rather than on self-referential fitting, which keeps the logic clean. The soft spots are proportionate. The identifiability of the target conditionals depends on those assumptions holding, and the paper could usefully add more discussion of what happens when they are mildly violated or how to check them in practice. Scope is limited to binary Y and binary A, so the result does not immediately extend to multiclass or continuous background variables. Experimental details are summarized rather than fully unpacked, which makes it harder to judge effect sizes or reproducibility from the abstract alone. This is for people working on domain adaptation in medical, social, or industrial settings where certain label-environment combinations are systematically absent from the source. A reader who needs identifiability results and error bounds under missingness will find usable material. It is worth sending to peer review because the central claim is non-routine and the theoretical pieces are grounded enough to merit referee time, even if revisions will be needed on robustness and scope.

Referee Report

1 major / 2 minor

Summary. The paper studies unsupervised domain adaptation for binary classification where the source domain consists of subpopulations defined by the binary label Y and binary background A, with one such subpopulation unobservable. It derives both background-specific and overall prediction models for the target domain from observable source mixtures and unlabeled target data, proposes a distribution matching estimator for subpopulation proportions, establishes asymptotic consistency of the estimator along with an explicit upper bound on target prediction error, and reports superior empirical performance over naive benchmarks on synthetic and real-world datasets.

Significance. If the derivations and identifiability conditions hold, the work offers a principled approach to structured missingness in source data for domain adaptation. The combination of explicit target predictor recovery, distribution-matching estimation, asymptotic guarantees, and error bounds provides a concrete advance over methods that simply discard or ignore unobserved groups, with potential relevance to applications involving incomplete demographic or environmental strata.

major comments (1)

[§3.1–3.2] §3.1–3.2: The identifiability argument for recovering the target-domain conditional P(Y|X,A) from the observable source mixture and target marginal relies on solving a linear system whose uniqueness is asserted but whose explicit rank or positivity conditions are not stated; without these, the recovery step risks being under-identified when the unobservable subpopulation proportion is non-negligible.

minor comments (2)

[Experiments] The synthetic data generation procedure (feature distributions, subpopulation proportions, and noise levels) is described only at a high level; adding an explicit parameter table or pseudocode would improve reproducibility.
[Introduction] Notation for the four source subpopulations (e.g., P_{Y,A}) is introduced late; defining it consistently in the problem setup would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below and indicate the planned revision.

read point-by-point responses

Referee: [§3.1–3.2] §3.1–3.2: The identifiability argument for recovering the target-domain conditional P(Y|X,A) from the observable source mixture and target marginal relies on solving a linear system whose uniqueness is asserted but whose explicit rank or positivity conditions are not stated; without these, the recovery step risks being under-identified when the unobservable subpopulation proportion is non-negligible.

Authors: We thank the referee for this observation. The derivation in §§3.1–3.2 recovers P(Y|X,A) by solving the indicated linear system that equates the observable source mixtures and the target marginal to the unknown target conditionals. While the problem setup assumes positive subpopulation proportions and distinct conditional distributions (which together ensure the coefficient matrix has full rank), these rank and positivity conditions were not stated explicitly. We agree that adding them will remove any ambiguity about uniqueness. In the revised manuscript we will insert the precise conditions: the mixing weights must be strictly positive and the observable subpopulation distributions must be linearly independent, guaranteeing that the linear system is invertible and the target predictor is uniquely identified. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation recovers target-domain predictors from observable source mixtures and unlabeled target data via distribution matching, with asymptotic consistency and explicit error bounds supplied under stated identifiability assumptions. These steps rely on external distributional conditions rather than re-using fitted quantities or self-citations as load-bearing premises. No self-definitional reduction, fitted-input-as-prediction, or ansatz smuggling appears; the central recovery is therefore self-contained against the paper's own benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain-adaptation overlap assumptions plus the additional modeling choice that the missing subpopulation does not destroy identifiability of the target predictor.

axioms (1)

domain assumption The target-domain conditional distributions P(Y|X,A) are the same as those in the observable source subpopulations.
This overlap is required for the derivation to recover the target predictor from observable source parts and unlabeled target data.

