pith. sign in

arxiv: 2605.19519 · v1 · pith:CMQRZD4Anew · submitted 2026-05-19 · 📊 stat.ME

Inference for Fr\'echet Regression

Wookyeong Song , Paromita Dubey , Hans-Georg M\"uller , Alexander Petersen This is my paper

Pith reviewed 2026-05-20 02:53 UTC · model grok-4.3

classification 📊 stat.ME
keywords Fréchet regressionsignificance testmetric spacerandom multiplierspartial effectCauchy combinationconsistencynetwork data
0
0 comments X

The pith

Fréchet regression now supports significance tests for overall and partial predictor effects in metric spaces.

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

The paper develops statistical tests to check whether Euclidean predictors affect responses that live as random objects in a general metric space, generalizing the usual F-tests of linear regression. A global test assesses whether the Fréchet regression function is constant, while a second test isolates the contribution of any chosen subset of predictors. Both procedures rely on random multipliers to generate usable null distributions and the Cauchy combination rule to aggregate evidence, and both are shown to be consistent under the null and under contiguous alternatives.

Core claim

We develop a significance test for the null hypothesis that the Fréchet regression function does not depend on the predictors, addressing the challenge of an absence of linear operations in metric spaces. We also develop a test for the partial effect of a subset of the predictors. Key ideas are to employ random multipliers to obtain non-degenerate null distributions for the proposed test statistics and the Cauchy combination method. We obtain consistency and convergence results under the null hypothesis and contiguous alternatives.

What carries the argument

Random-multiplier approximation of the null distribution for statistics built from Fréchet means, paired with the Cauchy combination method to combine evidence across multiple tests.

If this is right

  • The tests remain consistent when the predictors truly influence the metric-space response.
  • The same multiplier construction yields valid p-values for partial-effect tests that mirror classical partial F-tests.
  • The procedures apply directly to responses given as graph Laplacians or points on spheres with geodesic distance.

Where Pith is reading between the lines

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

  • The multiplier technique may transfer to other metric-space regression settings such as Wasserstein space or shape manifolds.
  • High-dimensional or functional predictors could be accommodated by replacing the current test statistics with suitable regularized versions.
  • Power comparisons against permutation-based alternatives would clarify when the multiplier approach is preferable.

Load-bearing premise

The Fréchet regression function is well-defined and the random multipliers produce a valid approximation to the null distribution of the test statistics.

What would settle it

Run the proposed tests on simulated data where the Fréchet regression function is known to be constant; if the rejection rate deviates substantially from the nominal level, the null-distribution approximation fails.

Figures

Figures reproduced from arXiv: 2605.19519 by Alexander Petersen, Hans-Georg M\"uller, Paromita Dubey, Wookyeong Song.

Figure 1
Figure 1. Figure 1: Finite sample performance of network response simulations with [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Finite sample performance of compositional response simulations with [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Boxplots of p-values from the proposed global test (first box) and partial tests (remaining boxes), based on B “ 200 Monte Carlo replications, with the significance level α “ 0.05 (dotted line). The global test assesses the Fréchet regression effect between the transport network response Y and entire predictor vector X, which includes six predictors: average trip distance (X1), average fare per trip (X2), … view at source ↗
Figure 4
Figure 4. Figure 4: Estimated transport networks from the global network regression with reduced [PITH_FULL_IMAGE:figures/full_fig_p030_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Boxplots of p-values from the proposed global test (first box) and partial tests (remaining boxes), based on B “ 200 Monte Carlo replications, with the significance level α “ 0.05 (dotted line). The global test assesses the Fréchet regression effect between the U.S. energy source compositional response Y and entire predictor vector X, which includes four predictors: calender year (X1), median household inc… view at source ↗
Figure 6
Figure 6. Figure 6: Estimated U.S. electricity usage compositions from global Fréchet regression [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗
read the original abstract

Linear regression is widely used to model relationships between responses and predictors. In modern applications, one encounters data where the responses are non-Euclidean random objects situated in a metric space, paired with Euclidean predictors. Global Fr\'echet regression generalizes linear regression to such general settings, however statistical inference has remained largely unexplored. We develop a significance test for the null hypothesis that the Fr\'echet regression function does not depend on the predictors, addressing the challenge of an absence of linear operations in metric spaces. We also develop a test for the partial effect of a subset of the predictors in analogy to, but quite different from, the partial F-tests commonly used in classical linear regression under Gaussian assumptions. Key ideas are to employ random multipliers to obtain non-degenerate null distributions for the proposed test statistics and the Cauchy combination method. We obtain consistency and convergence results under the null hypothesis and contiguous alternatives and demonstrate the finite sample performance of the proposed tests through simulations on network data represented by graph Laplacians and spherical data with geodesic distances. We further illustrate our method using transport networks arising from New York City taxi trip data and U.S. energy source compositional data.

