pith. sign in

arxiv: 2605.15524 · v1 · pith:VDSUBEEGnew · submitted 2026-05-15 · 💻 cs.LG · cs.AI· math.DG· math.ST· stat.TH

Neural Point-Forms

Bruno Trentini , Jacob Hume , Vincenzo Antonio Isoldi , Philipp Misof , Ekaterina S. Ivshina , Kelly Maggs This is my paper

Pith reviewed 2026-05-19 14:53 UTC · model grok-4.3

classification 💻 cs.LG cs.AImath.DGmath.STstat.TH
keywords neural point-formspoint cloudsdiffusion geometrydifferential formsgeometric featuresmanifold hypothesispermutation invarianceLaplacian methods
0
0 comments X

The pith

Neural point-forms represent point clouds as learned comparison matrices of differential forms.

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

Point cloud learning often misses higher-order tangency information that coordinates or distances do not capture directly. This paper introduces neural point-forms as learnable features that build discrete models for comparing differential forms using Laplacian techniques from diffusion geometry. In the continuum these become comparison matrices that describe interactions between feature forms and extrinsic tangency on submanifolds. The central step is a proof that the matrices remain consistent in the long run under standard sampling, bandwidth, density, and manifold assumptions. This construction supplies a compact, permutation-invariant neural layer whose output is the learned comparison matrix, with clearest gains on tasks sensitive to density or manifold structure.

Core claim

We introduce neural point-forms (NPFs) as a new family of principled learnable geometric features for point clouds. In the absence of a natural tangency structure, Laplacian-based techniques from Diffusion Geometry are used to build a discrete model for comparing differential forms on point clouds via inner products. In the continuum, submanifolds of a shared ambient feature space are represented as comparison matrices whose entries describe how pairs of feature forms interact with extrinsic tangency information. We prove the long-run consistency of these comparison matrices under standard sampling, bandwidth, density, and manifold-hypothesis assumptions. This yields a compact, efficient and

What carries the argument

The form-comparison matrix, a learned object whose entries encode inner-product interactions between pairs of feature forms and extrinsic tangency on the point cloud.

If this is right

  • The construction produces a compact, efficient, and permutation-invariant neural layer.
  • The layer outputs a learned form-comparison matrix rather than raw coordinates or distances.
  • The strongest performance gains occur on tasks whose labels depend on sampling density or manifold-like population geometry.
  • The representation remains competitive and interpretable on both synthetic and biologically relevant point-cloud data.

Where Pith is reading between the lines

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

  • The same consistency argument could be adapted to other discrete geometric data structures that admit a diffusion operator.
  • Explicit modeling of form comparisons may improve robustness when point clouds exhibit irregular or density-varying sampling.
  • The layer could be inserted into existing point-cloud architectures to add geometric awareness at modest computational cost.

Load-bearing premise

Comparison matrices remain consistent in the long run when point clouds are sampled from manifolds under standard assumptions on sampling, bandwidth, density, and the manifold hypothesis.

What would settle it

A direct numerical check that the entries of the learned comparison matrix fail to approach their continuum-limit values as the number of sampled points grows while the manifold hypothesis is violated.

Figures

Figures reproduced from arXiv: 2605.15524 by Bruno Trentini, Ekaterina S. Ivshina, Jacob Hume, Kelly Maggs, Philipp Misof, Vincenzo Antonio Isoldi.

Figure 1
Figure 1. Figure 1: Pairings of k-form restrictions distinguish manifolds by tangency information. Following on from the perspective in [28], the approach in this work is to use ambient differential forms as higher-order geometric features. Such features are activated or inactivated by the tangency and normal structure of a submanifold, as quantified by the inner products and norms of their restrictions ( [PITH_FULL_IMAGE:fi… view at source ↗
Figure 2
Figure 2. Figure 2: Tangency information is encoded by images of Gram fields and their approximations. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The second Gram field G (2) M is constructed as a compound matrix of the first GM. Gram fields The k-th Gram field is the mapping p ∈ M 7→ G (k) M (p) := h [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Circle one-form density-correction check. Top: point clouds sampled from von-Mises densities on S 1 as the concentration κ increases. Bottom left: ambient representatives of the two one-forms used in the inner-product estimator. Bottom right: MAE under nonuniform sampling shows that density correction substantially reduces the sampling-density bias. 1. (Gram field transform) The first step performs the map… view at source ↗
Figure 5
Figure 5. Figure 5: Qualitative diagnostics for learned point-form representations on controlled synthetic tasks. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: 6a: The pullback of a 1-form on R 2 (viewed as a vector field) to a 1-manifold. 6b: The pullback of a 2-form on R 3 (viewed as a normal field) to a 2-manifold. Riemannian metrics A Riemannian metric g on a manifold M corresponds to a family of inner products ⟨−, −⟩M(p) : TpM ⊗ TpM → R (37) parametrized by points p ∈ M. The inner product induces an isomorphism ♭ : TpM ♭ ⇄ ♯ T ∗ p M : ♯ (38) between tangent … view at source ↗
Figure 7
Figure 7. Figure 7: NPF Learning Pipeline. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
read the original abstract

