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arxiv: 2605.18587 · v1 · pith:YQ22GJTZnew · submitted 2026-05-18 · 🧬 q-bio.GN · cs.LG

PACE: Geometry-Aware Bridge Transport for Single-Cell Trajectory Inference

Chenglei Yu* , Chuanrui Wang* , Bangyan Liao , Tailin Wu This is my paper

Pith reviewed 2026-05-20 01:20 UTC · model grok-4.3

classification 🧬 q-bio.GN cs.LG
keywords single-cell trajectory inferenceoptimal transportRiemannian metricneural bridgesvelocity fielddevelopmental dynamicsasynchronous trajectories
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The pith

PACE recovers continuous cell trajectories from snapshots by building a time-varying anisotropic metric that favors local developmental directions.

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

Single-cell trajectory inference from destructive snapshots is ill-posed because neither cell correspondences nor continuous paths are observed. Existing methods couple cells by Euclidean proximity, which misaligns trajectories when development proceeds asynchronously. PACE instead constructs a state- and time-dependent anisotropic Riemannian metric that lowers transport cost along locally supported tangent directions and raises it for normal components. It alternates between refining cross-time couplings under the induced path cost and fitting endpoint-preserving neural bridges, then distills the result into a global continuous velocity field. On seven datasets and nine held-out reconstructions, this yields lower MMD and Wasserstein distances than prior baselines while also improving RNA-velocity alignment.

Core claim

PACE shows that a state- and time-dependent anisotropic Riemannian metric can be used to define path-action costs that enforce geometry-consistent couplings between snapshots; alternating optimization between these costs and neural bridge fitting then produces a distilled global velocity field that reconstructs held-out trajectories more accurately than Euclidean optimal transport or flow-based baselines.

What carries the argument

A state- and time-dependent anisotropic Riemannian metric that assigns low transport cost along locally supported tangent directions while penalizing normal velocity components.

If this is right

  • Reduces MMD, Wasserstein-1, and Wasserstein-2 distances by 23.7 percent on average across nine reconstruction experiments on seven datasets.
  • Improves alignment with measured RNA velocity by 15.4 percent on an embryoid body differentiation benchmark.
  • Recovers continuous dynamics without requiring explicit cell pairing, lineage tracing, or RNA-velocity supervision during training.

Where Pith is reading between the lines

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

  • The same metric construction could be tested on other asynchronous dynamical systems outside single-cell biology where local geometry is known to matter.
  • Datasets that supply ground-truth continuous trajectories would allow direct measurement of how much the anisotropy assumption improves path accuracy versus endpoint matching alone.
  • Combining the distilled velocity field with multi-omics measurements could yield joint trajectory models across transcriptomic and proteomic layers.

Load-bearing premise

That a suitable anisotropic metric reflecting local tangent directions of development can be constructed from the observed snapshots alone.

What would settle it

A controlled simulation in which true trajectories follow known curved paths but the learned metric forces straighter couplings would produce higher reconstruction errors than Euclidean baselines on the same data.

Figures

Figures reproduced from arXiv: 2605.18587 by Bangyan Liao, Chenglei Yu*, Chuanrui Wang*, Tailin Wu.

Figure 1
Figure 1. Figure 1: Overview of PACE. PACE uses local PCA to construct an anisotropic metric Gk(x, t) = I + αC(k) N (x, t), trains endpoint-preserving neural bridges under the corresponding path-action cost, iteratively refines cross-time couplings, and distills the learned bridge dynamics into a global velocity field for trajectory inference from unpaired snapshots. to non-gradient dynamics using approximate velocity informa… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the 2D benchmark datasets. Points are colored by observed time for Ocean [ [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Velocity-alignment diagnostics on Ocean [ [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: High-dimensional concentra￾tion diagnostics. Dashed lines mark norm CV = 0.3 and inter-time/within￾time ratio = 1.0. 6 11 16 Test Timepoint 0.1 0.2 0.3 0.4 0.5 MMD 6 11 16 Test Timepoint 0.15 0.20 0.25 0.30 0.35 W1 6 11 16 Test Timepoint 0.20 0.25 0.30 0.35 0.40 0.45 W2 PACE No Rematch =0 (Euclidean) KNN = All Spatial Kernel Temporal Kernel [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original abstract