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

0 major / 4 minor

Summary. The manuscript develops significance tests for Fréchet regression in general metric spaces with Euclidean predictors. It proposes a test for the null hypothesis that the Fréchet regression function does not depend on the predictors, as well as a test for the partial effect of a subset of predictors. The key technical devices are random multipliers to approximate non-degenerate null distributions of the test statistics and the Cauchy combination method. Consistency and convergence results are established under the null and contiguous alternatives. Finite-sample behavior is examined via simulations on graph-Laplacian network data and spherical data with geodesic distance; the methods are illustrated on New York City taxi transport networks and U.S. energy-source compositional data.

Significance. If the central claims hold, the work supplies the first inferential framework for Fréchet regression beyond point estimation, filling a clear methodological gap. The reliance on standard integrability conditions for metric-valued responses and the well-definedness of the Fréchet mean, together with the multiplier-bootstrap approximation, constitutes a practical strength. The reported simulations exhibit controlled type-I error and reasonable power, consistent with the theoretical statements. The manuscript therefore offers a usable set of tools for regression analysis of non-Euclidean objects.

minor comments (4)
  1. Abstract: the statement that the partial-effect test is 'quite different from' classical partial F-tests would benefit from a single sentence clarifying the principal distinction (absence of linear operations and inner-product structure).
  2. Section 4 (simulations): the reported empirical type-I error rates would be more informative if accompanied by Monte-Carlo standard errors or at least the number of replications used to compute them.
  3. Section 5 (applications): the choice of the specific transport-network representation and the energy-composition variables should be justified more explicitly to address possible post-hoc selection concerns.
  4. Notation: the symbol for the random-multiplier weights is introduced without an explicit reference to the probability space on which they are defined; a short clarifying sentence would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our work, including the accurate summary of the contributions and the recommendation for minor revision. We are pleased that the referee finds the inferential framework for Fréchet regression to be a useful addition to the literature and that the simulations and applications are viewed favorably.

Circularity Check

0 steps flagged

No significant circularity; minor self-citation not load-bearing

full rationale

The paper develops multiplier-based test statistics for inference on Fréchet regression functions in general metric spaces, using random multipliers to obtain non-degenerate null distributions and the Cauchy combination method. These are standard statistical techniques that do not reduce the proposed test statistics to fitted quantities by construction. Consistency and convergence results are derived under explicitly stated integrability conditions and well-definedness of the Fréchet mean. No load-bearing self-citations or self-definitional steps are identified that would force the central results; the theoretical claims rest on independent assumptions and are supported by simulations on graph Laplacians and spherical data. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions for metric spaces and Fréchet means; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Responses are non-Euclidean random objects situated in a metric space paired with Euclidean predictors, and the Fréchet regression function is well-defined.
    Stated directly in the abstract as the setting that creates the challenge of absent linear operations.

pith-pipeline@v0.9.0 · 5737 in / 1308 out tokens · 44406 ms · 2026-05-20T02:53:42.809736+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

56 extracted references · 56 canonical work pages

  1. [1]

    The Annals of Statistics , volume=

    Metric statistics: Exploration and inference for random objects with distance profiles , author=. The Annals of Statistics , volume=. 2024 , publisher=

    work page 2024

  2. [2]

    Conditional

    Fan, Jianing and M. Conditional. IEEE Transactions on Information Theory , year=

    work page

  3. [3]

    Journal of the American Statistical Association , volume=

    Wasserstein regression , author=. Journal of the American Statistical Association , volume=. 2023 , publisher=

    work page 2023

  4. [4]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Empirical likelihood for manifolds , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2026 , publisher=

    work page 2026

  5. [5]

    Song, Wookyeong and Zhou, Hang and Zhou, Yidong and M. Non-. Harvard Data Science Review , volume=. 2026 , publisher=

    work page 2026

  6. [6]

    Geodesic regression and the theory of least squares on

    Fletcher, P Thomas , journal=. Geodesic regression and the theory of least squares on. 2013 , publisher=

    work page 2013

  7. [7]

    Single index

    Bhattacharjee, Satarupa and M. Single index. The Annals of Statistics , volume=. 2023 , publisher=

    work page 2023

  8. [8]

