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Identifiability in Unlinked Linear Regression: Some Results and Open Problems

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that unlinked linear regression is strongly identifiable for a Gamma-plus-nonzero-mean-normal covariate vector, and shows that outside such parametric cases general identifiability fails.

desk verdict Solid identifiability note with several new results, but it overclaims impossibility in the abstract and contains a false general claim in the convolutions section that needs fixing. read the letter →

arxiv 2507.14986 v1 pith:BVHULBW5 submitted 2025-07-20 math.ST stat.TH

Fadoua Balabdaoui , Martin Slawski , Jonathan Steffani This is my paper · ORCID
classification math.STstat.TH MSC 62E1062J05
keywords unlinkedlinearregressionidentifiabilitydataintegrationindependentcomponentanalysischaracterizationproblemsGamma-normalmixturesfourthmomentsmodel
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

Unlinked linear regression asks whether the coefficient vector $\beta_0$ can be recovered when the response $Y$ and the covariates $X$ are observed in separate files, with no record-level link, and $Y$ is only assumed equal in distribution to $\beta_0^\top X + \epsilon$. The paper establishes that in the independent-but-not-identically-distributed case, no general identifiability theorem is possible: spherical symmetry makes every vector on a sphere a valid coefficient, and convolution relations among covariates create shifts. It then proves positive results: strong identifiability ($B_0 = \{\beta_0\}$) when one covariate is normal with nonzero mean and the others are Gamma with shape parameters whose subset sums are unique, and for $d=2$ a fourth-moment condition that bounds $B_0$ to at most eight vectors or, often, to sign flips alone. These results tell practitioners when unlinked-data regression is trustworthy and when it is inherently underdetermined.

What carries the argument

The set $B_0$ defined above is the object that carries the argument; levels of identifiability are literally the cardinality of $B_0$ being infinite, finite and larger than one, or a singleton. For the Gamma-normal theorem the load-bearing identity is the moment-generating-function equality $\prod_i M_{X_i}(\beta_i t) = \prod_i M_{X_i}(\tilde{\beta}_i t)$, which after dividing out the normal factor becomes a product of terms $(1 - t/R_i^{\pm})^{-\alpha_i}$; matching the smallest pole $R_{(1)}^+$ and then successively higher poles forces the coefficient sets and shape-exponent sums to agree, and the subset-sum uniqueness assumption converts exponent-sum equality into index-set equality. In the i.i.d. review, the classical Marcinkiewicz and Linnik theorems on identically distributed linear forms play the same role, and the $d=2$ theorem reduces identifiability to a quadratic equation in $a^4/r^4$ derived from second and fourth moments.

What would settle it

A direct check for $d=3$ with $\alpha_1=1$, $\alpha_2=\sqrt{2}$, $\lambda_1=\lambda_2=1$, and $X_3 \sim N(1,1)$: compare the moment-generating functions of $\beta^\top X$ and $\tilde{\beta}^\top X$ over a fine grid of $t$; since the subset-sum condition holds, the theorem predicts $\beta=\tilde{\beta}$, so any distinct pair with matching moment-generating functions would refute Theorem 5.

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Extended reading notes

Core claim

On the paper's own terms, the central object is the set $B_0 = \{\beta \in \mathbb{R}^d : \beta^\top X + \epsilon \stackrel{d}{=} \beta_0^\top X + \epsilon\}$, and identifiability is its cardinality. The main positive theorem (Theorem 5) says: if $X_d \sim N(\mu,\sigma^2)$ with $\mu \neq 0$ and $X_i \sim \mathrm{Gamma}(\alpha_i,\lambda_i)$ for $i<d$, all independent, and if equal sums of the shape parameters over subsets of $\{1,\dots,d-1\}$ force the subsets to be equal, then $B_0 = \{\beta_0\}$, so the regression vector is strongly identifiable. The proof matches poles of the moment-generating functions at the ratios $\lambda_i/|\beta_i|$, using the subset-sum condition to conclude coefficient-by-coefficient equality. Equally central is the negative message that outside such parametric settings identifiability fails in structured ways: spherical or elliptical symmetry yields the whole sphere or an ellipsoid section as $B_0$, and a covariate that is a convolution of others shifts the apparent coefficient. For $d=2$, the fourth-moment theorem shows that unless both centered standardized covariates have fourth moment $3$, $B_0$ is finite and often consists only of sign flips.