Single-cell trajectory inference from destructive time-course snapshots is fundamentally ill-posed: neither cross-time cell correspondences nor continuous trajectories are observed, so the snapshot distributions alone do not uniquely determine the underlying dynamics. Existing optimal transport and flow-based methods typically couple cells by Euclidean proximity at observed clock times, which can misalign trajectories when development is asynchronous and cells sampled at the same experimental time occupy different latent pseudotime stages. We propose PACE, a trajectory inference framework that recovers geometry-consistent continuous transport dynamics from destructive time-course snapshots through three coupled components. First, PACE constructs a state- and time-dependent anisotropic Riemannian metric that assigns low transport cost along locally supported tangent directions while penalizing normal velocity components. Second, it alternates between refining cross-time couplings under the induced path-action cost and fitting endpoint-preserving neural bridges between adjacent snapshots. Third, it distills the learned bridge dynamics into a global continuous-time velocity field over cellular states. Across seven controlled and biological datasets covering nine held-out reconstruction experiments, PACE achieves the strongest overall reconstruction performance, reducing MMD, Wasserstein-1 distance, and Wasserstein-2 distance by 23.7% on average relative to the strongest competing baseline. PACE also improves RNA-velocity alignment by 15.4% on an embryoid body differentiation benchmark, without requiring explicit cell pairing, lineage tracing, or RNA-velocity supervision during training. Code is available at https://github.com/AI4Science-WestlakeU/PACE.

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 proposes PACE, a trajectory inference method for single-cell data from destructive time-course snapshots. It constructs a state- and time-dependent anisotropic Riemannian metric that favors transport along locally supported tangent directions, alternates between optimizing cross-time couplings under the induced path cost and fitting endpoint-preserving neural bridges, and distills the dynamics into a global continuous-time velocity field. On seven controlled and biological datasets with nine held-out reconstruction experiments, it reports a 23.7% average reduction in MMD, Wasserstein-1, and Wasserstein-2 distances relative to the strongest baseline, plus a 15.4% improvement in RNA-velocity alignment on an embryoid-body benchmark, without requiring cell pairing or velocity supervision.

Significance. If the central geometry-consistency claim holds, PACE would address a key limitation of Euclidean OT and flow-based methods in asynchronous developmental settings by penalizing normal velocity components. The open-source code at the provided GitHub link is a clear strength for reproducibility and further testing. The empirical gains on held-out reconstruction tasks suggest practical utility, but only if the performance can be attributed to the Riemannian metric rather than the neural-bridge or distillation components alone.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (metric construction): the headline 23.7% average improvement is load-bearing for the geometry-awareness claim, yet the manuscript provides no explicit formula, algorithm, or pseudocode for estimating local tangent directions from snapshots (e.g., via local PCA or similar). Without this, it is impossible to verify that the anisotropic metric reliably penalizes normal components rather than reverting to near-Euclidean behavior under noise.
  2. [§4 and Table 2] §4 (experiments) and Table 2: the nine held-out reconstruction experiments report average percentage reductions but omit per-experiment error bars, standard deviations across random seeds, and an ablation that isolates the Riemannian metric from the neural-bridge fitting and distillation steps. This prevents assessment of whether the reported superiority is robust or driven by the geometry component.
minor comments (2)
  1. [§2] Notation for the path-action cost and the time-dependent metric tensor is introduced without a clear summary table relating symbols to their definitions, which would aid readability.
  2. [§4.3] The RNA-velocity alignment experiment on the embryoid-body benchmark is described only in the abstract; a dedicated subsection with the exact alignment metric and baseline details would strengthen the biological validation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments on our manuscript. We address each of the major comments below and have revised the manuscript to incorporate the suggested improvements for greater clarity and rigor.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (metric construction): the headline 23.7% average improvement is load-bearing for the geometry-awareness claim, yet the manuscript provides no explicit formula, algorithm, or pseudocode for estimating local tangent directions from snapshots (e.g., via local PCA or similar). Without this, it is impossible to verify that the anisotropic metric reliably penalizes normal components rather than reverting to near-Euclidean behavior under noise.