Point cloud learning often rests on the premise that observed samples are noisy traces of an underlying geometric object, such as a manifold embedded in a high-dimensional feature space. Yet much of this geometry is not captured directly by coordinates, pairwise distances, or learned graph neighborhoods alone. In the smooth setting, differential forms are devices to encode higher order tangency information. In this work, we introduce a new family of principled learnable geometric features for point clouds called neural point-forms (NPFs). In the absence of a natural tangency structure, we instead use Laplacian-based techniques from Diffusion Geometry to build a discrete model for comparing differential forms on point clouds via inner products. In the continuum, submanifolds of a shared ambient feature space are represented as comparison matrices, whose entries describe how pairs of feature forms interact with extrinsic tangency information. We make this intuition precise by proving the long-run consistency of comparison matrices under standard sampling, bandwidth, density, and manifold-hypothesis assumptions. This yields a compact, efficient and permutation-invariant neural layer whose output is a learned form-comparison matrix. Across synthetic and biologically relevant experiments, we show that NPFs provide a competitive, and interpretable representation, with the strongest benefits appearing when labels depend on sampling density, manifold-like structure, or response-relevant population geometry.

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 introduces neural point-forms (NPFs), a family of learnable geometric features for point clouds that use Laplacian-based inner-product models from diffusion geometry to represent comparisons of differential forms. It proves long-run consistency of the resulting form-comparison matrices under standard sampling, bandwidth, density, and manifold-hypothesis assumptions. This construction yields a compact, efficient, permutation-invariant neural layer. Experiments on synthetic and biologically relevant point-cloud tasks show competitive performance, with particular advantages when labels depend on sampling density or manifold-like structure.

Significance. If the consistency result holds with adequate rates and robustness, the work supplies a principled mechanism for injecting higher-order tangency information into point-cloud networks without relying solely on coordinates or graph neighborhoods. This could be valuable for tasks in which response-relevant geometry or non-uniform sampling is present, such as biological population data. The combination of a theoretical guarantee under standard assumptions and empirical competitiveness is a positive feature; the interpretability of the learned comparison matrices is an additional strength.

major comments (2)
  1. [Consistency proof / theoretical section] The central consistency claim (abstract and theoretical development): the proof invokes continuum limits under sampling, bandwidth, density, and manifold assumptions but supplies neither explicit convergence rates nor error bounds for finite noisy samples when the manifold hypothesis holds only approximately. This is load-bearing for the geometric interpretation of the neural layer, because without rates the discrete Laplacian inner-product model may not faithfully proxy differential forms on practical point clouds.
  2. [Experiments] Experimental validation (results section): the claim that NPFs show strongest benefits when labels depend on sampling density or manifold structure requires an ablation isolating the contribution of the learned form-comparison matrix versus the underlying Laplacian construction and bandwidth choice; without it the attribution of gains remains unclear.
minor comments (2)
  1. [Abstract] The abstract refers to 'standard' assumptions without listing them explicitly; a compact statement of the precise hypotheses used in the consistency theorem would improve readability.
  2. [Methods] Notation for the form-comparison matrix and its relation to the discrete inner product should be introduced with a small diagram or equation reference early in the methods to aid readers unfamiliar with diffusion geometry.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback on the manuscript. We address each major comment point by point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Consistency proof / theoretical section] The central consistency claim (abstract and theoretical development): the proof invokes continuum limits under sampling, bandwidth, density, and manifold assumptions but supplies neither explicit convergence rates nor error bounds for finite noisy samples when the manifold hypothesis holds only approximately. This is load-bearing for the geometric interpretation of the neural layer, because without rates the discrete Laplacian inner-product model may not faithfully proxy differential forms on practical point clouds.

    Authors: We acknowledge that the consistency theorem establishes long-run (asymptotic) convergence under the stated sampling, bandwidth, density, and manifold assumptions but does not supply explicit rates or finite-sample error bounds, particularly when the manifold hypothesis holds only approximately. This is a valid observation regarding the scope of the result. The paper focuses on proving consistency in the continuum limit to justify the discrete model as a proxy for differential-form comparisons; non-asymptotic analysis would require additional technical machinery and stronger assumptions. In the revised manuscript we will add a clarifying paragraph in the theoretical section that explicitly states the asymptotic character of the guarantee and discusses its implications for finite noisy data. revision: partial

  2. Referee: [Experiments] Experimental validation (results section): the claim that NPFs show strongest benefits when labels depend on sampling density or manifold structure requires an ablation isolating the contribution of the learned form-comparison matrix versus the underlying Laplacian construction and bandwidth choice; without it the attribution of gains remains unclear.