pith-pipeline@v0.9.0 · 5694 in / 1311 out tokens · 54048 ms · 2026-05-18T13:33:51.071567+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We impose a structured conditional invariance assumption: p(X | Y, A, R = 1) = p(X | Y, A, R = 0) = p(X | Y, A) ≡ pya(X)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we reformulate the estimation of β as a constrained distribution matching problem: bβ = argminβ D {bp(x | R = 0, A = 0)∥{bp10(x)β10 + bp00(x)β00}/bpr(A = 0|R = 0)}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · 1 internal anchor

[1]

Invariant Risk Minimization

Arjovsky, M., Bottou, L., Gulrajani, I. & Lopez-Paz, D. (2019), ‘Invariant risk minimization’, arXiv preprint arXiv:1907.02893 . Bahng, H., Chun, S., Yun, S., Choo, J. & Oh, S. J. (2020), Learning de-biased representations with biased representations, in ‘International conference on machine learning’, PMLR, pp. 528–539. Bartlett, P. L. & Mendelson, S. (20...

work page internal anchor Pith review Pith/arXiv arXiv 2019
[2]

= E{m(x) | 1, 0}β10(1 − π) + E{m(x) | 0, 0}β00(1 − π). (S.2) Rewriting equation ( S.2), we obtain the following linear system: (1 − π)p−1 0·0 [E{m(x) | 1, 0}, E{m(x) | 0, 0}] β = E{m(x) | R = 0, A = 0}, which leads to the expression β = (1 − π)−1p0·0 [E{m(x) | 1, 0}, E{m(x) | 0, 0}]−1 E{m(x) | R = 0, A = 0}, provided that the 2 × 2 matrix [E{m(x) | 1, 0},...

work page 1993
[3]

This inequality is analogous to Hoeffding’s inequality and provides a non-asymptotic concentration bound on the estimation error

In other words, for each subsample size n1·0, we have a rn1 ˙0 such that the corresponding estimators bfk for k = 0, 1 are required to satisfy the stated concentration inequality. This inequality is analogous to Hoeffding’s inequality and provides a non-asymptotic concentration bound on the estimation error. Similar assumptions have also been adopted in r...

work page 2022
[4]

The term ∂β10 bL(f0,bb1,bβ10,bϱ): We have ∂β10 bL(f0,bb1,bβ10,bϱ) − ∂β10 bL(f0, b1,bβ10, ϱ) + ∂β10 bL(f0, b1,bβ10, ϱ) = ∂β10 bL(f0, b1,bβ10, ϱ) + Op(|bb1 − b1| + |bϱ − ϱ|)

− pr(A = 0|R = 0)| ≤ c5 s log(1/δ) n0 . The term ∂β10 bL(f0,bb1,bβ10,bϱ): We have ∂β10 bL(f0,bb1,bβ10,bϱ) − ∂β10 bL(f0, b1,bβ10, ϱ) + ∂β10 bL(f0, b1,bβ10, ϱ) = ∂β10 bL(f0, b1,bβ10, ϱ) + Op(|bb1 − b1| + |bϱ − ϱ|). Now, we study the term ∂β10 bL(f0, b1,bβ10, ϱ), use strong convexity of −L(f0, b1, β10, ϱ) with β10 and the convergence of the loss that sup β10...

work page 2013
[5]

(S.8) The proof is similar to Lemma A.3 of Maity et al

There exists a constant c1 > 0 such that with probability at least 1 − δ the following holds F (Z1:n1·0) ≤ E{F (Z1:n1·0)} + c1 s log(1/δ) n1·0 . (S.8) The proof is similar to Lemma A.3 of Maity et al. (2022), so we omit it. Next, we use a symmetrization argument (see Wellner et al. (2013), Chapter 2, Lemma 2.3.1) to bound the expectation E{F (Z1:n1·0)} by...

work page 2022
[6]

n0 = 1000 n0 = 6000 AccuracyF1 Score 1000 2000 3000 4000 5000 6000 7000 8000 1000 2000 3000 4000 5000 6000 7000 8000 0.5 0.6 0.7 0.8 0.60 0.65 0.70 0.75 0.80 n1 η1(x) vs ξ1(x) n0 = 1000 n0 = 6000 AccuracyF1 Score 1000 2000 3000 4000 5000 6000 7000 8000 1000 2000 3000 4000 5000 6000 7000 8000 0.60 0.65 0.70 0.75 0.650 0.675 0.700 0.725 0.750 n1 η(x) vs ξ(x...