    Authors: We appreciate this observation. Section 3 describes the use of local PCA on cell neighborhoods within each snapshot to approximate tangent directions for the state- and time-dependent anisotropic metric. To improve verifiability and address the concern directly, we have added explicit pseudocode and the precise mathematical formula for the metric tensor construction in a new algorithm box in the revised Section 3. This addition clarifies how normal components are penalized and allows readers to assess behavior under noise. revision: yes

  2. Referee: [§4 and Table 2] §4 (experiments) and Table 2: the nine held-out reconstruction experiments report average percentage reductions but omit per-experiment error bars, standard deviations across random seeds, and an ablation that isolates the Riemannian metric from the neural-bridge fitting and distillation steps. This prevents assessment of whether the reported superiority is robust or driven by the geometry component.

    Authors: We agree that these details are important for assessing robustness. In the revised manuscript, we have updated Table 2 to report per-experiment means with error bars and standard deviations across five random seeds. We have also added an ablation study in Section 4.3 (with corresponding results in the supplement) that isolates the Riemannian metric by comparing full PACE against variants that disable the anisotropic component while retaining the neural-bridge fitting and distillation steps. The ablation confirms the geometry component drives a substantial portion of the gains. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The provided abstract and description outline a three-component framework: construction of a state- and time-dependent anisotropic Riemannian metric, alternation between cross-time couplings and neural bridge fitting, and distillation to a global velocity field. Performance is evaluated empirically via MMD, Wasserstein-1, and Wasserstein-2 distances against external baselines on held-out experiments, with no equations or self-referential definitions that reduce claimed gains to inputs by construction. No load-bearing self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work appear in the text. The chain remains self-contained with independent empirical support.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the ability to define and optimize under a custom geometry that is not derived from first principles in the abstract; neural bridge fitting introduces many implicit parameters whose values are learned from data.

free parameters (2)
  • neural bridge network parameters
    Endpoint-preserving neural bridges are fitted to data and therefore contain numerous learned weights and biases.
  • metric anisotropy parameters
    The state- and time-dependent anisotropic Riemannian metric requires choices or fitting of local tangent directions and penalty weights.
axioms (1)
  • domain assumption A state- and time-dependent anisotropic Riemannian metric exists that correctly captures locally supported tangent directions for cellular transport.
    Invoked in the first component of the framework as the basis for path-action cost.

pith-pipeline@v0.9.0 · 5804 in / 1401 out tokens · 51666 ms · 2026-05-20T01:20:07.309358+00:00 · methodology

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Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 1 internal anchor

  1. [1]

    Single-cell reconstruction of developmental trajectories during zebrafish embryogenesis.Science, 360(6392):eaar3131, 2018

    Jeffrey A Farrell, Yiqun Wang, Samantha J Riesenfeld, Karthik Shekhar, Aviv Regev, and Alexander F Schier. Single-cell reconstruction of developmental trajectories during zebrafish embryogenesis.Science, 360(6392):eaar3131, 2018

    work page 2018

  2. [2]

    Single-cell mapping of gene expression landscapes and lineage in the zebrafish embryo.Science, 360(6392):981–987, 2018

    Daniel E Wagner, Caleb Weinreb, Zach M Collins, James A Briggs, Sean G Megason, and Allon M Klein. Single-cell mapping of gene expression landscapes and lineage in the zebrafish embryo.Science, 360(6392):981–987, 2018

    work page 2018

  3. [3]

    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

  4. [4]

    Scaling single-cell genomics from phenomenology to mechanism

    Amos Tanay and Aviv Regev. Scaling single-cell genomics from phenomenology to mechanism. Nature, 541(7637):331–338, 2017

    work page 2017

  5. [5]

    Using single-cell genomics to understand developmental processes and cell fate decisions.Molecular systems biology, 14(4):MSB178046, 2018

    Jonathan A Griffiths, Antonio Scialdone, and John C Marioni. Using single-cell genomics to understand developmental processes and cell fate decisions.Molecular systems biology, 14(4):MSB178046, 2018

    work page 2018

  6. [6]

    The dynamics and regulators of cell fate decisions are revealed by pseudotemporal ordering of single cells

    Cole Trapnell, Davide Cacchiarelli, Jonna Grimsby, Prapti Pokharel, Shuqiang Li, Michael Morse, Niall J Lennon, Kenneth J Livak, Tarjei S Mikkelsen, and John L Rinn. The dynamics and regulators of cell fate decisions are revealed by pseudotemporal ordering of single cells. Nature biotechnology, 32(4):381–386, 2014

    work page 2014

  7. [7]

    Diffusion pseudotime robustly reconstructs lineage branching.Nature methods, 13(10):845– 848, 2016

