Title: Functional data analysis for multivariate distributions through Wasserstein slicing
Venue: NeurIPS 2025
Accepted: yes
Keywords: Distributional data analysis

multivariate distributional data

Radon transform

Sliced Wasserstein distance
Abstract: The modeling of samples of distributions is a major challenge since distributions do not form a vector space. While various approaches exist for univariate distributions, including transformations to a Hilbert space, far less is known about the multivariate case. We utilize a transformation approach to map multivariate distributions to a Hilbert space via a Wasserstein slicing method that is invertible. This approach combines functional data analysis tools, such as functional principal component analysis and modes of variation, with the facility to map back to interpretable distributions. We also provide convergence guarantees for the Hilbert space representations under a broad class of such transforms. The method is illustrated using joint systolic and diastolic blood pressure data.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose an embeddings from the space of multivariate probability distributions into an Hilbert space to enable principal component analysis in functional spaces. While several embeddings of this type exist, the authors emphasize that their proposed embedding is invertible, and therefore allows for the interpretation of modes of variations directly in the space of probability distributions. They also provide the asymptotic convergence of the plug-in empirical estimators of the principal components (i.e. modes of variations), and of the reconstructed density estimators. Their method is illustrated with toy examples and experiments on real-world datasets.\n\nSTRENGTHS AND WEAKNESSES:\nThe main advantage on using the proposed embedding for principal component analysis of probability distributions lies in its invertibility, which is particularly valuable for representing the modes of variation of PCA in the space of probability distributions. Additionally, the rate of convergence of the estimated truncated density is particularly relevant for assessing a reconstruction error of the method.\n\nMy major concern with this paper lies in its comparison to the existing literature. First, in my opinion, the comparison setting is not fair. Indeed, it is well known that performing PCA on densities without accounting for the density constraints leads to poor results. A more relevant baseline would be the classical tangent PCA approach, which consists in mapping the probability measures to the tangent space at the Wasserstein Fréchet mean denoted by $\\bar{f}$ (the reference distribution) using the Monge maps (i.e., optimal transport maps from $\\bar{f}$ to $f_i$), and then performing functional PCA in the space $L^2(\\bar{f})$. This is a standard technique in the analysis of the probability density spade endowed with the Wasserstein distance (see, e.g., [a, b]). Another fair baseline would be to apply a classical kernel mean embedding followed by PCA. Even without the invertibility property, I believe it remains worthwhile to compare the proposed method with these (linear Wasserstein PCA and kernel mean embedding PCA) more robust approaches.\n\nSecondly, a more in-depth analysis of the Gaussian case would have been valuable, especially given the existing literature on this setting. Although the proposed method assumes compact support of the measures, it is still possible to consider truncated Gaussians for comparison. For example, performing PCA directly on the parameters space (mean and covariance) could offer a fair baseline (e.g. with tangent PCA in the Bures Wasserstein space). In the experiments of Setting A, it could have been interesting to visualize the reconstructed $k$-th transformation mode of variation to assess whether the method recovers Gaussian-like behavior. In particular, in my opinion, the importance of embedding invertibility is not adequately emphasized in the experiments.\n\n[a] Wei Wang, Dejan Slepčev, Saurav Basu, John A. Ozolek, and Gustavo K. Rohde. A linear optimal transportation framework for quantifying and visualizing variations in sets of images. International Journal of Computer Vision, 101:254–269, 2013.\n\n[b] Seguy, Vivien, and Marco Cuturi. \"Principal geodesic analysis for probability measures under the optimal transport metric.\" Advances in Neural Information Processing Systems 28 (2015).\n\n\n\n------\n\nEDIT : I have increased my rating from 3 to 4 in light of the authors' response to their contributions, which I had certainly underestimated. I also appreciate that the authors plan to add a comparison with tangent PCA and extend the experiment with Gaussian distributions so that readers can better understand the geometry  of the proposed embedding.\n\nQUESTIONS:\n- Why is it necessary to assume that the support domain $D$ is convex?\n\n- In standard PCA, the explained variance curve typically exhibits an elbow. Did you observe a similar behavior in your experiments?\n\n- I may be missing something, but it is unclear to me whether the embedding proposed in this work is truly original. In what way does it differ from the embedding introduced in [9] for regression purposes?\n\n- Why use the log quantile density transformation instead of the quantile function directly?\n\n- In the Baltimore Longitudinal Study of Aging (BLSA) experiment, one could have approached each point cloud with a Gaussian, and then apply Wasserstein tangent PCA on the estimated covariance instead. How different would have been the results?\n\n- I understand that the family of modes of variation is parametrized by a variable $\\alpha$ within a compact interval $[-\\alpha_0,\\alpha_0]$, which is needed for the proofs. However, what happens as $\\alpha_0$ goes to infinity?\n\n- Typos. Line 183 \"convergene\". Line 503 : Theorem 6 and Corollary 7 do not exist.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}

