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arxiv: 2605.30266 · v1 · pith:XGITMKIFnew · submitted 2026-05-28 · 🧮 math.ST · stat.TH

Wasserstein Least Squares: A Canonical Regression Method for Probability Distributions

Uriel Mart\'inez Le\'on , Jonathan Niles-Weed This is my paper

Pith reviewed 2026-06-28 23:57 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords Wasserstein least squaresdistributional regressionoptimal transportWasserstein barycenterstemplate deformation modelconvex analysisparametric rates
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The pith

Wasserstein least squares extends classical least squares to probability distributions as its canonical convex-analytic counterpart and attains root-n estimation rates under a deformation model.

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

The paper introduces Wasserstein least squares for regressing vector covariates onto distribution-valued responses. It establishes this estimator as the direct analogue of Euclidean least squares by applying convex-analytic arguments that produce multimarginal and dual formulations. Under the template deformation model, where each observed distribution is a random push-forward of a fixed template, the method achieves the parametric convergence rate of n to the power of minus one half. This rate yields an exponential improvement on existing bounds for Wasserstein barycenters as a special case.

Core claim

Wasserstein least squares is the canonical extension of Euclidean least squares to the space of probability distributions from the perspective of convex analysis; this viewpoint gives rise to multimarginal and dual formulations of the Wasserstein least squares problem, extending a similar theory for Wasserstein barycenters. Under the template deformation model, estimation is possible at the n^{-1/2} rate, which produces an exponential improvement over prior rates for Wasserstein barycenters.

What carries the argument

Wasserstein least squares problem, defined by minimizing expected squared Wasserstein distance to a linear predictor in the space of measures and shown to be the convex-analytic lift of Euclidean least squares.

If this is right

  • The regression estimator converges at the parametric n^{-1/2} rate for the underlying map from covariates to distributions.
  • Wasserstein barycenters, recovered as the intercept-only case, inherit the same n^{-1/2} rate, an exponential improvement over previous bounds.
  • A particle-based heuristic permits computation on large data sets and yields new demographic insights from the RAND Health and Retirement Study.

Where Pith is reading between the lines

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

  • The random-effects interpretation may allow transfer of classical mixed-model diagnostics to the Wasserstein setting.
  • The multimarginal formulation could be adapted to other convex losses beyond squared distance in optimal transport regression.
  • Robustness checks on data lacking a clear template could quantify how much the parametric rate degrades outside the model.

Load-bearing premise

Observed distributions are generated as random deformations of one fixed template distribution.

What would settle it

A simulation in which distributions are generated without a common template yet the estimator still converges at rate n to the minus one half would falsify the necessity of the deformation model for the claimed rate.

Figures

Figures reproduced from arXiv: 2605.30266 by Jonathan Niles-Weed, Uriel Mart\'inez Le\'on.