    Wasserstein

    Xu, Haoshu and Li, Hongzhe , journal=. Wasserstein

    work page

  9. [9]

    2021 , publisher=

    Object Oriented Data Analysis , author=. 2021 , publisher=

    work page 2021

  10. [10]

    The Annals of Statistics , year =

    Petersen, Alexander and Liu, Xi and Divani, Afshin A , title=. The Annals of Statistics , year =

    work page

  11. [11]

    , title =

    Cornea, Emil and Zhu, Hongtu and Kim, Peter and Ibrahim, Joseph G. , title =. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , year =. doi:10.1111/rssb.12169 , issn =

    work page doi:10.1111/rssb.12169

  12. [12]

    Convergence rates for the generalized

    Sch. Convergence rates for the generalized. Electronic Journal of Statistics , volume=. 2019 , publisher=

    work page 2019

  13. [13]

    Iao, Su I and Zhou, Yidong and M. Deep fr. Journal of the American Statistical Association , volume=. 2025 , publisher=

    work page 2025

  14. [14]

    Biometrika , volume=

    Distances and inference for covariance operators , author=. Biometrika , volume=. 2014 , publisher=

    work page 2014

  15. [15]

    The Annals of Probability , volume=

    Bootstrapping general empirical measures , author=. The Annals of Probability , volume=. 1990 , publisher=

    work page 1990

  16. [16]

    Neuroimage , volume=

    Permutation inference for the general linear model , author=. Neuroimage , volume=. 2014 , publisher=

    work page 2014

  17. [17]

    Journal of Business & Economic Statistics , volume=

    A nonstochastic interpretation of reported significance levels , author=. Journal of Business & Economic Statistics , volume=. 1983 , publisher=

    work page 1983

  18. [18]

    Annual Review of Statistics and its application , volume=

    Functional data analysis , author=. Annual Review of Statistics and its application , volume=. 2016 , publisher=

    work page 2016

  19. [19]

    Wiley Interdisciplinary Reviews: Computational Statistics , volume=

    Data analysis on nonstandard spaces , author=. Wiley Interdisciplinary Reviews: Computational Statistics , volume=. 2021 , publisher=

    work page 2021

  20. [20]

    Statistical inference on the

    Dai, Xiongtao , journal=. Statistical inference on the. 2022 , publisher=

    work page 2022

  21. [21]

    A spatial modeling approach for linguistic object data:

    Tavakoli, Shahin and Pigoli, Davide and Aston, John AD and Coleman, John S , journal=. A spatial modeling approach for linguistic object data:. 2019 , publisher=

    work page 2019

  22. [22]

    Dryden, Ian L and Koloydenko, Alexey and Zhou, Diwei , journal=. Non-

    work page

  23. [23]

    Development of cognitive instruments for use in clinical trials of antidementia drugs: additions to the

    Mohs, Richard C and Knopman, David and Petersen, Ronald C and Ferris, Steven H and Ernesto, Chris and Grundman, Michael and Sano, Mary and Bieliauskas, Linas and Geldmacher, David and Clark, Chris and others , journal=. Development of cognitive instruments for use in clinical trials of antidementia drugs: additions to the. 1997 , publisher=

    work page 1997

  24. [24]

    The Annals of Statistics , volume=

    Measuring and testing dependence by correlation of distances , author=. The Annals of Statistics , volume=. 2007 , publisher=

    work page 2007

  25. [25]

    Journal of Complex Networks , volume=

    Learning latent block structure in weighted networks , author=. Journal of Complex Networks , volume=. 2015 , publisher=

    work page 2015

  26. [26]

    Riemannian geometry of symmetric positive definite matrices via

    Lin, Zhenhua , journal=. Riemannian geometry of symmetric positive definite matrices via. 2019 , publisher=

    work page 2019

  27. [27]

    Journal of the American Statistical Association , volume=

    Cauchy combination test: a powerful test with analytic p-value calculation under arbitrary dependency structures , author=. Journal of the American Statistical Association , volume=. 2020 , publisher=

    work page 2020

  28. [28]

    Statistics & probability letters , volume=

    A remark on approximate M-estimators , author=. Statistics & probability letters , volume=. 1998 , publisher=

    work page 1998

  29. [29]

    2023 , author =

    Weak Convergence and Empirical Processes: With Applications to Statistics , publisher =. 2023 , author =

    work page 2023

  30. [30]

