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arxiv: 2509.10415 · v2 · submitted 2025-09-12 · 🧮 math.NA · cs.NA

Multiscaling in Wasserstein Spaces

Wael Mattar , Nir Sharon This is my paper

Pith reviewed 2026-05-18 17:23 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords multiscale transformWasserstein spaceMcCann interpolantoptimality numbergeodesic structureprobability measuresanomaly detectionmeasure flows
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The pith

A multiscale transform for Wasserstein measures preserves geodesics via McCann interpolants and introduces an optimality number for scale-wise deviation measurement.

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

The paper develops a multiscale analysis method for sequences of probability measures equipped with the Wasserstein distance. It defines a refinement operator that upsamples measures while keeping their flow along geodesics. From this operator it derives the optimality number, which measures how far a sequence departs from being a geodesic at each resolution level. This framework applies equally to continuous densities and discrete point sets, and the authors show it remains stable with coefficients that decay geometrically. Readers may find it useful for tasks that involve tracking how data distributions evolve over time or under transformations.

Core claim

We construct a multiscale transform applicable to both absolutely continuous and discrete measures. Central to our approach is a refinement operator based on McCann's interpolants, which preserves the geodesic structure of measure flows and serves as an upsampling mechanism. Building on this, we introduce the optimality number, a scalar that quantifies deviations of a sequence from Wasserstein geodesicity across scales.

What carries the argument

The refinement operator based on McCann's interpolants, which acts as an upsampling mechanism that preserves geodesic structure in measure flows, enabling the multiscale transform and the computation of the optimality number.

If this is right

  • The multiscale transform is stable.
  • Coefficients exhibit geometric decay.
  • It enables denoising and anomaly detection in Gaussian flows.
  • Point cloud dynamics under vector fields can be analyzed at multiple scales.
  • Neural network learning trajectories admit a multiscale characterization.

Where Pith is reading between the lines

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

  • The optimality number might serve as a general diagnostic for irregular dynamics in measure-valued time series from any source.
  • The same refinement construction could extend to other optimal transport metrics if an analogous interpolant exists.
  • Applications to fluid simulations or population dynamics would test whether the geometric decay holds in practice.
  • Connecting the optimality number to classical irregularity measures in dynamical systems could link this work to existing anomaly tools.

Load-bearing premise

The refinement operator based on McCann's interpolants preserves the geodesic structure of measure flows for arbitrary sequences of measures and can be used as a stable upsampling mechanism without introducing artifacts that affect the optimality number at finer scales.

What would settle it

Apply the refinement operator to a sequence of discrete measures known to deviate from a Wasserstein geodesic, then recompute the optimality number at the finer scales; an unexpected increase due to the operator itself would show that artifacts are introduced.

Figures

Figures reproduced from arXiv: 2509.10415 by Nir Sharon, Wael Mattar.