    Laleh Haghverdi, Maren Büttner, F Alexander Wolf, Florian Buettner, and Fabian J Theis. Diffusion pseudotime robustly reconstructs lineage branching.Nature methods, 13(10):845– 848, 2016

    work page 2016

  8. [8]

    Characterization of cell fate probabilities in single-cell data with palantir.Nature biotechnology, 37(4):451–460, 2019

    Manu Setty, Vaidotas Kiseliovas, Jacob Levine, Adam Gayoso, Linas Mazutis, and Dana Pe’Er. Characterization of cell fate probabilities in single-cell data with palantir.Nature biotechnology, 37(4):451–460, 2019

    work page 2019

  9. [9]

    Aligning single-cell developmental and reprogramming trajectories identifies molecular determinants of myogenic reprogramming outcome.Cell Systems, 7(3):258–268, 2018

    Davide Cacchiarelli, Xiaojie Qiu, Sanjay Srivatsan, Anna Manfredi, Michael Ziller, Eliah Overbey, Antonio Grimaldi, Jonna Grimsby, Prapti Pokharel, Kenneth J Livak, et al. Aligning single-cell developmental and reprogramming trajectories identifies molecular determinants of myogenic reprogramming outcome.Cell Systems, 7(3):258–268, 2018. 10

    work page 2018

  10. [10]

    arXiv preprint arXiv:2102.09204 , year=

    Hugo Lavenant, Stephen Zhang, Young-Heon Kim, and Geoffrey Schiebinger. Towards a mathematical theory of trajectory inference.arXiv preprint arXiv:2102.09204, 2021

    work page arXiv 2021

  11. [11]

    Learning population-level diffusions with generative rnns

    Tatsunori Hashimoto, David Gifford, and Tommi Jaakkola. Learning population-level diffusions with generative rnns. InInternational Conference on Machine Learning, pages 2417–2426. PMLR, 2016

    work page 2016

  12. [12]

    Now Foundations and Trends, 2019

    Gabriel Peyré and Marco Cuturi.Computational optimal transport: With applications to data science. Now Foundations and Trends, 2019

    work page 2019

  13. [13]

    Fundamental limits on dynamic inference from single-cell snapshots.Proceedings of the National Academy of Sciences, 115(10):E2467–E2476, 2018

    Caleb Weinreb, Samuel Wolock, Betsabeh K Tusi, Merav Socolovsky, and Allon M Klein. Fundamental limits on dynamic inference from single-cell snapshots.Proceedings of the National Academy of Sciences, 115(10):E2467–E2476, 2018

    work page 2018

  14. [14]

    Action matching: Learning stochastic dynamics from samples

    Kirill Neklyudov, Rob Brekelmans, Daniel Severo, and Alireza Makhzani. Action matching: Learning stochastic dynamics from samples. InInternational conference on machine learning, pages 25858–25889. PMLR, 2023

    work page 2023

  15. [15]

    Improving and generalizing flow-based generative models with minibatch optimal transport

    Alexander Tong, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector-Brooks, Kilian Fatras, Guy Wolf, and Yoshua Bengio. Conditional flow matching: Simulation-free dynamic optimal transport.arXiv preprint arXiv:2302.00482, 2(3), 2023

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  16. [16]

    Trajecto- rynet: A dynamic optimal transport network for modeling cellular dynamics

    Alexander Tong, Jessie Huang, Guy Wolf, David Van Dijk, and Smita Krishnaswamy. Trajecto- rynet: A dynamic optimal transport network for modeling cellular dynamics. InInternational conference on machine learning, pages 9526–9536. PMLR, 2020

    work page 2020

  17. [17]

    Manifold interpolating optimal-transport flows for trajectory inference.Advances in neural information processing systems, 35:29705–29718, 2022

    Guillaume Huguet, Daniel Sumner Magruder, Alexander Tong, Oluwadamilola Fasina, Manik Kuchroo, Guy Wolf, and Smita Krishnaswamy. Manifold interpolating optimal-transport flows for trajectory inference.Advances in neural information processing systems, 35:29705–29718, 2022

    work page 2022

  18. [18]

    Vi- sualizing structure and transitions in high-dimensional biological data.Nature biotechnology, 37(12):1482–1492, 2019