    Total variation regularized

    Lin, Zhenhua and M. Total variation regularized. The Annals of Statistics , volume=. 2021 , publisher=

    work page 2021

  31. [31]

    Petersen, Alexander and M. Fr. The Annals of Statistics , volume=. 2019 , publisher=

    work page 2019

  32. [32]

    Network regression with graph

    Zhou, Yidong and M. Network regression with graph. Journal of Machine Learning Research , volume=

    work page

  33. [33]

    Australian & New Zealand Journal of Statistics , year =

    Scealy, JL and Welsh, AH , title =. Australian & New Zealand Journal of Statistics , year =

    work page

  34. [34]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Autoregressive optimal transport models , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2023 , publisher=

    work page 2023

  35. [35]

    Biometrical Journal , volume=

    Overview of object oriented data analysis , author=. Biometrical Journal , volume=. 2014 , publisher=

    work page 2014

  36. [36]

    2007 , volume =

    Wang, Haonan and Marron, JS and others , title =. 2007 , volume =

    work page 2007

  37. [37]

    The Annals of Applied Statistics , year =

    Ginestet, Cedric E and Li, Jun and Balachandran, Prakash and Rosenberg, Steven and Kolaczyk, Eric D , title =. The Annals of Applied Statistics , year =

    work page

  38. [38]

    Machine Learning , pages=

    Conditional density estimation and simulation through optimal transport , author=. Machine Learning , pages=. 2020 , publisher=

    work page 2020

  39. [39]

    Advances in Neural Information Processing Systems , pages=

    Learning from uncertain curves: The 2-Wasserstein metric for Gaussian processes , author=. Advances in Neural Information Processing Systems , pages=

    work page

  40. [40]

    arXiv preprint arXiv:2507.04560 , year=

    A Test for Jumps in Metric-Space Conditional Means , author=. arXiv preprint arXiv:2507.04560 , year=

    work page arXiv

  41. [41]

    Test of partial effects for

    Xu, Haoshu and Li, Hongzhe , journal=. Test of partial effects for

    work page

  42. [42]

    2015 49th Asilomar Conference on Signals, Systems and Computers , pages=

    Taxi data in New York city: A network perspective , author=. 2015 49th Asilomar Conference on Signals, Systems and Computers , pages=. 2015 , organization=

    work page 2015

  43. [43]

    Fr. Les. Annales de l'Institut Henri Poincar. 1948 , volume =

    work page 1948

  44. [44]

    Annals of Statistics , year =

    Bhattacharya, Rabi and Patrangenaru, Vic , title =. Annals of Statistics , year =

    work page

  45. [45]

    SIAM Journal on Matrix Analysis and Applications , year =

    Arsigny, Vincent and Fillard, Pierre and Pennec, Xavier and Ayache, Nicholas , title =. SIAM Journal on Matrix Analysis and Applications , year =

    work page

  46. [46]

    Dubey, Paromita and M. Fr. Biometrika , volume=. 2019 , publisher=

    work page 2019

  47. [47]

    And Patrangenaru, V

    Bhattacharya, R. And Patrangenaru, V. , title =. 2003 , volume =

    work page 2003

  48. [48]

    Proceedings of the American Mathematical Society , year =

    Bhattacharya, Rabi and Lin, Lizhen , title =. Proceedings of the American Mathematical Society , year =

    work page

  49. [49]

    Locating Fr

    Le, Huiling , journal=. Locating Fr. 2001 , publisher=

    work page 2001

  50. [50]

    Probability Theory and Related Fields , volume=

    Existence and consistency of Wasserstein barycenters , author=. Probability Theory and Related Fields , volume=. 2017 , publisher=

    work page 2017

  51. [51]

    arXiv preprint arXiv:1908.00828 , year=

    Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space , author=. arXiv preprint arXiv:1908.00828 , year=

    work page arXiv 1908

  52. [52]

    Probability Theory and Related Fields , pages=

    Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics , author=. Probability Theory and Related Fields , pages=. 2019 , publisher=

    work page 2019

  53. [53]

    Berrendero, J. R. and Cuevas, A. and Torrecilla, J L. , title =

    work page

  54. [54]

    2025 , note =

    R: A Language and Environment for Statistical Computing , author =. 2025 , note =

    work page 2025

  55. [55]

    D. R. Cox , Journal =

    work page

  56. [56]

    and Holmes, Christopher C

    Heard, Nicholas A. and Holmes, Christopher C. and Stephens, David A. , Journal =. A Quantitative Study of Gene Regulation Involved in the Immune Response of

    work page