Title: Theoretical and Practical Analysis of Fréchet Regression via Comparison Geometry
Venue: NeurIPS 2025
Accepted: no
Keywords: Fréchet regression
Abstract: Fréchet regression extends classical regression methods to non-Euclidean metric spaces, enabling the analysis of data relationships on complex structures such as manifolds and graphs. This work establishes a rigorous theoretical analysis for Fréchet regression through the lens of comparison geometry which leads to important considerations for its use in practice. The analysis provides key results on the existence, uniqueness, and stability of the Fréchet mean, along with statistical guarantees for nonparametric regression, including exponential concentration bounds and convergence rates. Additionally, insights into angle stability reveal the interplay between curvature of the manifold and the behavior of the regression estimator in these non-Euclidean contexts. Empirical experiments validate the theoretical findings, demonstrating the effectiveness of proposed hyperbolic mappings, particularly for data with heteroscedasticity, and highlighting the practical usefulness of these results.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper explores the statistical properties of Fréchet regression in CAT($K$) space, a metric space with an upper bound on curvature. The main results of the paper are non-asymptotic convergence results (Section 3.2) and angle stability (Section 3.3) of the nonparametric (kernel-based) conditional Fréchet mean estimator.\n\nSTRENGTHS AND WEAKNESSES:\n### Strength:\n* The paper analyzes the convergence of the kernel Fréchet regression estimator. The analysis of Fréchet regression has mainly been conducted when the covariates are univariate rather than multivariate [1, 2]. On the other hand, this work presents the results in the multivariate case.\n* To the best of my knowledge, Section 3.3 is new, as I have not seen any literature analyzing the angle property of Fréchet regression.\n\n### Weaknesses\n* Several statements are not clearly written or are misleading. To list them:\n    * Definition 3 is not the definition of strong convexity, but rather the definition of the so-called Variance Inequality (see, for example, Definitions 2.3.3 and 2.3.5 of [3]), which is weaker than strong convexity. The definition in the paper coincides with the standard strong convexity in Riemannian manifolds only if $p$ is the minimizer of $f$.\n    * In Theorem 1, Line 144: what is $z$? Does it mean $\\mu$ instead of $z$?\n    * In Theorem 1, $\\alpha$ is the strong convexity constant of what? It seems like the authors meant the distance function, but this is not clearly mentioned.\n    * In Theorem 1, the statement is written for general CAT($K$) spaces, but it later says $m$ denotes the dimension of the manifold. Does the result only hold for CAT($K$) manifolds instead of general CAT($K$) spaces?\n    * What is “mild regularity condition” specifically in Theorem 2, Line 165?\n    * In Theorem 3, Line 178: the statement “does not differ too much” is vague. Under which topology?\n    * What is “small” mathematically in Lemma 7, Line 217?\n    * For Proposition 3 and Theorem 4, it does not say which Wasserstein metric is used (e.g., 2-Wasserstein? Or any $p$-Wasserstein?). It is not written in the proof either.\n    * In Section 3.4, the notion of tangent space on general CAT($K$) space is not clearly stated. One should introduce a concept like the tangent cone as in [4]. Without introducing the notion of the tangent cone, it is not clear what is meant by the gradient and Hessian in Proposition 3.4.\n\n* Proofs are not written in full detail and omit important theoretical steps. For example:\n    * Line 596 should be $\\alpha \\leq \\sin(2\\sqrt{K}R)/2\\sqrt{K}R$. In addition, while I think Line 596 is true (after fixing the typo), I don’t think this result is trivial enough to omit justification. The authors should either provide the proof or give a reference.\n    * I think references for Line 612 are needed as well. For example, as mentioned above, what $m$ means is not clear when $M$ is a general CAT($K$) space instead of a manifold. I suspect this quantity can be replaced by a metric entropy-related quantity for general metric spaces, but to assure this, the details of this step should be stated precisely.\n    * In Line 631, why is the integral taken over the real line instead of $M$?\n    * As mentioned above, which Wasserstein metric is used in Line 773?\n    * Proposition 4 is written with respect to general CAT($K$) space, but the proof uses Riemannian geometry formulas. Whether such formulas can be extended to CAT($K$) space seem nontrivial. I believe one can make the similar statement in general CAT($K$) spaces, but to that end one would need to define quantities like tangent cone, gradient, and Hessian for general CAT($K$) spaces.