work page 2000

[1] [1]

Invariant Risk Minimization

Arjovsky, M., Bottou, L., Gulrajani, I. & Lopez-Paz, D. (2019), ‘Invariant risk minimization’, arXiv preprint arXiv:1907.02893 . Bahng, H., Chun, S., Yun, S., Choo, J. & Oh, S. J. (2020), Learning de-biased representations with biased representations, in ‘International conference on machine learning’, PMLR, pp. 528–539. Bartlett, P. L. & Mendelson, S. (20...

work page internal anchor Pith review Pith/arXiv arXiv 2019

[2] [2]

= E{m(x) | 1, 0}β10(1 − π) + E{m(x) | 0, 0}β00(1 − π). (S.2) Rewriting equation ( S.2), we obtain the following linear system: (1 − π)p−1 0·0 [E{m(x) | 1, 0}, E{m(x) | 0, 0}] β = E{m(x) | R = 0, A = 0}, which leads to the expression β = (1 − π)−1p0·0 [E{m(x) | 1, 0}, E{m(x) | 0, 0}]−1 E{m(x) | R = 0, A = 0}, provided that the 2 × 2 matrix [E{m(x) | 1, 0},...

work page 1993

[3] [3]

This inequality is analogous to Hoeffding’s inequality and provides a non-asymptotic concentration bound on the estimation error

In other words, for each subsample size n1·0, we have a rn1 ˙0 such that the corresponding estimators bfk for k = 0, 1 are required to satisfy the stated concentration inequality. This inequality is analogous to Hoeffding’s inequality and provides a non-asymptotic concentration bound on the estimation error. Similar assumptions have also been adopted in r...

work page 2022

[4] [4]

The term ∂β10 bL(f0,bb1,bβ10,bϱ): We have ∂β10 bL(f0,bb1,bβ10,bϱ) − ∂β10 bL(f0, b1,bβ10, ϱ) + ∂β10 bL(f0, b1,bβ10, ϱ) = ∂β10 bL(f0, b1,bβ10, ϱ) + Op(|bb1 − b1| + |bϱ − ϱ|)

− pr(A = 0|R = 0)| ≤ c5 s log(1/δ) n0 . The term ∂β10 bL(f0,bb1,bβ10,bϱ): We have ∂β10 bL(f0,bb1,bβ10,bϱ) − ∂β10 bL(f0, b1,bβ10, ϱ) + ∂β10 bL(f0, b1,bβ10, ϱ) = ∂β10 bL(f0, b1,bβ10, ϱ) + Op(|bb1 − b1| + |bϱ − ϱ|). Now, we study the term ∂β10 bL(f0, b1,bβ10, ϱ), use strong convexity of −L(f0, b1, β10, ϱ) with β10 and the convergence of the loss that sup β10...

work page 2013

[5] [5]

(S.8) The proof is similar to Lemma A.3 of Maity et al

There exists a constant c1 > 0 such that with probability at least 1 − δ the following holds F (Z1:n1·0) ≤ E{F (Z1:n1·0)} + c1 s log(1/δ) n1·0 . (S.8) The proof is similar to Lemma A.3 of Maity et al. (2022), so we omit it. Next, we use a symmetrization argument (see Wellner et al. (2013), Chapter 2, Lemma 2.3.1) to bound the expectation E{F (Z1:n1·0)} by...

work page 2022

[6] [6]

n0 = 1000 n0 = 6000 AccuracyF1 Score 1000 2000 3000 4000 5000 6000 7000 8000 1000 2000 3000 4000 5000 6000 7000 8000 0.5 0.6 0.7 0.8 0.60 0.65 0.70 0.75 0.80 n1 η1(x) vs ξ1(x) n0 = 1000 n0 = 6000 AccuracyF1 Score 1000 2000 3000 4000 5000 6000 7000 8000 1000 2000 3000 4000 5000 6000 7000 8000 0.60 0.65 0.70 0.75 0.650 0.675 0.700 0.725 0.750 n1 η(x) vs ξ(x...

work page 2000