    Kevin R Moon, David Van Dijk, Zheng Wang, Scott Gigante, Daniel B Burkhardt, William S Chen, Kristina Yim, Antonia van den Elzen, Matthew J Hirn, Ronald R Coifman, et al. Vi- sualizing structure and transitions in high-dimensional biological data.Nature biotechnology, 37(12):1482–1492, 2019

    work page 2019

  19. [19]

    Paga: graph abstraction reconciles clustering with trajectory inference through a topology preserving map of single cells

    F Alexander Wolf, Fiona K Hamey, Mireya Plass, Jordi Solana, Joakim S Dahlin, Berthold Göttgens, Nikolaus Rajewsky, Lukas Simon, and Fabian J Theis. Paga: graph abstraction reconciles clustering with trajectory inference through a topology preserving map of single cells. Genome biology, 20(1):59, 2019

    work page 2019

  20. [20]

    Concepts and limitations for learning developmental trajectories from single cell genomics.Development, 146(12):dev170506, 2019

    Sophie Tritschler, Maren Büttner, David S Fischer, Marius Lange, V olker Bergen, Heiko Lickert, and Fabian J Theis. Concepts and limitations for learning developmental trajectories from single cell genomics.Development, 146(12):dev170506, 2019

    work page 2019

  21. [21]

    Reprogram- ming roadmap reveals route to human induced trophoblast stem cells.Nature, 586(7827):101– 107, 2020

    Xiaodong Liu, John F Ouyang, Fernando J Rossello, Jia Ping Tan, Kathryn C Davidson, Daniela S Valdes, Jan Schroeder, Yu BY Sun, Joseph Chen, Anja S Knaupp, et al. Reprogram- ming roadmap reveals route to human induced trophoblast stem cells.Nature, 586(7827):101– 107, 2020

    work page 2020

  22. [22]

    Rna velocity of single cells.Nature, 560(7719):494–498, 2018

    Gioele La Manno, Ruslan Soldatov, Amit Zeisel, Emelie Braun, Hannah Hochgerner, Viktor Petukhov, Katja Lidschreiber, Maria E Kastriti, Peter Lönnerberg, Alessandro Furlan, et al. Rna velocity of single cells.Nature, 560(7719):494–498, 2018

    work page 2018

  23. [23]

    General- izing rna velocity to transient cell states through dynamical modeling.Nature biotechnology, 38(12):1408–1414, 2020

    V olker Bergen, Marius Lange, Stefan Peidli, F Alexander Wolf, and Fabian J Theis. General- izing rna velocity to transient cell states through dynamical modeling.Nature biotechnology, 38(12):1408–1414, 2020

    work page 2020

  24. [24]

    Cellrank for directed single-cell fate mapping.Nature methods, 19(2):159–170, 2022

    Marius Lange, V olker Bergen, Michal Klein, Manu Setty, Bernhard Reuter, Mostafa Bakhti, Heiko Lickert, Meshal Ansari, Janine Schniering, Herbert B Schiller, et al. Cellrank for directed single-cell fate mapping.Nature methods, 19(2):159–170, 2022

    work page 2022

  25. [25]

    A comparison of single-cell trajectory inference methods.Nature biotechnology, 37(5):547–554, 2019

    Wouter Saelens, Robrecht Cannoodt, Helena Todorov, and Yvan Saeys. A comparison of single-cell trajectory inference methods.Nature biotechnology, 37(5):547–554, 2019. 11

    work page 2019

  26. [26]

    Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. InThe Eleventh International Conference on Learning Representations, 2023

    work page 2023

  27. [27]

    Diffusion schrödinger bridge with applications to score-based generative modeling.Advances in neural information processing systems, 34:17695–17709, 2021

    Valentin De Bortoli, James Thornton, Jeremy Heng, and Arnaud Doucet. Diffusion schrödinger bridge with applications to score-based generative modeling.Advances in neural information processing systems, 34:17695–17709, 2021

    work page 2021

  28. [28]

    Diffusion schrödinger bridge matching.Advances in neural information processing systems, 36:62183–62223, 2023

    Yuyang Shi, Valentin De Bortoli, Andrew Campbell, and Arnaud Doucet. Diffusion schrödinger bridge matching.Advances in neural information processing systems, 36:62183–62223, 2023

    work page 2023

  29. [29]

    Deep momentum multi-marginal schrödinger bridge.Advances in Neural Information Processing Systems, 36:57058–57086, 2023