\n\n* I find the results of Section 3.2 duplicative of previous works, in particular [2]. While [2] analyzes the problem under 1-dimensional covariates, the extension to multivariate regression seems the same as in the canonical kernel regression problem (if I am thinking wrong here, can you provide the specific technical difficulties?). For example, Theorem 3 of this paper seems to duplicate Theorems 1 and 2 of [2] in the 1-dimensional case. \n\n* I find Section 3.3 theoretically interesting, but I am not sure how one should interpret the result in practice. I have not seen an analysis where $\\angle_{\\mu}(u,v)$ is the quantity of interest. Is there any practical example where such quantities should be taken into account?\n\n* Some references are missing. For example, [2, 4, 5] seem relevant and should be included.\n\n[1] Zhenhua Lin and Hans-Georg Müller. Total Variation Regularized Fréchet Regression for Metric-Space Valued Data. Annals of Statistics, 2021.\n\n[2] Christof Schötz. Nonparametric Regression in Nonstandard Spaces. ArXiv preprint https://arxiv.org/abs/2012.13332, 2020.\n\n[3] Austin J. Stromme. Wasserstein barycenters : Statistics and Optimization. Graduate Thesis, 2020.\n\n[4] Thibaut Le Gouic, Quentin Paris, Philippe Rigollet, and Austin J. Stromme. Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space. Journal of the European Mathematical Society, 2019.\n\n[5] Victor-Emmanuel Brunel and Jordan Serres. Concentration of empirical barycenters in metric spaces. 35th International Conference on Algorithmic Learning Theory. 2024.\n\nQUESTIONS:\n* Can authors answer questions I have written in the weaknesses section?\n\n* Can authors characterize the difference between Theorem 1 of this paper and Corollary 11, Theorem 18 of [5]?\n\n* Can authors illustrate the novelty of the work compared to [2]?\n\nLIMITATIONS:\nThe authors expressed limitations of the work. I have stated additional limitations that I found in the weaknesses section.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-21"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a rigorous theoretical foundation for Fréchet regression, which generalizes regression to non-Euclidean metric spaces (such as manifolds). The key innovation is the use of comparison geometry, particularly CAT(K) spaces (geodesic metric spaces with curvature bounded above by K), to study or prove:\n\n1. the existence, uniqueness, stability of the Fréchet mean and associated regression estimators according to the sign of K.\n\n2. the sample Fréchet mean converges exponentially fast to the population mean; pointwise consistency of nonparametric Fréchet regression, and the convergence rate O(h_n^{2\\beta}+(nh_n^d)^{-1}), a usual trade-off case from Euclidean nonparametric statistics carried over to the CAT(K) setting.\n\n3. angle stability and local geometric effects.\n\nThe paper also provides some real world data experiments.\n\nSTRENGTHS AND WEAKNESSES:\nThe strength of this paper is to use comparison geometry, particularly CAT(K) spaces, to study the Fréchet regression and establish statistical guarantees (see summary). Particularly, they provided rates of convergence compared to to the existing work Chen & Müller (2022).\n\nThe weakness of this paper is as follows. \n\n1. As the paper appears to be mainly theocratically focused paper, the use of CAT(K) space is more tailored to suit the need for proofs while the fact that the data is assumed to subject to this fixed curvature K geometry is restrictive, that is not data adaptive.\n\n2. Moreover, although there are real world data experiments, at least the paper could discuss whether the data could be used to directly learn the curvature or to determine if the CAT(K) condition is satisfied, or roughly satisfied. To this end, is there any oracle approach to determine if the data fits in this CAT(K) model, such that even if the data may not satisfy this CAT(K) model, any data preprocessing or transformation could be applied to remedy this?\n\nQUESTIONS:\nPlease answer the question in Section weakness. Moreover, what does approximate equality sign mean, say in equation 9? It seems there is no rigorous explanation for it throughout the paper.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper is concerned with Fréchet regression on $\\text{CAT}(K)$ spaces. It is divided into two parts: a theoretical part and an experimental part. The theoretical part starts with some classical results on the Fréchet mean in $\\text{CAT}(K)$  spaces, then establishes several other results in the context of $\\text{CAT}(K)$ spaces, in particular (i) a concentration result for the sample Fréchet mean, (ii) pointwise consistency and (iii) convergence rate of the nonparametric Fréchet regression estimator, and (iv) angle stability for the conditional Fréchet mean. The experiments section compares, in terms of mean squared error, the estimation performance of Fréchet regression on positively curved and negatively curved spaces, showing the superiority of hyperbolic coordinates. This is shown on synthetic data as well as on real-world datasets.