Load-bearing premise

The load-bearing premise of the positive theorem is the subset-sum uniqueness condition on the Gamma shape parameters: matching pole orders in the moment-generating-function proof concludes $\beta_i = \tilde{\beta}_i$ only because equal sums of $\alpha$'s over subsets are assumed to force equal index sets, and if two different subsets sum to the same value the argument degrades to identification up to permutation.

Editorial extensions

If this is right

  • Under the Theorem 5 conditions, an unlinked-data regression estimate is uniquely anchored: any consistent estimator of $\beta_0$ is estimating the true coefficient, not an equivalent one.
  • For independent covariates with different distributions, unlinked regression is identifiable only when the design prevents symmetries: if $X$ is spherically or elliptically symmetric, every $\beta$ on the corresponding sphere or ellipse is indistinguishable, so no method can single out $\beta_0$.
  • If any covariate is a convolution of other covariates (for example, $X_3 \stackrel{d}{=} X_1 + X_2$ for Gammas), the coefficient vector is identified only up to shifts, so one must either exclude such redundancy or accept set-valued inference.
  • In the $d=2$ case, computing the fourth moments of the centered standardized covariates gives a practical criterion: if not both kurtoses equal $3$, the solution set is finite and usually just sign flips, so sign is the only ambiguity.
  • The multi-response variant with $m \ge 2$ response variables inherits a general weak identifiability result from overcomplete independent component analysis: with non-Gaussian independent sources and pairwise linearly independent columns, $B_0$ consists only of permutations and sign flips of the true matrix.

Reading between the lines

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

  • Because randomly drawn continuous positive shape parameters have no equal subset sums with probability one, Theorem 5 covers 'almost every' Gamma-plus-nonzero-normal design, so its restrictive-looking condition is generic rather than exceptional.
  • The $d=2$ fourth-moment criterion could be converted into a finite-sample pre-test: estimate $m_1$, $m_2$, and $c$ from the observed data, compute $w_1$ and $w_2$, and declare identifiability up to sign when $\min\{w_1,w_2\}<1$; the paper proves only the population version.
  • A natural next step is to test the paper's closing conjecture — weak identifiability for independent unit-variance components with at most one Gaussian and a minimal representation — by searching for counterexamples among discrete or mixture distributions, where the moment-generating-function and moment arguments used here do not directly apply.
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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 / 6 minor

Summary. The paper studies identifiability of the regression coefficient vector in unlinked linear regression, where Y is distributed as β⊤X + ε but the data on X and Y are not paired. It reviews classical results for i.i.d. components (Marcinkiewicz, Linnik, Ghurye-Olkin), then turns to the non-i.i.d. case. The paper gives counterexamples showing that general analogues of the i.i.d. results fail (spherical symmetry, convolution invariance), proves a positive strong-identifiability result for independent Gaussian-plus-Gamma covariates under a subset-sum uniqueness condition (Theorem 5), and derives a fourth-moment condition for two-dimensional covariates that bounds the identifiability set (Theorem 6). It also discusses connections to independent component analysis and states an informal conjecture.

Significance. If the positive results hold, the paper provides a useful map of identifiability phenomena in unlinked linear regression and extends earlier work of Azadkia and Balabdaoui. The fourth-moment result for d=2 is elegant and checkable from the first four moments of the covariates, and the review of the i.i.d. case is well organized. The ICA connection is illuminating. However, the paper's negative message is weakened by a false convolution-invariance claim with a concrete counterexample, and the abstract overstates the scope of the negative findings by claiming impossibility rather than presenting counterexamples.