    Authors: We agree that a targeted ablation would strengthen attribution of the observed gains. The current experiments compare NPFs against coordinate- and graph-based baselines, but do not isolate the learned comparison matrix from the fixed Laplacian operator and bandwidth selection. We will add an ablation study in the revised results section that (i) compares learned versus fixed form-comparison matrices while holding the Laplacian construction constant and (ii) reports sensitivity to bandwidth choice on the synthetic and biological tasks. This will clarify the source of the performance differences when labels depend on density or manifold structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; consistency follows from invoked assumptions

full rationale

The derivation proceeds by constructing a discrete Laplacian-based inner-product model for differential forms on point clouds using established diffusion geometry techniques, then proving long-run consistency of the resulting comparison matrices under standard sampling, bandwidth, density, and manifold-hypothesis assumptions. This proof is presented as a direct consequence of those external assumptions rather than a self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation. The neural layer is obtained as the output of this construction, with no equations or steps shown to reduce tautologically to the inputs by construction. The central claim therefore retains independent mathematical content.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 2 invented entities

The central construction rests on diffusion-geometry Laplacians and manifold assumptions drawn from prior literature, plus the new entities of neural point-forms and comparison matrices introduced without external falsifiable handles beyond the claimed proof.

free parameters (1)
  • bandwidth parameter
    Appears in the Laplacian construction and consistency statement; its selection or fitting is required for the discrete model.
axioms (1)
  • domain assumption standard sampling, bandwidth, density, and manifold-hypothesis assumptions
    Invoked to establish long-run consistency of the comparison matrices.
invented entities (2)
  • neural point-forms (NPFs) no independent evidence
    purpose: Learnable geometric features that encode higher-order tangency information on point clouds
    New family of features defined via the comparison matrices.
  • form-comparison matrix no independent evidence
    purpose: Compact representation of how pairs of feature forms interact with extrinsic tangency
    Output of the neural layer and central object of the consistency proof.

pith-pipeline@v0.9.0 · 5780 in / 1512 out tokens · 123012 ms · 2026-05-19T14:53:28.984988+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

106 extracted references · 106 canonical work pages · 7 internal anchors

  1. [1]

    Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges

    Michael M. Bronstein, Joan Bruna, Taco Cohen, and Petar Veliˇckovi´c. Geometric deep learning: Grids, groups, graphs, geodesics, and gauges.arXiv:2104.13478, 2021

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  2. [2]

    Schoenholz, Patrick F

    Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural message passing for quantum chemistry. InICML, 2017

    work page 2017

  3. [3]

    Relational inductive biases, deep learning, and graph networks

    Peter W. Battaglia et al. Relational inductive biases, deep learning, and graph networks.arXiv:1806.01261, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  4. [4]

    Hamilton, Rex Ying, and Jure Leskovec

    William L. Hamilton, Rex Ying, and Jure Leskovec. Inductive representation learning on large graphs,

    work page

  5. [5]

    URLhttps://arxiv.org/abs/1706.02216

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Qi, Jean-Emmanuel Deschaud, Beatriz Marcotegui, François Goulette, and Leonidas J

    Hugues Thomas, Charles R. Qi, Jean-Emmanuel Deschaud, Beatriz Marcotegui, François Goulette, and Leonidas J. Guibas. KPConv: Flexible and deformable convolution for point clouds. InICCV, 2019

    work page 2019

  7. [7]

    How Powerful are Graph Neural Networks? InInternational Conference on Learning Representations, 2019

    Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How Powerful are Graph Neural Networks? InInternational Conference on Learning Representations, 2019. URL https://openreview.net/for um?id=ryGs6iA5Km

    work page 2019

  8. [8]

    Graph attention networks

    Petar Veliˇckovi´c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph attention networks. InInternational Conference on Learning Representations, 2018. URL https: //openreview.net/forum?id=rJXMpikCZ

    work page 2018

  9. [9]

    A generalization of transformer networks to graphs, 2021

    Vijay Prakash Dwivedi and Xavier Bresson. A generalization of transformer networks to graphs, 2021. URLhttps://arxiv.org/abs/2012.09699

    work page arXiv 2021

  10. [10]

    Garg, Stefanie Jegelka, and Tommi Jaakkola

    Vikas K. Garg, Stefanie Jegelka, and Tommi Jaakkola. Generalization and representational limits of graph neural networks, 2020. URLhttps://arxiv.org/abs/2002.06157

    work page arXiv 2020

  11. [11]

    Position: Topological deep learning is the new frontier for relational learning

    Theodore Papamarkou et al. Position: Topological deep learning is the new frontier for relational learning. InICML, 2024

    work page 2024

  12. [12]