comments: SUMMARY:
This paper addresses the challenge of analyzing samples of multivariate distributions by proposing a novel, invertible transformation that maps them into a Hilbert space using a Wasserstein slicing approach. This enables the application of functional data analysis tools—such as functional PCA, modes of variation, and regression—while preserving the ability to recover the original distributions. Theoretical convergence results are established, and the method is demonstrated using joint systolic and diastolic blood pressure data.

STRENGTHS AND WEAKNESSES:
This paper extends the univariate log quantile density and log hazard transformation approach to multivariate distributions, enabling the use of functional data analysis techniques—such as FPCA, functional modes of variation, and regression—within a Hilbert space framework. The original densities are recovered via a regularized inverse map, and convergence results are established under regularity conditions. The paper also addresses practical challenges related to estimating multivariate distributions from sampled data. 

Here are some comments based on my reading. 

First, how to handle density functions with multiple isolated support sets? 

Second, another limitation of the Radon transformation is the curse of dimensionality.

QUESTIONS:
no questions.

LIMITATIONS:
yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new method to analyze multivariate probability distributions using functional analysis. The key idea is to slice a distribution into many directions using Radon transform and then apply log quantile/hazard transformation to each slice. Such resulting transformation is invertible so it could be mapped back to the original space for interpretations. The authors also prove theoretical guarantees on asy. convergence and did experiments on simulation data to justify the effectiveness of the proposed method.\n\nSTRENGTHS AND WEAKNESSES:\nStrength: The proposed method is novel and it allows for invertible mapping that allows classical approaches like pca, regression to be applied meaningfully. This does address the limit of current approach.\n\nWeakness: One weakness is that the proposed method by authors might suffer from high dimensional space, as Radon transform requires sample from an informative direction to get a meaningful slice. The experiments are also too simple. I also feel that the paper mainly focuses on combining two existing approaches for application, although it provides some theoretical analysis on asymptotic convergence, the depth of theory may not fully meet the expectation and emphasis typically preferred by NeurIPS.\n\nQUESTIONS:\n1.  How can be the method applied to high dimensional space if it is difficult to get a good slice from Radon transform? How many samples of slice do you need to make sure the method is working?\n\n2. Can the author explain more on the implications of assumptions made for convergence? I'm not really familiar with this proof and it could be helpful to explain more get me have a better understanding.\n\n3. The asymptoic convergence is based on inf sample size. How it applies to finite sample size? I think this is important as in practice, you have limited data usually.\n\n4. Line 160 mentioned \"data application play an important role\". This looks like disconnected from Section 4.2. Can you explain the important role specifically?\n\n5. I'm not familiar with functional approaches on PCA, but I think the experiment comparison on the standard FPCA and the proposed method seems to be not enough. I asked LLM if there are other methods, it gives me \"Wasserstein geodesic PCA or tangent space PCA\". Can the author explain it to me if it could be possible to extend the experiment sections and if not why the above methods are not available for comparison?\n\nLIMITATIONS:\nsee above", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe submission presents an invertible map from the space of multivariate probability distributions to a Hilbert space,  a method to determine modes of variation, and a means of computing these empirically from data. The result builds primarily on the univariate  Hilbert space embedding [33] by Petersen and M'\"uller, and draws significantly on [9] for the multivariate case.\n\nSection 2.2 reviews the Radon transform. Since the Radon transform is unstable, in eq. 4 a low pass filter is applied, and the resulting function renormalized to be a valid density function (eq. 5).  