Figure 1
Figure 1. Figure 1: Illustration of the discrete ⊖ operator and McCann’s average. On the left, the original source and target measures with uniform distribution over 40 and 20 points in R 2 , respectively. On the middle, the gray vectors depict the optimal transport plan for the quadratic cost between the measures. The difference ⊖ encodes these vectors in addition to the masses transported along each vector, 1/40 in this cas… view at source ↗
Figure 2
Figure 2. Figure 2: Analysis of the smooth Gaussian curve {µbt}. On the left, a curve of Gaussian measures with parameters that vary smoothly. On the right, norms of the detail coefficients ψ (ℓ) obtained by the elementary multiscale representation (23). Note the decay of the maximal norm with each layer of details. The color coding in both figures correspond to each other. to create a discrepancy between {µbt} and the geodes… view at source ↗
Figure 3
Figure 3. Figure 3: Analysis of a noisy Gaussian curve. On the left, the smooth curve of Gaussian measures {µbt} but with parameters contaminated with noise. On the right, norms of the detail coefficients ψ (ℓ) obtained by the elementary multiscale representation (23). The norms show no geometric decay, and, they have high values even on high scales. This indicates the noisy texture of the curve. The color coding in both figu… view at source ↗
Figure 4
Figure 4. Figure 4: Denoised Gaussian curve. On the left, the result of denoising the curve that appears in [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Anomaly detection in Gaussian curve. The locations of two jump dis￾continuities of a Gaussian curve are revealed by the elementary multiscale trans￾form (23). k = 0 ω = 0 k = 0.25 ω = 0.5013 k = 0.5 ω = 0.9934 k = 0.75 ω = 1.4762 k = 1 ω = 1.9503 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Weighted averages between a curve connecting two measures and their geodesic. Five members of the family µ [k] t of (47) are illustrated with their respective optimality numbers. Now, we compute the weighted averages between the geodesic {µt} and {µbt} which both connect the initial and the final measures µ0 and µ1. Namely, define the family of curves µ [k] by (47) µ [k] t = (1 − k)µt + kµbt , (k, t) ∈ [0,… view at source ↗
Figure 7
Figure 7. Figure 7: Maximal error against different detail layers ℓ on the logarithmic scale. The geometric decay of the maximal norm of the detail coefficient of five members of the family µ [k] t of (47). 6.2. Curves of point clouds. In this section we demonstrate the elementary multiscale transform for discrete measures with free support on an example from physics. Sequences in this subsection form a point cloud that evolv… view at source ↗
Figure 8
Figure 8. Figure 8: Multiscaling a geodesic of discrete measures in P2(R 2 ). On the left, the trajectories of the 10 particles along the electric field (48). On the right, norms of the 6 layers of detail coefficients obtained by the multiscaling (23) of the geodesic. Because some particles began their movement near the positive charge, the detail norms are salient on the left endpoint of the pyramid representation. The optim… view at source ↗
Figure 9
Figure 9. Figure 9: Multiscaling a noisy sequence of discrete measures in P2(R 2 ). On the left, the trajectories of the 10 particles along the electric field (48). On the right, norms of the 6 layers of detail coefficients obtained by the multiscaling (23) of the clouds. Because all particles are pushed farther from both charges, the detail coefficients exhibit geometric decay along the time axis. Due to the added noise, the… view at source ↗
Figure 10
Figure 10. Figure 10: The coarse approximations of the curves appearing in Figures 8 and 9. 11 point clouds each consisting of 10 atoms with uniform distribution. The takeaway message of the two experiments presented in this section is as follows. The elementary multiscale transform (23) can be used to study how smooth point clouds evolve over time. In particular, the faster the detail coefficients decay in scale, the smoother… view at source ↗
Figure 11
Figure 11. Figure 11: Analysis of the learning dynamics of a neural network. The heat map on the left depicts the mean probability of predicting the digit “3” by the end of each epoch, over 161 epochs. On the right, the norms of the detail coefficients (23) over 4 layers. In the early stages of learning, the distribution is more or less uniform, and as learning advances, the distribution converges to Dirac’s measure over the s… view at source ↗
read the original abstract

We present a novel multiscale framework for analyzing sequences of probability measures in Wasserstein spaces over Euclidean domains. Exploiting the intrinsic geometry of optimal transport, we construct a multiscale transform applicable to both absolutely continuous and discrete measures. Central to our approach is a refinement operator based on McCann's interpolants, which preserves the geodesic structure of measure flows and serves as an upsampling mechanism. Building on this, we introduce the optimality number, a scalar that quantifies deviations of a sequence from Wasserstein geodesicity across scales, enabling the detection of irregular dynamics and anomalies. We establish key theoretical guarantees, including stability of the transform and geometric decay of coefficients, ensuring robustness and interpretability of the multiscale representation. Finally, we demonstrate the versatility of our methodology through numerical experiments: denoising and anomaly detection in Gaussian flows, analysis of point cloud dynamics under vector fields, and the multiscale characterization of neural network learning trajectories.

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 constructs a multiscale transform for sequences of probability measures in Wasserstein spaces over Euclidean domains, applicable to both absolutely continuous and discrete measures. It introduces a refinement operator based on McCann interpolants that is claimed to preserve geodesic structure and function as stable upsampling, defines an optimality number to quantify deviations from Wasserstein geodesicity across scales, establishes stability of the transform together with geometric decay of coefficients, and illustrates the framework via numerical experiments on denoising/anomaly detection in Gaussian flows, point-cloud dynamics, and neural-network learning trajectories.