    Simplicial neural networks

    Stefania Ebli, Michaël Defferrard, and Gard Spreemann. Simplicial neural networks. InNeurIPS Workshop on Topological Data Analysis, 2020

    work page 2020

  13. [13]

    Simplicial 2-complex convolutional neural networks

    Eric Bunch, Qian You, Glenn Fung, and Vikas Singh. Simplicial 2-complex convolutional neural networks. InNeurIPS Workshop on Topological Data Analysis, 2020

    work page 2020

  14. [14]

    Bronstein

    Cristian Bodnar, Fabrizio Frasca, Yu Guang Wang, Nina Otter, Guido Montúfar, Pietro Lió, and Michael M. Bronstein. Weisfeiler and lehman go topological: Message passing simplicial networks. InICML, 2021

    work page 2021

  15. [15]

    Mitchell Roddenberry, Nicholas Glaze, and Santiago Segarra

    T. Mitchell Roddenberry, Nicholas Glaze, and Santiago Segarra. Principled simplicial neural networks for trajectory prediction. InICML, 2021

    work page 2021

  16. [16]

    Hypergraph Neural Networks

    Yifan Feng, Haoxuan You, Zizhao Zhang, Rongrong Ji, and Yue Gao. Hypergraph neural networks, 2019. URLhttps://arxiv.org/abs/1809.09401

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  17. [17]

    Bronstein

    Cristian Bodnar, Fabrizio Frasca, Nina Otter, Yu Guang Wang, Pietro Lió, Guido Montúfar, and Michael M. Bronstein. Weisfeiler and lehman go cellular: Cw networks. InNeurIPS, 2021. 10

    work page 2021

  18. [18]

    Dey, Soham Mukherjee, Shreyas N

    Mustafa Hajij et al. Topological deep learning: Going beyond graph data.arXiv:2206.00606, 2022

    work page arXiv 2022

  19. [19]

    Bronstein

    Cristian Bodnar, Francesco Di Giovanni, Benjamin Chamberlain, Pietro Lió, and Michael M. Bronstein. Neural sheaf diffusion: A topological perspective on heterophily and oversmoothing in gnns. InNeurIPS, 2022

    work page 2022

  20. [20]

    On the sheafification of higher-order message passing.arXiv preprint arXiv:2509.23020, 2025

    Jacob Hume and Pietro Liò. On the sheafification of higher-order message passing.arXiv preprint arXiv:2509.23020, 2025

    work page arXiv 2025

  21. [21]

    Topotune : A framework for generalized combinatorial complex neural networks, 2025

    Mathilde Papillon, Guillermo Bernárdez, Claudio Battiloro, and Nina Miolane. Topotune : A framework for generalized combinatorial complex neural networks, 2025. URL https://arxiv.org/abs/2410.0 6530

    work page 2025

  22. [22]

    Qi, Hao Su, Kaichun Mo, and Leonidas J

    Charles R. Qi, Hao Su, Kaichun Mo, and Leonidas J. Guibas. PointNet: Deep learning on point sets for 3D classification and segmentation. InCVPR, 2017

    work page 2017

  23. [23]

    Qi, Li Yi, Hao Su, and Leonidas J

    Charles R. Qi, Li Yi, Hao Su, and Leonidas J. Guibas. PointNet++: Deep hierarchical feature learning on point sets in a metric space. InNeurIPS, 2017

    work page 2017

  24. [24]

    Sarma, Michael M

    Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E. Sarma, Michael M. Bronstein, and Justin M. Solomon. Dynamic graph CNN for learning on point clouds.ACM Transactions on Graphics, 38(5), 2019

    work page 2019

  25. [25]

    Hengshuang Zhao, Li Jiang, Jiaya Jia, Philip H. S. Torr, and Vladlen Koltun. Point transformer. InICCV, 2021

    work page 2021

  26. [26]

    Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds

    Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Ri- ley. Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds. arXiv:1802.08219, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  27. [27]

    Fuchs, Daniel E

    Fabian B. Fuchs, Daniel E. Worrall, V olker Fischer, and Max Welling. SE(3)-transformers: 3D roto- translation equivariant attention networks. InNeurIPS, 2020

    work page 2020

  28. [28]

    Ilyes Batatia, David Peter Kovacs, Gregor N. C. Simm, Christoph Ortner, and Gábor Csányi. MACE: Higher order equivariant message passing neural networks for fast and accurate force fields. InNeurIPS, 2022

    work page 2022

  29. [29]

    Simplicial representation learning with neural k-forms,

    Kelly Maggs, Celia Hacker, and Bastian Rieck. Simplicial representation learning with neural k-forms,

    work page

  30. [30]