The review is very similar to that in [9].\n\nSection 3.3 introduces the Hilbert Space embedding, which is the univariate embedding of the  Radon transform. Section 4.1 recalls the sliced Wasserstein distance, as similarly reviewed in [9, Section 3], and relates it to the Hilbert space embedding of Section 3.3.\n\nSection 4.2 recalls from [33] the method of obtaining \"modes of variation\" of the Hilbert space embeddings.\n\nIn practice, it is usual to have access to data, and not closed-form densities. Thus in Section 5, approximate densities are obtained via a kernel density estimate on the data, and the modes of variation are computed on the basis of these estimates. \n\nSection 6 describes convergence of the modes of variation, under the estimation strategies and with the assumptions made in the paper.  I did not verify the claims. \n\nSection 7 describes synthetic experiments against functional PCA directly on kernel density estimates, treating them as elements of L2, and comparing with RLQD. The fraction of variance explained is higher for the new method.\n\nSection 8 shows a 2-D blood pressure distribution experiment, where the first two components are sufficient to reconstruct most of the variation, and reveal the nature of changes in blood pressure distribution with age.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n----\n\nThe paper clearly presents the sliced Wasserstein distance, and demonstrates how to use the Hilbert Space distribution representation implicit in this distance in order to discover modes of variation. The procedure is shown to give a small gain in explaining variance on synthetic data, compared with functional PCA on the density functions. On real data experiments, the resulting modes of variation are interpretable. \n\nThe main contribution of this submission appears to be primarily the calculation of the modes of variation in Section (4.2) their  corresponding empirical estimates in Section 5, the convergence results for these modes in Section 6, and the experimental evaluation of the proposed method. \n\nThe generalization of the modes of variation in [33] to the multivariate representation in [9], and the corresponding empirical estimates, are a sufficient contribution for NeurIPS. \n\nWeaknesses (theory)\n-----\n\nThe abstract makes stronger claims than the paper warrants. First, the claim that the multivariate transformation is \"introduced\" in this submission is overly strong in light of the results of [9] (which the authors cite) - see  the paper summary above for details. Second, \"functional regression\" is claimed as a contribution, however this does not appear in the paper. If it did, then a NeurIPS submission should demonstrate a thorough experimental comparison against relevant ML methods that solve the same problem, such as\n\nZoltán Szabó, Bharath K. Sriperumbudur, Barnabás Póczos, Arthur Gretton\nLearning Theory for Distribution Regression\nJMLR 17(152):1−40, 2016\n\nwhich the authors of [33] are aware of.\n\nThe paper suffers from lossy compression of some aspects of the manuscripts [33,9].   The paper also reads like a longer journal paper rearranged for conference submission. This was not always done with sufficient care, for instance with an entire Appendix H Temperature case study never mentioned in the main document.\n\nWeaknesses (experiment)\n----\n\nFigure 2: performance for RLQD and FPCA seems very similar for cases A,B and K=1,2. Only for K=3 is there a larger difference. Thus the advantage on synthetic data seems marginal.\n\nFigure 3: the argument is made that RLQD is useful as it provides an interpretable functional PCA decomposition of the densities. However, for the blood pressure dataset, would FPCA provide an equally interpretable visualization? On real data is there any significant difference in variance explained between RLQD and FPCA?  It is notable that this is omitted, which weakens the claims made for the method.\n\nQUESTIONS:\nl. 79: should be l_[z,\\theta} ?  It seems this is an error copied over from [9 p. 5].   In any case not sure why notation l_{z,\\theta} is actually needed in the neurips submission.\n\nThe notation J_1 in eq. 4 is undefined (we are told J in eq. 3 denotes the Fourier transform). I assume this notation is taken from [9 p. 5], i.e. the 1-D Fourier transform?\n\nReferences [36] and [37] are the same.\n\nLIMITATIONS:\nnot applicable", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-05"}