Significance. If the central claims hold, the work supplies a new scalar diagnostic (the optimality number) and an upsampling mechanism grounded in optimal transport geometry that could be useful for detecting irregular dynamics in measure-valued sequences. The extension to discrete measures and the reported geometric decay would be concrete strengths if the proofs are robust to non-uniqueness of couplings.

major comments (2)
  1. [§3.2] §3.2 (refinement operator): The construction uses McCann interpolants along optimal plans, but for discrete measures the optimal coupling is frequently non-unique. The manuscript does not specify a canonical selection rule nor prove that the optimality number is invariant under different choices of the coupling. This directly affects the claim that the operator preserves geodesic structure for arbitrary sequences and serves as a stable upsampling mechanism without introducing scale-dependent artifacts.
  2. [Theorem 5.3] Theorem 5.3 (geometric decay): The decay estimate for the optimality-number coefficients is stated to hold uniformly, yet it appears to rest on the refinement operator being well-defined and geodesic-preserving for discrete measures. If different couplings produce different interpolated measures at intermediate scales, the decay rate may depend on the (unspecified) choice and the theorem would require an additional invariance argument.
minor comments (2)
  1. [§4] Notation for the optimality number is introduced without an explicit formula in the main text; a displayed equation would improve readability.
  2. [Figure 4] Figure 4 (point-cloud experiment) lacks error bars or multiple random seeds; the reported optimality numbers appear sensitive to initialization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the refinement operator and the geometric decay result. We address the two major comments point by point below and will revise the manuscript accordingly to strengthen the treatment of discrete measures.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (refinement operator): The construction uses McCann interpolants along optimal plans, but for discrete measures the optimal coupling is frequently non-unique. The manuscript does not specify a canonical selection rule nor prove that the optimality number is invariant under different choices of the coupling. This directly affects the claim that the operator preserves geodesic structure for arbitrary sequences and serves as a stable upsampling mechanism without introducing scale-dependent artifacts.

    Authors: We agree that non-uniqueness of optimal couplings must be handled explicitly for discrete measures. The current manuscript assumes an optimal plan exists but does not detail a selection procedure. In the revision we will add a canonical selection rule in §3.2 (for example, the coupling obtained as the zero-temperature limit of entropic optimal transport, or the one minimizing a secondary quadratic cost among all optimal plans). We will also insert a short invariance lemma showing that the optimality number depends only on the Wasserstein distances between the interpolated marginals and is therefore independent of the particular choice of optimal coupling. revision: yes

  2. Referee: [Theorem 5.3] Theorem 5.3 (geometric decay): The decay estimate for the optimality-number coefficients is stated to hold uniformly, yet it appears to rest on the refinement operator being well-defined and geodesic-preserving for discrete measures. If different couplings produce different interpolated measures at intermediate scales, the decay rate may depend on the (unspecified) choice and the theorem would require an additional invariance argument.

    Authors: We acknowledge that the proof of Theorem 5.3 implicitly relies on the refinement operator being unambiguously defined. The revised manuscript will include the invariance lemma mentioned above and will use it to show that the coefficients entering the geometric decay bound are the same for any choice of optimal coupling. With this addition the uniform decay statement remains valid for both absolutely continuous and discrete measures. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from Wasserstein geometry and McCann interpolants

full rationale

The paper constructs the multiscale transform and optimality number directly from the intrinsic geometry of optimal transport and McCann's interpolants, which are standard external results. No equation reduces the optimality number to a fitted parameter by construction, no self-citation chain bears the central load, and the refinement operator is asserted to preserve geodesicity without the claim itself being defined in terms of the output scalar. The framework supplies independent stability guarantees and numerical validation, making the derivation non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on standard properties of Wasserstein space and McCann interpolation (prior literature) plus two new constructs: the refinement operator and the optimality number. No free parameters are explicitly fitted in the abstract description. The optimality number is an invented scalar whose independent evidence would come from its behavior on known geodesic sequences.