    ArXiv preprint available athttps://arxiv.org/abs/2312.08515

    Accepted to theInternational Conference of Learning and Representations 2024. ArXiv preprint available athttps://arxiv.org/abs/2312.08515

    work page arXiv 2024

  31. [31]

    Computing Diffusion Geometry, February 2026

    Iolo Jones and David Lanners. Computing Diffusion Geometry, February 2026. URL http://arxiv.or g/abs/2602.06006. arXiv:2602.06006 [math]

    work page arXiv 2026

  32. [32]

    Spectral exterior calculus.Communications on Pure and Applied Mathematics, 73(4):689–770, 2020

    Tyrus Berry and Dimitrios Giannakis. Spectral exterior calculus.Communications on Pure and Applied Mathematics, 73(4):689–770, 2020. doi: https://doi.org/10.1002/cpa.21885. URL https://onlinelibr ary.wiley.com/doi/abs/10.1002/cpa.21885

    work page doi:10.1002/cpa.21885 2020

  33. [33]

    Diffusion geometry, 2024

    Iolo Jones. Diffusion geometry, 2024. URLhttps://arxiv.org/abs/2405.10858

    work page arXiv 2024

  34. [34]

    Coifman and Stéphane Lafon

    Ronald R. Coifman and Stéphane Lafon. Diffusion maps.Applied and Computational Harmonic Analysis, 21(1):5–30, 2006. doi: 10.1016/j.acha.2006.04.006. URLhttps://doi.org/10.1016/j.acha.2006. 04.006

    work page doi:10.1016/j.acha.2006.04.006 2006

  35. [35]

    Variable bandwidth diffusion kernels.Applied and Computational Harmonic Analysis, 40(1):68–96, January 2016

    Tyrus Berry and John Harlim. Variable bandwidth diffusion kernels.Applied and Computational Harmonic Analysis, 40(1):68–96, January 2016. ISSN 1063-5203. doi: 10.1016/j.acha.2015.01.001. URL http://dx.doi.org/10.1016/j.acha.2015.01.001

    work page doi:10.1016/j.acha.2015.01.001 2016

  36. [36]

    Towards a theoretical foundation for laplacian-based manifold methods

    Mikhail Belkin and Partha Niyogi. Towards a theoretical foundation for laplacian-based manifold methods. Journal of Computer and System Sciences, 74(8):1289–1308, December 2008. ISSN 0022-0000. doi: 10.1016/j.jcss.2007.08.006. URLhttp://dx.doi.org/10.1016/j.jcss.2007.08.006

    work page doi:10.1016/j.jcss.2007.08.006 2008

  37. [37]

    From graph to manifold Laplacian: The convergence rate.Applied and Computational Harmonic Analysis, 21(1):128–134, 2006

    Amit Singer. From graph to manifold Laplacian: The convergence rate.Applied and Computational Harmonic Analysis, 21(1):128–134, 2006

    work page 2006

  38. [38]

    Diffusion Processes on Implicit Manifolds

    Victor Kawasaki-Borruat, Clara Grotehans, Pierre Vandergheynst, and Adam Gosztolai. Diffusion processes on implicit manifolds, 2026. URLhttps://arxiv.org/abs/2604.07213

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  39. [39]

    Manifold diffusion geometry: Curvature, tangent spaces, and dimension, 2026

    Iolo Jones. Manifold diffusion geometry: Curvature, tangent spaces, and dimension, 2026. URL https: //arxiv.org/abs/2411.04100. 11

    work page arXiv 2026

  40. [40]

    Hiponet: A multi-view simplicial complex network for high-dimensional point clouds and single-cell data

    Siddharth Viswanath, Aarthi Madhu, Dhananjay Bhaskar, et al. Hiponet: A multi-view simplicial complex network for high-dimensional point clouds and single-cell data. InNeurIPS, 2025. arXiv:2502.07746

    work page arXiv 2025

  41. [41]

    María Ramos Zapatero, Alexander Tong, James W. Opzoomer, Rhianna O’Sullivan, Ferran Cardoso Ro- driguez, Jahangir Sufi, Petra Vlckova, Callum Nattress, Xiao Qin, Jeroen Claus, Daniel Hochhauser, Smita Krishnaswamy, and Christopher J. Tape. Trellis tree-based analysis reveals stromal regulation of patient-derived organoid drug responses.Cell, 186(25):5606–...

    work page doi:10.1016/j.cell.2023.11.005 2023

  42. [42]

    Kastriti, Peter Lonnerberg, Alessandro Furlan, Jean Fan, Lars E

    Gioele La Manno, Ruslan Soldatov, Amit Zeisel, Emelie Braun, Hannah Hochgerner, Viktor Petukhov, Katja Lidschreiber, Maria E. Kastriti, Peter Lonnerberg, Alessandro Furlan, Jean Fan, Lars E. Borm, Zehua Liu, David van Bruggen, Jimin Guo, Xiaoling He, Roger Barker, Erik Sundstrom, Goncalo Castelo-Branco, Patrick Cramer, Igor Adameyko, Sten Linnarsson, and ...

