Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance

Tanguy, Eloi; Flamary, Rémi; Delon, Julie

Comptes Rendus. Mathématique, Tome 362 (2024) no. G10, pp. 1121-1129

This paper deals with the reconstruction of a discrete measure ${\gamma }_Z$ on ${ℝ}^d$ from the knowledge of its pushforward measures $P_i\#{\gamma }_Z$ by linear applications $P_i:{ℝ}^d\to {ℝ}^{d_i}$ (for instance projections onto subspaces). The measure ${\gamma }_Z$ being fixed, assuming that the rows of the matrices $P_i$ are independent realizations of laws which do not give mass to hyperplanes, we show that if ${\sum }_i{d}_i>d$, this reconstruction problem has almost certainly a unique solution. This holds for any number of points in ${\gamma }_Z$. A direct consequence of this result is an almost-sure separability property on the empirical Sliced Wasserstein distance.

On s’intéresse dans cet article au problème de reconstruction d’une mesure discrète ${\gamma }_Z$ sur ${ℝ}^d$ connaissant ses images par des applications linéaires $P_i:{ℝ}^d\to {ℝ}^{d_i}$ (par exemple des projections sur des sous-espaces). La mesure ${\gamma }_Z$ étant fixée, en supposant que les lignes des matrices $P_i$ sont des réalisations indépendantes de lois ne donnant pas de masse aux hyperplans, on montre que si ${\sum }_i{d}_i>d$, ce problème de reconstruction a presque sûrement une unique solution, et ceci quelque soit le nombre de points dans ${\gamma }_Z$. Ce résultat permet de démontrer une propriété de séparabilité presque sûre pour la distance de Sliced–Wasserstein empirique.

Classification : 28E99, 15A29
Keywords: Reconstruction, Inverse Problems, Discrete Measures
Mots-clés : Reconstruction, problèmes inverses, mesures discrètes

[1] Bonneel, Nicolas; Rabin, Julien; Peyré, Gabriel; Pfister, Hanspeter Sliced and Radon Wasserstein barycenters of measures, J. Math. Imaging Vis., Volume 51 (2015) no. 1, pp. 22-45 | DOI | MR | Zbl

[2] Chafaï, Dajlil Random projections, marginals, and moments (https://djalil.chafai.net/docs/projections.pdf)

[3] Cuadras, C. M. Probability distributions with given multivariate marginals and given dependence structure, J. Multivariate Anal., Volume 42 (1992) no. 1, pp. 51-66 | DOI | MR | Zbl

[4] Cramér, H.; Wold, H. Some Theorems on Distribution Functions, J. Lond. Math. Soc., Volume 11 (1936) no. 4, pp. 290-294 | DOI | MR | Zbl

[5] Dall’Aglio, Giorgio; Kotz, Samuel; Salinetti, Gabriella Advances in probability distributions with given marginals. Beyond the copulas, Mathematics and its Applications, 67, Springer, 2012 | DOI

[6] Deshpande, Ishan; Zhang, Ziyu; Schwing, Alexander G. Generative Modeling Using the Sliced Wasserstein Distance, 2018 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2018, Salt Lake City, UT, USA, June 18-22, 2018, Computer Vision Foundation / IEEE Computer Society (2018), pp. 3483-3491 | DOI

[7] Fernández-Bertolin, Aingeru; Jaming, Philippe; Gröchenig, Karlheinz Determining point distributions from their projections, 2017 International Conference on Sampling Theory and Applications (SampTA) (2017), pp. 164-168 | DOI

[8] Gardner, Richard J.; Gritzmann, Peter Uniqueness and complexity in discrete tomography, Discrete tomography (Applied and Numerical Harmonic Analysis), Birkhäuser, 1999, pp. 85-113 | DOI | MR | Zbl

[9] Gladkov, Nikita A.; Kolesnikov, Alexander V.; Zimin, Alexander P. On multistochastic Monge-Kantorovich problem, bitwise operations, and fractals, Calc. Var. Partial Differ. Equ., Volume 58 (2019) no. 5, 173, 33 pages | DOI | MR | Zbl

[10] Heppes, A. On the determination of probability distributions of more dimensions by their projections, Acta Math. Acad. Sci. Hung., Volume 7 (1956), pp. 403-410 | DOI | MR | Zbl

[11] Joe, Harry Parametric families of multivariate distributions with given margins, J. Multivariate Anal., Volume 46 (1993) no. 2, pp. 262-282 | DOI | MR | Zbl

[12] Karras, Tero; Aila, Timo; Laine, Samuli; Lehtinen, Jaakko Progressive Growing of GANs for Improved Quality, Stability, and Variation, International Conference on Learning Representations (2018)

[13] Kellerer, Hans G. Verteilungsfunktionen mit gegebenen Marginalverteilungen, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 3 (1964), pp. 247-270 | DOI | MR | Zbl

[14] Kazi-Tani, Nabil; Rullière, Didier On a construction of multivariate distributions given some multidimensional marginals, Adv. Appl. Probab., Volume 51 (2019) no. 2, pp. 487-513 | DOI | MR | Zbl

[15] Nadjahi, Kimia Sliced-Wasserstein distance for large-scale machine learning: theory, methodology and extensions, Ph. D. Thesis, Institut polytechnique de Paris (2021)

[16] Peyré, Gabriel; Cuturi, Marco Computational Optimal Transport: With Applications to Data Science, Found. Trends Mach. Learn., Volume 11 (2019) no. 5-6, pp. 355-607 | DOI | Zbl

[17] Rabin, Julien; Peyré, Gabriel; Delon, Julie; Bernot, Marc Wasserstein Barycenter and Its Application to Texture Mixing, Scale Space and Variational Methods in Computer Vision (Bruckstein, Alfred M.; ter Haar Romeny, Bart M.; Bronstein, Alexander M.; Bronstein, Michael M., eds.), Springer, 2012, pp. 435-446 | DOI

[18] Rényi, A. On projections of probability distributions, Acta Math. Acad. Sci. Hung., Volume 3 (1952), pp. 131-142 | DOI | MR | Zbl

[19] Tanguy, Eloi Convergence of sgd for training neural networks with sliced Wasserstein losses (2023) (https://arxiv.org/abs/2307.11714)

[20] Tachella, Julián; Chen, Dongdong; Davies, Mike Sensing theorems for unsupervised learning in linear inverse problems, J. Mach. Learn. Res., Volume 24 (2023), 39, 45 pages | DOI | MR | Zbl

[21] Tanguy, Eloi; Flamary, Rémi; Delon, Julie Properties of discrete sliced Wasserstein losses (2023)

[22] Wu, Jiqing; Huang, Zhiwu; Acharya, Dinesh; Li, Wen; Thoma, Janine; Paudel, Danda Pani; Van Gool, Luc Sliced Wasserstein Generative Models, 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2019), pp. 3708-3717 | DOI