Figure 1
Figure 1. Figure 1: (left) shows its corner plot: diagonal panels display the marginal dis￾tribution of each coefficient, and off-diagonal panels reveal genuine correlations among demographic effects. On the other hand, as we describe in Section A, Fr´echet regression also implicitly yields a joint distribution over the coefficient vec￾tor: the predicted distribution corresponding to a covariate vector x has quantile function… view at source ↗
Figure 2
Figure 2. Figure 2: Sequential conditioning (BMI ∈ [29, 33] at age 50; three age-60 scenarios), cohorts 1935–39 and 1940– 44, both sexes. Columns: worsening (BMI > 33, red), stable (BMI 29–33, blue), improved (BMI < 29, green). Rows: observed trajectories (top), Wasserstein least squares (middle), Fr´echet (bot￾tom). Shaded bands: 75% and 99% prediction intervals; box plots: matched empirical distribution. Wasserstein least s… view at source ↗
Figure 3
Figure 3. Figure 3: Random transport maps ∇ϕi from five noise families (coloured) applied to the template Q⋆ x (black dashed). Top row: the maps T(y); shaded band shows the C3 curvature bounds. Bot￾tom row: deviation T(y) − y from the identity, illustrating C2 (E[T(y)] = y). The two templates are: (i) Univariate (d = 1, n = 50), with Q⋆ x = N (t, 1 + t 2 ) and covariate x = (1, t) ⊤, t ∼ U[−2, 2]; the variance is U-shaped, gr… view at source ↗
Figure 4
Figure 4. Figure 4: Univariate: estimated densities at t ∈ {−1.8, −0.5, 0.5, 1.8}. True template Q⋆ xi (black dashed), noisy observations νi (grey), Wasserstein least squares (blue), Fr´echet (red dashed). Wasserstein least squares recovers the true template across the covariate range under additive noise. The U-shaped variance (univariate), Fr´echet regression underestimates spread at the extremes. Both templates share a fea… view at source ↗
Figure 5
Figure 5. Figure 5: Covariance trajectory t 7→ Σ(t) projected onto three coordinate pairs of the SPD cone. Wasserstein least squares (blue) traces the curved true path (black); Fr´echet-GD (red) fol￾lows a straight line.Bivariate Gaussian experiment: Wasserstein least squares recovers the quadratic covariance trajectory Σ(t) = A + t(B + B⊤) + t 2C. The log-Euclidean view (right) makes the curvature explicit; Fr´echet regressi… view at source ↗
Figure 6
Figure 6. Figure 6: Predictive parity between Wasserstein least squares and Fr´echet regression on the BMI data. Left: LOO W2 error his￾tograms (top) and paired cell-level comparison (bottom); the me￾dian difference is 0.025 BMI units and 61% of cells favour Fr´echet, confirming that neither method dominates on raw out-of-sample fit. Right: Fitted density heatmaps for male subjects, cohort 1935– 1939; both models produce virt… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of coefficient distributions from Wasser￾stein least squares and Fr´echet regression. (a) Marginal distribu￾tions of the random coefficient vector β ∼ Qb estimated via Wasser￾stein least squares. Each panel shows the empirical distribution of one coefficient component from the M = 2000 particle representa￾tion of Qb. Histogram with kernel density estimate overlay; vertical black dashed line at z… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of joint distribution structures between Wasserstein least squares and Fr´echet regression. To the left: Corner plot showing the joint distribution structure of β ∼ Qb. Diagonal panels: marginal distributions (histogram with KDE) for each coefficient; black vertical line at zero, red vertical line at the mean. Lower triangle: bivariate scatter plots showing pairwise joint distributions with cont… view at source ↗
Figure 9
Figure 9. Figure 9: Marginal versus conditional standard deviation (cohort 1940, female). The shaded region represents the variance explained by knowing β0. cohort effect—across the entire distribution. In contrast, interrogating the Wasser￾stein least squares particle cloud reveals distinct minority trajectories [PITH_FULL_IMAGE:figures/full_fig_p039_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Analysis of particles with βage2 > 0 (accelerating BMI growth). Using M = 20,000 particles, approximately 10% exhibit positive quadratic age coefficients, representing individuals whose BMI continues to accelerate with age rather than plateau or de￾cline. (a) Marginal distribution of βage2 ; vertical dashed line in￾dicates zero, solid line shows the mean. Shaded region highlights particles with βage2 > 0.… view at source ↗
Figure 11
Figure 11. Figure 11: Concave/monotone vs. convex BMI trajec￾tories, cohort 1940–44, female. Top row: individual ob￾served BMI trajectories (thin lines) with group median (thick line); dashed line at BMI = 30. Bottom row: detrended residuals after removing each individual’s personal linear trend, with smoothed median and interquartile range (IQR) band (25th–75th percentile); dashed line at zero (no curvature). Left (blue): the… view at source ↗
Figure 12
Figure 12. Figure 12: Analysis of particles with βcohort < 0 (reverse co￾hort effect). Using M = 20,000 particles, approximately 6% exhibit negative cohort coefficients, representing individuals for whom later birth cohorts have lower BMI—opposite of the typ￾ical obesity epidemic trend. (a) Marginal distribution of βcohort; vertical dashed line indicates zero. Shaded region highlights parti￾cles with βcohort < 0. (b) Condition… view at source ↗
Figure 13
Figure 13. Figure 13: Joint distribution of coefficients colored by extreme values (M = 20,000 particles). (a) Scatter plot of (β0, βage) colored by βage2 . Particles with βage2 > 0 (red/orange) tend to cluster in specific regions of the (β0, βage) space, revealing the correlation structure among coefficients. (b) Same scatter plot colored by βcohort. Particles with βcohort < 0 (red) show distinct patterns in the joint distrib… view at source ↗
Figure 14
Figure 14. Figure 14: Observed BMI trajectories (thin lines) over￾laid on model-implied 80 % prediction bands (shaded) for individuals at the obesity threshold (BMI = 31 at age 50), ages 50–75. Prediction bands are the 10th–90th percentile of {x(age)⊤βm}M′ m=1, where βm are the M′ ≤ M particles (Wasser￾stein least squares) or quantile-level coefficient vectors (Fr´echet) that survive the conditioning criterion yb(50) ∈ [30, 32… view at source ↗
Figure 15
Figure 15. Figure 15: Probability of obesity (BMI ≥ 30) as a function of age for females in cohort 1945, comparing Wasserstein least squares and Fr´echet regression under single and se￾quential conditioning. (Panel 1) Wasserstein least squares, single conditioning. Conditioning on BMI at age 50 produces smooth, clinically meaningful probability curves. Individuals start￾ing below the threshold (BMI50 = 28, green) show a modest… view at source ↗
Figure 16
Figure 16. Figure 16: Predicted BMI trajectories under single con￾ditioning (BMI ∈ [29, 33] at age 50), cohorts 1935–39 and 1940–44, both sexes. The top row shows individual observed BMI trajectories from the matched HRS cohort for Wasserstein least squares (left, blue) and Fr´echet (right, red). The bottom row shows the Wasserstein least squares (left) and Fr´echet (right) prediction intervals together with empirical box plot… view at source ↗
Figure 17
Figure 17. Figure 17: Predicted BMI trajectories under sequential conditioning (BMI ∈ [29, 33] at age 50 followed by three scenarios at age 60), cohorts 1935–39 and 1940–44, both sexes. Columns correspond to three second-observation scenarios: worsening (BMI > 33 at 60, red), stable (BMI 29–33 at 60, blue), and improved (BMI < 29 at 60, green). The top row shows individ￾ual observed trajectories from the matched HRS cohort; th… view at source ↗
Figure 18
Figure 18. Figure 18: Template recovery under sinusoidal deformation (k = 1.2, n = 50). Density heatmaps of the true Q⋆ x (left), Wasserstein least squares fit (center), and Fr´echet fit (right) over the covari￾ate range t ∈ [−2, 2]. Wasserstein least squares reproduces the U-shaped variance of the true template; Fr´echet produces nearly uniform spread across all t. uniformly from [−2, 2]; each response νi = (∇ϕi)#Q⋆ xi is app… view at source ↗
Figure 19
Figure 19. Figure 19: Wasserstein least squares recovers Q⋆ un￾der affine and non-linear noise; Fr´echet regression is structurally misspecified. DGP: νi = (∇ϕi)#Q⋆ xi , Q⋆ xi ∼ N (t, 1+t 2 ), n = 50, five noise models (see [PITH_FULL_IMAGE:figures/full_fig_p054_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Wasserstein least squares recovers a quadratic covariance trajectory on the SPD manifold; Fr´echet-GD is structurally misspecified. DGP: νi = (∇ϕi)#Q⋆ xi , Q⋆ xi ∼ N (µ(t), Σ(t)) with Σ(t) = A + t(B+B⊤) + t 2C (quadratic on the SPD manifold), n = 50, three noise models (see [PITH_FULL_IMAGE:figures/full_fig_p057_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: shows the median W2 2 error and interquartile range on log–log axes together with n −1/2 and n −1 reference lines. All three curves decay monotonically. The location-scale and radial models display a rate closer to n −1 over this range. 10 25 50 100 200 500 n (number of observations) 10 3 10 2 10 1 1 n i W2 2 (Q xi , Q xi ) Additive ( = = 1) Radial ( = 0.70) Location-Scale ( = 0.40) n 1/2 n 1 [PITH_FULL_… view at source ↗
read the original abstract