    work page doi:10.1038/s41586-018-0414-6 2018

  43. [43]

    Maria Ramos, Alexander Tong, Jahangir Sufi, Petra Vlckova, Ferran Cardoso Rodriguez, Callum Nattress, Xiao Qin, Daniel Hochhauser, Smita Krishnaswamy, and Christopher J. Tape. Cancer-associated fibroblasts regulate patient-derived organoid drug responses. Mendeley Data, Version 1, 2022. URL https: //data.mendeley.com/datasets/hc8gxwks3p/1

    work page 2022

  44. [44]

    Deeper insights into graph convolutional networks for semi-supervised learning

    Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into graph convolutional networks for semi-supervised learning. InAAAI, 2018

    work page 2018

  45. [45]

    Graph neural networks exponentially lose expressive power for node classification

    Kenta Oono and Taiji Suzuki. Graph neural networks exponentially lose expressive power for node classification. InICLR, 2020

    work page 2020

  46. [46]

    A note on over-smoothing for graph neural networks.arXiv preprint arXiv:2006.13318, 2020

    Chen Cai and Yusu Wang. A note on over-smoothing for graph neural networks.arXiv:2006.13318, 2020

    work page arXiv 2006

  47. [47]

    A survey on oversmoothing in graph neural networks.arXiv preprint arXiv:2303.10993, 2023

    T. Konstantin Rusch, Michael M. Bronstein, and Siddhartha Mishra. A survey on oversmoothing in graph neural networks.arXiv:2303.10993, 2023

    work page arXiv 2023

  48. [48]

    On the bottleneck of graph neural networks and its practical implications

    Uri Alon and Eran Yahav. On the bottleneck of graph neural networks and its practical implications. In ICLR, 2021

    work page 2021

  49. [49]

    Chamberlain, Xiaowen Dong, and Michael M

    Jake Topping, Francesco Di Giovanni, Benjamin P. Chamberlain, Xiaowen Dong, and Michael M. Bronstein. Understanding over-squashing and bottlenecks on graphs via curvature. InICLR, 2022

    work page 2022

  50. [50]

    On over-squashing in message passing neural networks: The impact of width, depth, and topology

    Francesco Di Giovanni, Lorenzo Giusti, Federico Barbero, Giulia Luise, Pietro Lio, and Michael Bronstein. On over-squashing in message passing neural networks: The impact of width, depth, and topology. In ICML, 2023

    work page 2023

  51. [51]

    World Models

    David Ha and Jürgen Schmidhuber. World models.arXiv preprint arXiv:1803.10122, 2018. doi: 10.48550/arXiv.1803.10122. URLhttps://arxiv.org/abs/1803.10122

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1803.10122 2018

  52. [52]

    Re- Act: Synergizing reasoning and acting in language models

    Shunyu Yao, Jeffrey Zhao, Dian Yu, Nan Du, Izhak Shafran, Karthik Narasimhan, and Yuan Cao. Re- Act: Synergizing reasoning and acting in language models. InInternational Conference on Learning Representations, 2023. URLhttps://openreview.net/forum?id=WE_vluYUL-X

    work page 2023

  53. [53]

    Bernstein

    Joon Sung Park, Joseph C. O’Brien, Carrie J. Cai, Meredith Ringel Morris, Percy Liang, and Michael S. Bernstein. Generative agents: Interactive simulacra of human behavior. InProceedings of the 36th Annual ACM Symposium on User Interface Software and Technology, pages 1–22. Association for Computing Machinery, 2023. doi: 10.1145/3586183.3606763. URL https...

    work page doi:10.1145/3586183.3606763 2023

  54. [54]

    PhD thesis, California Institute of Technology, 2003

    Anil Nirmal Hirani.Discrete Exterior Calculus. PhD thesis, California Institute of Technology, 2003. URLhttps://thesis.library.caltech.edu/1885/

    work page 2003

  55. [55]

    Hirani, and Jerrold E

    Mathieu Desbrun, Anil N. Hirani, and Jerrold E. Marsden. Discrete exterior calculus for variational problems in computer vision and graphics. InProceedings of the 42nd IEEE Conference on Decision and Control, volume 5, pages 4902–4907. IEEE, 2003. doi: 10.1109/CDC.2003.1272393. URL https://authors.library.caltech.edu/records/8ada4-c7c35

    work page doi:10.1109/cdc.2003.1272393 2003

  56. [56]