axioms (1)
  • domain assumption McCann's interpolants preserve geodesic structure in the Wasserstein space for sequences of measures
    Invoked as the basis for the refinement operator that serves as upsampling while preserving geodesicity.
invented entities (1)
  • optimality number no independent evidence
    purpose: Scalar quantifying deviations of a measure sequence from Wasserstein geodesicity across scales
    New diagnostic introduced to detect irregular dynamics and anomalies; no independent falsifiable prediction outside the paper is stated in the abstract.

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    Relation between the paper passage and the cited Recognition theorem.

    Central to our approach is a refinement operator based on McCann's interpolants, which preserves the geodesic structure of measure flows and serves as an upsampling mechanism... optimality number, a scalar that quantifies deviations of a sequence from Wasserstein geodesicity across scales

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Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

  1. [1]

    Barycenters in the Wasserstein space.SIAM Journal on Mathematical Analysis, 43(2):904–924, 2011

    Martial Agueh and Guillaume Carlier. Barycenters in the Wasserstein space.SIAM Journal on Mathematical Analysis, 43(2):904–924, 2011

    work page 2011

  2. [2]

    Springer Science & Business Media, 2008

    Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar´ e.Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media, 2008

    work page 2008

  3. [3]

    Subdivision scheme for discrete probability measure- valued data.Applied Mathematics Letters, 158:109233, 2024

    Jean Baccou and Jacques Liandrat. Subdivision scheme for discrete probability measure- valued data.Applied Mathematics Letters, 158:109233, 2024

    work page 2024

  4. [4]

    Efficient trajectory inference in Wasserstein space using consecutive averaging

    Amartya Banerjee, Harlin Lee, Nir Sharon, and Caroline Moosm¨ uller. Efficient trajectory inference in Wasserstein space using consecutive averaging. InThe 28th International Con- ference on Artificial Intelligence and Statistics, 2025

    work page 2025

  5. [5]

    Statistical data analysis in the Wasserstein space.ESAIM: Proceedings and Surveys, 68:1–19, 2020

    J´ er´ emie Bigot. Statistical data analysis in the Wasserstein space.ESAIM: Proceedings and Surveys, 68:1–19, 2020

    work page 2020

  6. [6]

    Fast and scalable Wasserstein-1 neural optimal transport solver for single-cell perturbation prediction.Bioinformatics, 41(Sup- plement 1):i513–i522, 2025

    Yanshuo Chen, Zhengmian Hu, Wei Chen, and Heng Huang. Fast and scalable Wasserstein-1 neural optimal transport solver for single-cell perturbation prediction.Bioinformatics, 41(Sup- plement 1):i513–i522, 2025

    work page 2025

  7. [7]

    Wasserstein regression.Journal of the American Statistical Association, 118(542):869–882, 2023

    Yaqing Chen, Zhenhua Lin, and Hans-Georg M¨ uller. Wasserstein regression.Journal of the American Statistical Association, 118(542):869–882, 2023

    work page 2023

  8. [8]

    Fast and smooth interpolation on Wasserstein space

    Sinho Chewi, Julien Clancy, Thibaut Le Gouic, Philippe Rigollet, George Stepaniants, and Austin Stromme. Fast and smooth interpolation on Wasserstein space. InInternational Conference on Artificial Intelligence and Statistics, pages 3061–3069. PMLR, 2021

    work page 2021

  9. [9]

    SIAM, 1992

    Ingrid Daubechies.Ten lectures on wavelets. SIAM, 1992

    work page 1992

  10. [10]

    Gromov-Wasserstein optimal transport to align single-cell multi-omics data.BioRxiv, pages 2020–04, 2020

    Pinar Demetci, Rebecca Santorella, Bj¨ orn Sandstede, William Stafford Noble, and Ritambhara Singh. Gromov-Wasserstein optimal transport to align single-cell multi-omics data.BioRxiv, pages 2020–04, 2020

    work page 2020

  11. [11]