We perform a mathematical and statistical analysis of the Wasserstein least squares problem, a regression method for vector-valued covariates and distribution-valued responses. Our proposal contrasts with other distributional regression methods by having a direct interpretation in terms of random variables, as a nonparametric analogue of the classic random-effects model. On the mathematical side, we use a strategy of Lavenant (2024) to show that Wasserstein least squares is the canonical extension of Euclidean least squares to the space of probability distributions from the perspective of convex analysis; this viewpoint gives rise to multimarginal and dual formulations of the Wasserstein least squares problem, extending a similar theory for Wasserstein barycenters. We perform a statistical analysis of the Wasserstein least squares problem under the template deformation model, showing, surprisingly, that estimation is possible at the n^{-1/2} rate. As a special case, we obtain improved rates of estimation for Wasserstein barycenters, which are an exponential improvement over those established by Ahidar-Coutrix, Le Gouic and Paris (2020). Finally, we propose a heuristic particle method for Wasserstein least squares and use it to conduct a novel analysis of large-scale demographic data from the RAND Health and Retirement Study.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces Wasserstein least squares as a regression method mapping vector covariates to distribution-valued responses. It uses a convex-analysis strategy from Lavenant (2024) to establish this as the canonical extension of Euclidean least squares, yielding multimarginal and dual formulations that extend barycenter theory. Under the template deformation model, the estimator achieves the parametric n^{-1/2} rate; as a special case this yields exponentially faster rates for Wasserstein barycenters than those in Ahidar-Coutrix et al. (2020). A particle heuristic is proposed and applied to RAND Health and Retirement Study demographic data.