    Retrieval- augmented generation for knowledge-intensive NLP tasks

    Patrick Lewis, Ethan Perez, Aleksandra Piktus, Fabio Petroni, Vladimir Karpukhin, Naman Goyal, Heinrich Küttler, Mike Lewis, Wen-tau Yih, Tim Rocktäschel, Sebastian Riedel, and Douwe Kiela. Retrieval- augmented generation for knowledge-intensive NLP tasks. InAdvances in Neural Information Processing Systems, volume 33, pages 9459–9474, 2020. URL https://p...

    work page 2020

  57. [57]

    , date =

    John M. Lee.Introduction to Riemannian Manifolds. Springer International Publishing, 2018. ISBN 9783319917559. doi: 10.1007/978-3-319-91755-9. URL http://dx.doi.org/10.1007/978-3-319 -91755-9

    work page doi:10.1007/978-3-319-91755-9 2018

  58. [58]

    Springer International Publishing, 2017

    Jürgen Jost.Riemannian Geometry and Geometric Analysis. Springer International Publishing, 2017. ISBN 9783319618609. doi: 10.1007/978-3-319-61860-9. URL http://dx.doi.org/10.1007/978-3 -319-61860-9

    work page doi:10.1007/978-3-319-61860-9 2017

  59. [59]

    Carr \’e du champ flow matching: better quality-generalisation tradeoff in generative models

    Jacob Bamberger, Iolo Jones, Dennis Duncan, Michael M Bronstein, Pierre Vandergheynst, and Adam Gosztolai. Carr \’e du champ flow matching: better quality-generalisation tradeoff in generative models. arXiv preprint arXiv:2510.05930, 2025

    work page arXiv 2025

  60. [60]

    An algorithm for finding intrinsic dimensionality of data.IEEE Transactions on computers, 100(2):176–183, 1971

    Keinosuke Fukunaga and David R Olsen. An algorithm for finding intrinsic dimensionality of data.IEEE Transactions on computers, 100(2):176–183, 1971

    work page 1971

  61. [61]

    Daniel Ting, Ling Huang, and Michael I. Jordan. An analysis of the convergence of graph Laplacians. In ICML, 2010

    work page 2010

  62. [62]

    Leakage in data mining: formulation, detection, and avoidance

    Shachar Kaufman, Saharon Rosset, and Claudia Perlich. Leakage in data mining: formulation, detection, and avoidance. InProceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 556–563, 2011. doi: 10.1145/2020408.2020496. URL https: //doi.org/10.1145/2020408.2020496

    work page doi:10.1145/2020408.2020496 2011

  63. [63]

    Leakage and the reproducibility crisis in machine-learning- based science.Patterns, 4(9):100804, 2023

    Sayash Kapoor and Arvind Narayanan. Leakage and the reproducibility crisis in machine-learning- based science.Patterns, 4(9):100804, 2023. doi: 10.1016/j.patter.2023.100804. URL https: //doi.org/10.1016/j.patter.2023.100804

    work page doi:10.1016/j.patter.2023.100804 2023

  64. [64]

    David R. Cox. The regression analysis of binary sequences.Journal of the Royal Statistical Society: Series B, 20(2):215–242, 1958

    work page 1958

  65. [65]

    Random forests

    Leo Breiman. Random forests.Machine Learning, 45(1):5–32, 2001. doi: 10.1023/A:1010933404324

    work page doi:10.1023/a:1010933404324 2001

  66. [66]

    Support-vector networks.Machine Learning, 20:273– 297, 1995

    Corinna Cortes and Vladimir Vapnik. Support-vector networks.Machine Learning, 20(3):273–297, 1995. doi: 10.1007/BF00994018

    work page doi:10.1007/bf00994018 1995

  67. [67]

    Learning representations by back- propagating errors.Nature, 323(6088):533–536, 1986

    David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by back- propagating errors.Nature, 323(6088):533–536, 1986

    work page 1986

  68. [68]

    Kipf and Max Welling

    Thomas N. Kipf and Max Welling. Semi-Supervised Classification with Graph Convolutional Networks. InInternational Conference on Learning Representations, 2017. URL https://openreview.net/for um?id=SJU4ayYgl

    work page 2017

  69. [69]

    Arnold.Finite element exterior calculus

    Douglas N. Arnold.Finite element exterior calculus. SIAM, 2018

    work page 2018

  70. [70]

    Bendall, Erin F

    Sean C. Bendall, Erin F. Simonds, Peng Qiu, El-ad D. Amir, Peter O. Krutzik, Rachel Finck, Robert V . Bruggner, Rachel Melamed, Angelica Trejo, Olga I. Ornatsky, Robert S. Balderas, Sylvia K. Plevritis, Karen Sachs, Dana Pe’er, Scott D. Tanner, and Garry P. Nolan. Single-cell mass cytometry of differential immune and drug responses across a human hematopo...

    work page doi:10.1126/science.1198704 2011

  71. [71]