    Tianrong Chen, Guan-Horng Liu, Molei Tao, and Evangelos Theodorou. Deep momentum multi-marginal schrödinger bridge.Advances in Neural Information Processing Systems, 36:57058–57086, 2023

    work page 2023

  30. [30]

    Bronstein, Joey Bose, and Alexander Tong

    Katarina Petrovi´c, Lazar Atanackovic, Viggo Moro, Kacper Kapu´sniak, Ismail Ilkan Ceylan, Michael M. Bronstein, Joey Bose, and Alexander Tong. Curly flow matching for learning non-gradient field dynamics. InThe Thirty-ninth Annual Conference on Neural Information Processing Systems, 2026

    work page 2026

  31. [31]

    A geometric take on metric learning

    Søren Hauberg, Oren Freifeld, and Michael Black. A geometric take on metric learning. Advances in Neural Information Processing Systems, 25, 2012

    work page 2012

  32. [32]

    Latent space oddity: on the curvature of deep generative models.arXiv preprint arXiv:1710.11379, 2017

    Georgios Arvanitidis, Lars Kai Hansen, and Søren Hauberg. Latent space oddity: on the curvature of deep generative models.arXiv preprint arXiv:1710.11379, 2017

    work page arXiv 2017

  33. [33]

    Ricky T. Q. Chen and Yaron Lipman. Flow matching on general geometries. InThe Twelfth International Conference on Learning Representations, 2024

    work page 2024

  34. [34]

    Metric flow matching for smooth interpolations on the data manifold.Advances in Neural Information Processing Systems, 37:135011–135042, 2024

    Kacper Kapu ´sniak, Peter Potaptchik, Teodora Reu, Leo Zhang, Alexander Tong, Michael Bronstein, Avishek J Bose, and Francesco Di Giovanni. Metric flow matching for smooth interpolations on the data manifold.Advances in Neural Information Processing Systems, 37:135011–135042, 2024

    work page 2024

  35. [35]

    Multi-marginal schrödinger bridges with iterative reference refinement

    Yunyi Shen, Renato Berlinghieri, and Tamara Broderick. Multi-marginal schrödinger bridges with iterative reference refinement. InThe 28th International Conference on Artificial Intelli- gence and Statistics, 2025

    work page 2025

  36. [36]

    Multimodal single cell data integration challenge: results and lessons learned.BioRxiv, pages 2022–04, 2022

    Christopher Lance, Malte D Luecken, Daniel B Burkhardt, Robrecht Cannoodt, Pia Rauten- strauch, Anna Laddach, Aidyn Ubingazhibov, Zhi-Jie Cao, Kaiwen Deng, Sumeer Khan, et al. Multimodal single cell data integration challenge: results and lessons learned.BioRxiv, pages 2022–04, 2022

    work page 2022

  37. [37]

    Multi-marginal flow matching with adversarially learnt interpolants

    Oskar Kviman, Kirill Tamogashev, Nicola Branchini, Víctor Elvira, Jens Lagergren, and Nikolay Malkin. Multi-marginal flow matching with adversarially learnt interpolants. InThe Fourteenth International Conference on Learning Representations, 2026

    work page 2026

  38. [38]

    The hycom (hybrid coordinate ocean model) data assimilative system.Journal of Marine Systems, 65(1-4):60–83, 2007

    Eric P Chassignet, Harley E Hurlburt, Ole Martin Smedstad, George R Halliwell, Patrick J Hogan, Alan J Wallcraft, Remy Baraille, and Rainer Bleck. The hycom (hybrid coordinate ocean model) data assimilative system.Journal of Marine Systems, 65(1-4):60–83, 2007

    work page 2007

  39. [39]

    Number 89

    Michel Ledoux.The concentration of measure phenomenon. Number 89. American Mathemati- cal Soc., 2001

    work page 2001

  40. [40]

    Cambridge university press, 2018

    Roman Vershynin.High-dimensional probability: An introduction with applications in data science, volume 47. Cambridge university press, 2018

    work page 2018

  41. [41]

    nearest neighbor

    Kevin Beyer, Jonathan Goldstein, Raghu Ramakrishnan, and Uri Shaft. When is “nearest neighbor” meaningful? InInternational conference on database theory, pages 217–235. Springer, 1999. 12 Technical appendices and supplementary material A. Benchmark and Metric Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

    work page arXiv 1999