major comments (3)
  1. [Abstract] The abstract states that when the covariate components have different distributions, 'we show that it is not possible to prove similar theorems in the general case.' This is a meta-mathematical impossibility claim, but the paper only provides counterexamples (spherical symmetry, convolutions) showing that certain natural generalizations fail. Counterexamples do not establish that no theorem of a similar kind can be proved. The abstract, and the parallel sentence in §1, should be rephrased to say that the paper demonstrates obstacles to direct generalization by exhibiting counterexamples, rather than claiming a proof of impossibility.
  2. [§4.3] The convolution-invariance claim is false as stated. If Xj is equal in distribution to Σ_{i∈I} α_i X_i, the paper asserts that B0 contains β0+δ with δj = -β0j and δi = αiβ0j for i∈I. This does not hold in general: after the substitution, the remainder R = Σ_{k≠j} β0k X_k is generally not independent of the replacement combination when R contains any Xi with i∈I. For a concrete counterexample, take X1,X2∼Exp(1) independent and X3∼Gamma(2,1) independent, so X3 d= X1+X2. Let β0=(1,0,1). The construction yields β=(2,1,0). But β0^T X = X1+X3 has moment-generating function (1−t)^{-3}, whereas β^T X = 2X1+X2 has MGF ((1−2t)(1−t))^{-1}; the t^2 coefficients are 6 and 7, respectively, so these distributions are not equal. The invariance is valid only when β0i=0 for all i∈I, so that the remainder is independent of both Xj and the combination. The specific illustrative example with β0=(0,0,1) works, but the general construction and the surrounding discussion must be corrected.
  3. [§4.5] The paragraph 'Extension to more than two random variables' asserts, without proof, that applying Theorem 6 to subsets yields an upper bound of 3!·2^3 = 48 solutions for d=3 and, recursively, d!·2^d for general d. This is not established: Theorem 6 concerns the identifiability set for a fixed two-dimensional response, and the reasoning does not show that a candidate solution in higher dimension must be controlled by the two-dimensional marginal conditions in a way that yields the claimed global bound. The moment equations couple all coefficients through the full projection β^T X. Either a rigorous proof should be supplied, or the paragraph should be explicitly labeled as a heuristic sketch or open problem.
minor comments (6)
  1. [Theorem 6 statement] In the definition of B0 within Theorem 6, 'aX1 + bX1' should read 'aX1 + bX2'; the subsequent proof uses the correct form.
  2. [Theorem 4] The theorem states that there exists ρ>0 with B0 = Sρ, but if the true regression vector β0 equals zero, the sphere has radius zero. The statement should allow ρ≥0 or explicitly exclude the degenerate case β0=0, which is consistent with the proof (ρ = ||β0||).
  3. [Lemma 1 proof] In the appendix proof of Lemma 1, the displayed equivalence is missing a factor of Φε on the right-hand side inside the statement; it should read Φ_{β⊤0X} Φε = Φ_{β⊤1X} Φε, so that cancellation of Φε is justified.
  4. [Theorem 5 proof] The pole-matching argument in the proof of Theorem 5 does not explicitly handle the case where one side has no positive coefficients (c+ = 0) or no negative coefficients (c− = 0), even though such cases can occur. The contradiction can be obtained by taking the limit at the smallest pole of the other side, but this case should be stated, since the current text assumes that both R+(1) and R−(1) exist.
  5. [Theorem 3 note] The note at the end of Theorem 3 that the assertion 'continues to hold if X is replaced by M X' for an invertible matrix M is unclear: if M is not diagonal, the components of M X are generally not independent, so the proof does not apply. Please clarify the intended statement or remove the note.
  6. [Footnote] The footnote contains the placeholder 'Supported by XXX'; this should be filled in before publication.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorems 3–6 are derived from stated assumptions with complete proofs; self-citations are contextual and never load-bearing.

full rationale

All new results are proved from scratch in the appendix. Theorem 3 follows by rescaling scale-family variables to an i.i.d. sample and applying Marcinkiewicz's theorem; Theorem 4 and Corollary 1 use the defining invariance of spherical/elliptical symmetry; Theorem 5 works directly on moment-generating functions, matching Gamma poles R±(i) on both sides (Eqs. 10–12), and the subset-sum condition on the shape parameters converts matching exponent sums into matching index sets, yielding βi = β̃i for i < d and then βd = β̃d from φ(t) = 1 since μ ≠ 0; Theorem 6 reduces β^T X d= β0^T X to the second- and fourth-moment equalities (13)–(15) and solves the resulting quadratics. None of these steps invokes the theorem being proved, fits a parameter to data, or imports a result from this paper's authors as a premise. The external theorems used (Marcinkiewicz 1939; Linnik 1953; Kagan–Linnik–Rao 1973 via Eriksson–Koivunen; Cambanis–Huang–Simons 1981) are classical published benchmarks and thus genuine independent support. Self-citations to [2], [3], [37], [38] appear only in the introduction, the review of the i.i.d. case, and motivation; the two examples taken from [2] are re-derived by new methods (Theorem 6 Example 1 and Theorem 5), not imported as premises. The review half is explicitly labeled as a review, so restating [2]'s results there is restatement, not circular reasoning. Two non-circularity caveats flagged per the review rule: (i) the §4.3 convolution claim that Xj d= Σ αi Xi forces β0+δ ∈ B0 is false as stated — replacing Xj by the dependent combination changes the joint law whenever some β0i for i ∈ I is nonzero, so the reduction is valid only when the remaining part is independent of the combination; the paper's own example satisfies this, and the flaw is a correctness risk, not a circular step; (ii) the abstract's "not possible to prove similar theorems in the general case" is stronger than the exhibited counterexamples establish. These issues do not raise the circularity score because no step in the derivation chain is equivalent to its own input by definition, by fit, or by self-citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The proofs rely on standard independence and moment assumptions, two classical characterization theorems used as external results, and one ad hoc subset-sum condition introduced for the proof of Theorem 5.