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is overall well written and the exposition is clear. The results presented in sections 3.2. and 3.3. of the theoretical part are interesting and implications of these results for practical purposes are included at the end of each subsection. \n\nHowever, the importance of the results of sections 3.4. and 3.5. is less clear to me, as detailed in the questions below. Moreover, it is not always clear in my opinion which results are already known, and which are new, and how the new results relate to the existing literature. In particular it seems to me that Theorem 1 is closely related to Theorem 5.8. of [a], giving a concentration inequality for sample Fréchet mean in CAT(K) spaces for $K>0$, however [1] is not cited in the paper. Finally, the experiments section is very short, and a bit disappointing in my opinion: the experiments concern the estimation precision of the Fréchet regression estimator in positive and negative curvature, but nothing is shown to illustrate (at least some of) the precise results given in the theoretical section. As it is, the theoretical part and the experiments part seem somewhat unrelated.\n\n[a] V. E. Brunel and J. Serres, Concentration of empirical barycenters in metric spaces. International Conference on Algorithmic Learning Theory. PMLR, 2024.\n\n\n----\n\nEDIT : I have increased my rating according to the authors' clarifications regarding their contributions. I believe that providing information on results already known in the literature concerning Fréchet regression will add significant value to the paper.I would also like to thank the authors for their careful response to the reviewers' comments, which have helped to clarify the impact of their results.\n\nQUESTIONS:\n- It is not clear in my opinion which results were known, and which are new, and how the new results relate to the existing literature. It is stated that the lemmas in Section 3.1. follow from previous work. What about the Lemmas 6 and 7 on angle comparison and continuity in $\\text{CAT}(K)$ spaces, and Lemma 8 on angle comparison in a Riemannian manifold? Concerning the main results on concentration, consistency and rates of convergence of Fréchet regression in $\\text{CAT}(K)$ spaces, how do they relate to the literature? Were similar results known in other more restrictive contexts, or weaker results in the same context? In particular, how does Theorem 1 relate to Theorem 5.8. of [a], that gives a concentration inequality for sample Fréchet mean in $\\text{CAT}(K)$ spaces for $K>0$ ?\n- Section 3.4: I am not sure that I understand the point of the results in this section. It seems to me that Lemma 8 is a general Riemannian geometry result, not particularly linked to Fréchet regression, and not used in any other result of the paper. As for Proposition 4, it is written “By expanding the Fréchet functional in the tangent space via the exponential map, one can gain insights into the functional’s curvature and higher-order properties. » However here the curvature term is not explicit, neither is it explicitly computed in the proof if I am not mistaken. Thus, it is not clear to me what the reader could use this formula for.\n- Experiments do not really illustrate the theoretical results: they only show better estimation in negative curvature than in positive curvature for the MSE criterion. Would it be possible to illustrate some of the formulas given in the theorems ? \n- How restrictive is the assumption of Hölder continuity for $\\mu^*(x)$ in Theorem 3? More precisely, under what conditions on the distributions of $X$ and $Y$ can this regularity be derive?\n\n\nMinor comments:\n- I don’t understand the inner product in last formula of Proposition 5\n- Section 2 : definition 7 requires M to be a Riemannian manifold and not just a geodesic metric space like stated at the beginning of the section.\n- In Lemma 8, why not use $U$ and $V$ in all the formula, including in the $O(...)$ term ?\n- It could be more clearly stated that the predictor live in the Euclidean space.\n- Why not use the same notations in Proposition 5 and its proof ? ($\\Delta$ and $\\Pi$)\n- Typos. In the inequalities line 514, a term is missing on the third line. Typo in the equation of line 536. \"Proof of Proposition 4\" in line 563 should refer to Lemma 4 and 5? Theorem 5 line 160 should be \"Lemma\". In eq.(6), $f(x)$ should be $f(Y)$. In the second equality of equation line 631, the integral should be taken over $\\mathcal{M}$; and again line 633. Typo in eq. line 450. Appendix D is not referenced in the text.