    The MNIST database of handwritten digit images for machine learning research

    Li Deng. The MNIST database of handwritten digit images for machine learning research. IEEE signal processing magazine, 29(6):141–142, 2012

    work page 2012

  12. [12]

    Interpolating wavelet transforms.Preprint, Department of Statistics, Stan- ford University, 2(3):1–54, 1992

    David L Donoho. Interpolating wavelet transforms.Preprint, Department of Statistics, Stan- ford University, 2(3):1–54, 1992. 24 W. MATTAR AND N. SHARON

    work page 1992

  13. [13]

    Dowson and Basil V

    David C. Dowson and Basil V. Landau. The Fr´ echet distance between multivariate normal distributions.Journal of multivariate analysis, 12(3):450–455, 1982

    work page 1982

  14. [14]

    Subdivision schemes in CAGD.Advances in numerical analysis, 2:36–104, 1992

    Nira Dyn. Subdivision schemes in CAGD.Advances in numerical analysis, 2:36–104, 1992

    work page 1992

  15. [15]

    Subdivision schemes in geometric modelling.Acta Numerica, 11:73–144, 2002

    Nira Dyn and David Levin. Subdivision schemes in geometric modelling.Acta Numerica, 11:73–144, 2002

    work page 2002

  16. [16]

    Manifold-valued subdivision schemes based on geodesic inductive averaging.Journal of Computational and Applied Mathematics, 311:54–67, 2017

    Nira Dyn and Nir Sharon. Manifold-valued subdivision schemes based on geodesic inductive averaging.Journal of Computational and Applied Mathematics, 311:54–67, 2017

    work page 2017

  17. [17]

    Subdivision schemes in metric spaces.preprint arXiv:2509.08070, 2025

    Nira Dyn and Nir Sharon. Subdivision schemes in metric spaces.preprint arXiv:2509.08070, 2025

    work page arXiv 2025

  18. [18]

    Nira Dyn and Xiaosheng Zhuang. Linear multiscale transforms based on even-reversible subdi- vision operators.Excursions in Harmonic Analysis, Volume 6: In Honor of John Benedetto’s 80th Birthday, pages 297–319, 2021

    work page 2021

  19. [19]

    Conditional Wasserstein barycenters and interpola- tion/extrapolation of distributions.IEEE Transactions on Information Theory, 2024

    Jianing Fan and Hans-Georg M¨ uller. Conditional Wasserstein barycenters and interpola- tion/extrapolation of distributions.IEEE Transactions on Information Theory, 2024

    work page 2024

  20. [20]

    Stability of manifold-valued subdivision schemes and multiscale transforma- tions.Constructive approximation, 32:569–596, 2010

    Philipp Grohs. Stability of manifold-valued subdivision schemes and multiscale transforma- tions.Constructive approximation, 32:569–596, 2010

    work page 2010

  21. [21]

    Improved training of Wasserstein gans.Advances in neural information processing systems, 30, 2017

    Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville. Improved training of Wasserstein gans.Advances in neural information processing systems, 30, 2017

    work page 2017

  22. [22]

    Manifold learning in Wasserstein space.SIAM Journal on Mathematical Analysis, 57(3):2983–3029, 2025

    Keaton Hamm, Caroline Moosm¨ uller, Bernhard Schmitzer, and Matthew Thorpe. Manifold learning in Wasserstein space.SIAM Journal on Mathematical Analysis, 57(3):2983–3029, 2025

    work page 2025

  23. [23]

    GENOT: Entropic (Gromov) Wasserstein flow matching with applications to single-cell genomics.Advances in Neural Information Processing Systems, 37:103897–103944, 2024

    Dominik Klein, Th´ eo Uscidda, Fabian Theis, and Marco Cuturi. GENOT: Entropic (Gromov) Wasserstein flow matching with applications to single-cell genomics.Advances in Neural Information Processing Systems, 37:103897–103944, 2024

    work page 2024

  24. [24]

    Jeffrey M Lane and Richard F Riesenfeld. A theoretical development for the computer gener- ation and display of piecewise polynomial surfaces.IEEE Transactions on Pattern Analysis and Machine Intelligence, pages 35–46, 1980

    work page 1980

  25. [25]