assumptions (6)
  • domain assumption The noise epsilon in the ULR model satisfies the condition of Lemma 1: its characteristic function has no open interval of zeros.
    This condition is used in Lemma 1 to drop the noise and reduce B0 to equality of the un-noised linear forms. All later identifiability results rely on this reduction. It holds for Gaussian, Laplace, uniform, and other standard noise distributions, but is a genuine assumption on the noise.
  • domain assumption The covariate components X1,...,Xd are independent.
    Independence is assumed for the positive results in Section 4 (Theorems 3, 5, 6). The paper does not give positive identifiability results for dependent, non-elliptical covariates, and the final conjecture explicitly requires independence.
  • ad hoc to paper In Theorem 5, the Gamma shape parameters alpha_i satisfy the subset-sum uniqueness condition: equality of sums over subsets I and J implies I=J.
    This condition is introduced specifically to make the moment-generating-function pole matching work; it forces equality of the index sets at each pole, yielding strong identifiability. It is not a natural modeling assumption and is violated by common choices such as alpha_1 = alpha_2.
  • domain assumption In Theorem 3, the base density f has finite moments of all orders and is not a Gaussian density.
    The full moment condition is needed to apply Marcinkiewicz's theorem; the paper notes that without it, stable distributions such as the Cauchy create additional non-identifiability.
  • domain assumption In Theorem 6, X1 and X2 are centered, unit variance, have finite fourth moments, and not both fourth moments equal 3.
    The fourth-moment approach requires finite fourth moments; the exclusion of m1=m2=3 removes the Gaussian-like case in which the sphere of solutions makes identifiability fail.
  • standard math Classical characterization theorems (Marcinkiewicz 1939, Linnik 1953, Kagan-Linnik-Rao 1973) are valid and applicable.
    Marcinkiewicz's theorem underpins Theorem 3; Linnik's theorem is reviewed in Section 3; Kagan-Linnik-Rao Theorem 10.3.5 is used in the ICA section. These are accepted external results.

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Pith. "Pith review of Identifiability in Unlinked Linear Regression: Some Results and Open Problems." pith.science (2026). https://pith.science/paper/BVHULBW5

@misc{pith2026250714986,
  author       = {Pith},
  title        = {Pith review of: Identifiability in Unlinked Linear Regression: Some Results and Open Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BVHULBW5}},
  note         = {Machine review of arXiv:2507.14986}
}
abstract

A tacit assumption in classical linear regression problems is the full knowledge of the existing link between the covariates and responses. In Unlinked Linear Regression (ULR) this link is either partially or completely missing. While the reasons causing such missingness can be different, a common challenge in statistical inference is the potential non-identifiability of the regression parameter. In this note, we review the existing literature on identifiability when the $d \ge 2$ components of the vector of covariates are independent and identically distributed. When these components have different distributions, we show that it is not possible to prove similar theorems in the general case. Nevertheless, we prove some identifiability results, either under additional parametric assumptions for $d \ge 2$ or conditions on the fourth moments in the case $d=2$. Finally, we draw some interesting connections between the ULR and the well established field of Independent Component Analysis (ICA).

Figures

Figures reproduced from arXiv: 2507.14986 by the authors.

Figure 1
Figure 1. Illustration of Unlinked Linear Regression. Left: The setup motivating the problem – two [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Illustration of Theorem 2 for d = 3 and two pairs of coefficients. Left: ξ0 = 1 is the only root of τ ; since ξ0 is simple and an odd integer, the conditions of the theorem are satisfied. Right: ξ0 = .5 is the only root of τ , but condition (B) in the theorem is not met since the integral part of ξ0 is even. with strict inequality for at least one k ∈ {1, . . . , d−1} [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

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