    Crowell, Lukas M

    Malgorzata Nowicka, Carsten Krieg, Helena L. Crowell, Lukas M. Weber, Felix J. Hartmann, Silvia Guglietta, Burkhard Becher, Mitchell P. Levesque, and Mark D. Robinson. Cytof workflow: differential discovery in high-throughput high-dimensional cytometry datasets.F1000Research, 6:748, 2017. doi: 10.12688/f1000research.11622.3. URLhttps://doi.org/10.12688/f1...

    work page doi:10.12688/f1000research.11622.3 2017

  72. [72]

    Francies, Joshua M

    Marc van de Wetering, Hayley E. Francies, Joshua M. Francis, Gergana Bounova, Francesco Iorio, Apollo Pronk, Winan van Houdt, Joost van Gorp, Amaro Taylor-Weiner, Lennart Kester, et al. Prospective derivation of a living organoid biobank of colorectal cancer patients.Cell, 161(4):933–945, 2015. doi: 10.1016/j.cell.2015.03.053. URLhttps://doi.org/10.1016/j...

    work page doi:10.1016/j.cell.2015.03.053 2015

  73. [73]

    Patient-derived organoids model treatment response of metastatic gastrointestinal cancers.Science, 359(6378):920–926, 2018

    Georgios Vlachogiannis, Somaieh Hedayat, Alexandra Vatsiou, Yann Jamin, Javier Fernandez-Mateos, Khalid Khan, Andrea Lampis, Kevin Eason, Ian Huntingford, Rachel Burke, et al. Patient-derived organoids model treatment response of metastatic gastrointestinal cancers.Science, 359(6378):920–926, 2018. doi: 10.1126/science.aao2774. URLhttps://doi.org/10.1126/...

    work page doi:10.1126/science.aao2774 2018

  74. [74]

    Nicholson, Ambereen Ali, Nancy A

    Donald W. Nicholson, Ambereen Ali, Nancy A. Thornberry, John P. Vaillancourt, Connie K. Ding, Michel Gallant, Yves Gareau, Patrick R. Griffin, Marc Labelle, Yuri A. Lazebnik, Neil A. Munday, Sayyaparaju M. Raju, Mark E. Smulson, Ting-Ting Yamin, Violeta L. Yu, and Douglas K. Miller. Identification and inhibition of the ice/ced-3 protease necessary for mam...

    work page doi:10.1038/376037a0 1995

  75. [75]

    Quan, Karen O’Rourke, Stephanie Desnoyers, Zhenying Zeng, Deborah R

    Muneesh Tewari, Leslie T. Quan, Karen O’Rourke, Stephanie Desnoyers, Zhenying Zeng, Deborah R. Beidler, Guy G. Poirier, Guy S. Salvesen, and Vishva M. Dixit. Yama/cpp32 beta, a mammalian homolog of ced-3, is a crma-inhibitable protease that cleaves the death substrate poly(adp-ribose) polymerase.Cell, 81(5):801–809, 1995. doi: 10.1016/0092-8674(95)90541-3...

    work page doi:10.1016/0092-8674(95)90541-3 1995

  76. [76]

    Jackson and Jiri Bartek

    Stephen P. Jackson and Jiri Bartek. The dna-damage response in human biology and disease.Nature, 461: 1071–1078, 2009. doi: 10.1038/nature08467. URLhttps://doi.org/10.1038/nature08467

    work page doi:10.1038/nature08467 2009

  77. [77]

    Lord and Alan Ashworth

    Christopher J. Lord and Alan Ashworth. The dna damage response and cancer therapy.Nature, 481: 287–294, 2012. doi: 10.1038/nature10760. URLhttps://doi.org/10.1038/nature10760

    work page doi:10.1038/nature10760 2012

  78. [78]

    Cell cycle, cdks and cancer: a changing paradigm.Nature Reviews Cancer, 9:153–166, 2009

    Marcos Malumbres and Mariano Barbacid. Cell cycle, cdks and cancer: a changing paradigm.Nature Reviews Cancer, 9:153–166, 2009. doi: 10.1038/nrc2602. URL https://doi.org/10.1038/nrc2602

    work page doi:10.1038/nrc2602 2009

  79. [79]

    Hallmarks of cancer: new dimensions.Cancer Discovery, 12(1):31–46, 2022

    Douglas Hanahan. Hallmarks of cancer: new dimensions.Cancer Discovery, 12(1):31–46, 2022. doi: 10.1158/2159-8290.CD-21-1059. URLhttps://doi.org/10.1158/2159-8290.CD-21-1059

    work page doi:10.1158/2159-8290.cd-21-1059 2022

  80. [80]

    The biology and function of fibroblasts in cancer.Nature Reviews Cancer, 16:582–598,

    Raghu Kalluri. The biology and function of fibroblasts in cancer.Nature Reviews Cancer, 16:582–598,

    work page

Showing first 80 references.