    Pyramid transform of manifold data via subdivision operators

    Wael Mattar and Nir Sharon. Pyramid transform of manifold data via subdivision operators. IMA Journal of Numerical Analysis, 43(1):387–413, 2023

    work page 2023

  26. [26]

    A convexity principle for interacting gases.Advances in mathematics, 128(1):153–179, 1997

    Robert J McCann. A convexity principle for interacting gases.Advances in mathematics, 128(1):153–179, 1997

    work page 1997

  27. [27]

    The geometry of dissipative evolution equations: The porous medium equation

    Felix Otto. The geometry of dissipative evolution equations: The porous medium equation. Communications in Partial Differential Equations, 26(1-2):101–174, 2001

    work page 2001

  28. [28]

    Springer Nature, 2020

    Victor M Panaretos and Yoav Zemel.An invitation to statistics in Wasserstein space. Springer Nature, 2020

    work page 2020

  29. [29]

    Computational optimal transport: With applications to data science.Foundations and Trends in Machine Learning, 11(5-6):355–607, 2019

    Gabriel Peyr´ e, Marco Cuturi, et al. Computational optimal transport: With applications to data science.Foundations and Trends in Machine Learning, 11(5-6):355–607, 2019

    work page 2019

  30. [30]

    Wasserstein barycenter and its application to texture mixing

    Julien Rabin, Gabriel Peyr´ e, Julie Delon, and Marc Bernot. Wasserstein barycenter and its application to texture mixing. InInternational conference on scale space and variational methods in computer vision, pages 435–446. Springer, 2011

    work page 2011

  31. [31]

    Mul- tiscale representations for manifold-valued data.Multiscale Modeling & Simulation, 4(4):1201– 1232, 2005

    Inam Ur Rahman, Iddo Drori, Victoria C Stodden, David L Donoho, and Peter Schr¨ oder. Mul- tiscale representations for manifold-valued data.Multiscale Modeling & Simulation, 4(4):1201– 1232, 2005

    work page 2005

  32. [32]

    Introduction to Optimal Transport Theory

    Filippo Santambrogio. Introduction to optimal transport theory.preprint arXiv:1009.3856, 2010

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  33. [33]

    Optimal transport for applied mathematicians.Birk¨ auser, NY, 55(58- 63):94, 2015

    Filippo Santambrogio. Optimal transport for applied mathematicians.Birk¨ auser, NY, 55(58- 63):94, 2015. MULTISCALING IN W ASSERSTEIN SPACES 25

    work page 2015

  34. [34]

    Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming.Cell, 176(4):928–943, 2019

    Geoffrey Schiebinger, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subramanian, Aryeh Solomon, Joshua Gould, Siyan Liu, Stacie Lin, Peter Berube, et al. Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming.Cell, 176(4):928–943, 2019

    work page 2019

  35. [35]

    American Mathematical Soc., 2021

    C´ edric Villani.Topics in optimal transportation, volume 58. American Mathematical Soc., 2021

    work page 2021

  36. [36]

    Geometric subdivision and multiscale transforms

    Johannes Wallner. Geometric subdivision and multiscale transforms. InHandbook of Varia- tional Methods for Nonlinear Geometric Data, pages 121–152. Springer, 2020

    work page 2020

  37. [37]

    Application of opti- mal transport and the quadratic Wasserstein metric to full-waveform inversion.Geophysics, 83(1):R43–R62, 2018

    Yunan Yang, Bj¨ orn Engquist, Junzhe Sun, and Brittany F Hamfeldt. Application of opti- mal transport and the quadratic Wasserstein metric to full-waveform inversion.Geophysics, 83(1):R43–R62, 2018. Appendix We prove the inequalities (38). Letµ, ν∈ P p(Rd) for somep >1 andψ, eψ:R d →R d be two measurable Lipschitz maps. We have Wp(ψ#µ,eψ#µ)≤ ∥ψ− eψ∥Lp(µ) ...

    work page 2018