Significance. The convex-analytic characterization supplies a principled, optimization-based foundation for distributional regression that may unify existing approaches. The n^{-1/2} rate under the deformation model is a notable improvement in a structured setting and directly improves barycenter estimation when the model holds. The empirical section provides a concrete demonstration on large-scale data, though the practical scope is limited by the modeling assumption.

major comments (2)
  1. [Statistical analysis section] Statistical analysis section (abstract and corresponding main-text section): the n^{-1/2} rate and the exponential improvement over Ahidar-Coutrix et al. (2020) are obtained exclusively under the template deformation model; the manuscript should state explicitly whether any general (nonparametric) rate is available or whether the model is indispensable for the claimed rate, as this assumption is load-bearing for the headline statistical result.
  2. [Mathematical analysis section] Mathematical analysis section: the multimarginal and dual formulations are asserted to follow from Lavenant (2024); the paper should include a self-contained verification that the Wasserstein least-squares functional is convex (or strictly convex under stated conditions) and that the dual problem recovers the same minimizer, citing the precise equation or proposition where this is shown.
minor comments (2)
  1. [Application section] The abstract states that the n^{-1/2} claim rests on the template deformation model; the main text should add a short paragraph clarifying how this model is checked or motivated for the RAND data application.
  2. Notation for the multimarginal formulation should be aligned with the barycenter literature to facilitate comparison; a short remark contrasting the two problems would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the two constructive major comments. Both points can be addressed by targeted revisions that clarify the scope of the statistical results and strengthen the self-contained mathematical presentation. We outline our responses below.

read point-by-point responses

  1. Referee: [Statistical analysis section] Statistical analysis section (abstract and corresponding main-text section): the n^{-1/2} rate and the exponential improvement over Ahidar-Coutrix et al. (2020) are obtained exclusively under the template deformation model; the manuscript should state explicitly whether any general (nonparametric) rate is available or whether the model is indispensable for the claimed rate, as this assumption is load-bearing for the headline statistical result.

    Authors: We agree that the n^{-1/2} rate (and the resulting exponential improvement for barycenters) is derived exclusively under the template deformation model; no general nonparametric rate is claimed or derived in the paper. The model is indispensable for the parametric rate. We will revise the abstract and the statistical analysis section to state this explicitly, making clear that the headline rate requires the deformation assumption and that the analysis does not provide rates outside this structured setting. revision: yes

  2. Referee: [Mathematical analysis section] Mathematical analysis section: the multimarginal and dual formulations are asserted to follow from Lavenant (2024); the paper should include a self-contained verification that the Wasserstein least-squares functional is convex (or strictly convex under stated conditions) and that the dual problem recovers the same minimizer, citing the precise equation or proposition where this is shown.

    Authors: We will add a short self-contained verification in the mathematical analysis section. This will adapt the convex-analytic arguments of Lavenant (2024) to the Wasserstein least-squares functional, explicitly showing convexity (and strict convexity under the stated conditions on the cost) and verifying that the dual recovers the same minimizer. The added text will cite the precise propositions from Lavenant (2024) that are being specialized, while keeping the argument self-contained for the reader. revision: yes

Circularity Check

0 steps flagged

No circularity: external strategy and explicit model assumption yield independent content

full rationale

The paper invokes an external strategy from Lavenant (2024) for the convex-analysis claim that Wasserstein least squares is canonical, and derives the n^{-1/2} rate explicitly under the stated template deformation model. Neither step reduces a prediction to a fitted quantity by construction, nor relies on self-citation load-bearing or ansatz smuggling. The comparison to Ahidar-Coutrix et al. (2020) is a benchmark contrast, not a definitional reduction. The derivation chain therefore contains independent mathematical and statistical content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the template deformation model for the statistical rates and on the convex-analysis strategy of Lavenant (2024) for the canonical characterization; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption The template deformation model holds for the observed distributions.
    Invoked to obtain the n^{-1/2} estimation rate in the statistical analysis.
  • domain assumption Lavenant (2024) strategy applies to the Wasserstein least squares functional.
    Used to establish the canonical extension and multimarginal/dual formulations.

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discussion (0)

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Reference graph

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