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors study Frechet regression (a setting where data labels are lying in a non-Euclidean space) and provide a theoretical analysis of it via comparison geometry (i.e., via comparing with known spaces with constant curvature K). The main focus in on CAT(K) spaces, which have thinner triangle when compared with a space with constant curvature K. \n \n \n In particular, the prove statistical convergence results for this problem. In Theorem 1, they prove the distance between the population mean and the empirical mean is small with high probability, for large sample sizes. Next, in Proposition 2, they prove the same results  for L_p norms. Later in Theorem 2 they prove that almost surely the empirical regressor will converge to the population regressor, and finally in Theorem 3 they prove a non asymptotic convergence rate for population risk in Fechet regression. They also provide convergence for angles in Theorem 4\n \n They conclude the paper with experiments.\n\nSTRENGTHS AND WEAKNESSES:\nPros:\n\n- very well written paper\n- nice theoretical results\n- relevant to the conference (specifically geometry/ML intersection)\n \n\nCons:\n\n- barely motivated\n- some assumptions are not well explained (why they make sense?)\n - only focus on CAT(K)\n\nQUESTIONS:\nThis is an interesting paper about regression when labels are non-Euclidean. The authors derive statistical convergence (high probability, L_p, non-asymptotic, etc.) bounds on this problem. They all match our previous understanding from Euclidean spaces, thus extending them to beyond such settings. \n\n\n### Major Comments/Question:\n\n - Missing concrete applications: the paper is not well motivated about why we need going beyond Euclidean labels. Please provide more evidence for that. \n \n - Given a space, how can we ever verify if it is CAT(K)? It looks this assumption is theoretically plausible but hard to see or verify. Please provide evidence why assuming we have CAT(K) is reasonable. \n \n - Section 3.4 is a bit awkward. I can barely understand what the purpose of it is. It seems the authors are using Taylor series but I don't get why. Please explain it. \n \n \n### Other Comments:\n\n- In Definition 1, how do you make sure the comparison triangle exists? How do you find the so-called \"corresponding\" points for x and y in comparison triangle?\n\n- In Definition 3, shouldn't the parameter $\\lambda$ depend also on $f$? They way the definition phrased look like it doesn't. \n\n - In Line 77, are you implicitly assuming that the number K is fixed? Otherwise, what do you mean by comparison triangle there?\n \n \n - Section 3 provides an interesting series of lemmas as background. This is interesting but in my opinion it would be great if you can include exact references for them (where exactly they are appeared). This helps the reader if they want to trace back the previous results. \n \n - When the kernel in Line162 satisfy Assumption 1? Does it always do that? Can you provide a proof? This is essential because it is difficult then to find cases satisfying Assumption 1. \n \n - In Theorem 4, isn't it the case that achieving a small epsilon needs exponentially many samples in dimension $d$ since the optimal transport measure estimation suffers from the curse of dimensionality? In that case the results of Theorem 4 are barely applicable in practice in high dimensions. Can we somehow extend them to a better distance such as sliced Wasserstein?\n\n- In Equation 10, people usually continue analysis by optimizing the bandwidth and obtaining the final rate as a function of sample size. I suggest the authors to do it to make their result more readable.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
In this submission, the author(s) leverage the comparison geometry to understand the important Fréchet regression problem, which provides a rigorous theoretical analysis and statistical guarantees for Fréchet nonparametric regression, including exponential concentration bounds, convergence rates and insights into Alexandrov angle stability. Some numerical validations are also provided to support the theoretical findings.

STRENGTHS AND WEAKNESSES:
Strengths: The manuscript is well-written and a rigorous theoretical analysis and statistical guarantees for Fréchet nonparametric regression based on the comparison geometry.

Weaknesses: I don't see any obvious weakness in the manuscript.

QUESTIONS:
Good work, I don't have other questions.

LIMITATIONS:
The manuscript considers the exact manifold in the manuscript, which may be impractical. So the noised manifold may be a more practical setup and deserves a further investigation in the future. But this is beyond the scope of the current manuscript. So I brought it here just for the potential considerations in the future for the author(s).
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03