pith. sign in

arxiv: 2509.03758 · v4 · submitted 2025-09-03 · 💻 cs.LG · cs.NA· math.NA

A Data-Driven Interpolation Method on Smooth Manifolds via Diffusion Processes and Voronoi Tessellations

Alvaro Almeida Gomez This is my paper

Pith reviewed 2026-05-18 18:47 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NA
keywords manifold interpolationdiffusion processesVoronoi tessellationsNadaraya-Watson estimatortotal variationcompressed sensingLaplace-Beltrami operatorcomputational tomography
0
0 comments X

The pith

A Voronoi tessellation sets diffusion bandwidths so the resulting manifold interpolant has vanishing gradients at every sample point.

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

The paper develops a data-driven way to extend pointwise samples into a continuous function on a smooth manifold by using Voronoi cells to choose the bandwidth of a diffusion kernel inside a Nadaraya-Watson estimator. The construction needs no training phase and no preprocessing, and its inference cost grows only linearly with the number of points. A reader would care because the interpolant is proved to have zero gradient at the given samples and, with high probability for large point sets, to remove high-frequency content from the underlying signal. The same construction is shown to minimize a total-variation-type energy and therefore supplies a closed-form solution to a compressed-sensing problem whose forward operator is simply the identity.

Core claim

The interpolant is obtained by running a diffusion process whose bandwidth at each location is taken from the Voronoi cell of the nearest sample point on the manifold. This choice forces the gradient of the interpolant to vanish exactly at every sample point. As the number of samples tends to infinity the interpolant damps high-frequency modes with high probability. The same object also minimizes a total-variation-type energy and therefore constitutes a closed-form analytic approximation to the compressed-sensing problem with identity forward operator.

What carries the argument

Voronoi-tessellation bandwidth selection inside the diffusion kernel of a Nadaraya-Watson estimator on the manifold.

Load-bearing premise

The given point samples must be distributed so that their Voronoi tessellations accurately reflect the intrinsic geometry of the manifold and the diffusion process must closely approximate the Laplace-Beltrami operator.

What would settle it

A direct numerical computation on the unit sphere with evenly spaced points that shows the constructed function has a nonzero gradient at any sample point would falsify the vanishing-gradient property.

Figures

Figures reproduced from arXiv: 2509.03758 by Alvaro Almeida Gomez.

Figure 1
Figure 1. Figure 1: The logarithmic spiral curve M and the logarithmic function f(x) defined in Eq. (13). The purpose of this experiment is twofold. First, we investigate how well the method can learn the function outside the logarithmic spiral curve, i.e., over the two-dimensional cube [−exp(1), exp(1)]2 , and what kinds of patterns the learning procedure can extract from the dataset. Second, we study how accurately the meth… view at source ↗
Figure 2
Figure 2. Figure 2: The first column (left) shows the dataset, the middle column displays the learned function [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The first column (left) shows the dataset, the middle column displays the learned function [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: From left to right: Shepp-Logan phantom, human head, and human abdomen. The human images were [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sinograms of tomographic projections of the Shepp–Logan phantom. Columns (left to right): (i) sparse [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Tomographic reconstruction of the Shepp–Logan phantom. Columns (left to right): (i) sparse projections [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Sinograms of tomographic projections of a human head. Columns (left to right): (i) sparse projections [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Tomographic reconstruction of a human head. Columns (left to right): (i) sparse projections from the [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Sinograms of tomographic projections of a human abdomen. Columns (left to right): (i) sparse projections [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Tomographic reconstruction of a human abdomen. Columns (left to right): (i) sparse projections from the [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
read the original abstract

We propose a data-driven interpolation method for approximating real-valued functions on smooth manifolds, based on the Laplace--Beltrami operator and Voronoi tessellations. Given pointwise evaluations, the method constructs a continuous extension by exploiting diffusion processes and the intrinsic geometry of the data. The approach builds on the Nadaraya--Watson kernel regression estimator, where the bandwidth is determined by Voronoi tessellations of the manifold. It is fully data-driven and requires neither a training phase nor any preprocessing prior to inference. The computational complexity of the inference step scales linearly with the number of sample points, leading to substantial gains in scalability compared to classical methods such as neural networks, radial basis function networks, and Gaussian process regression. We show that the resulting interpolant has vanishing gradient at the sample points and, with high probability as the number of samples increases, suppresses high-frequency components of the signal. Moreover, the method can be interpreted as minimizing a total variation--type energy, providing a closed-form analytical approximation to a compressed sensing problem with identity forward operator. We illustrate the performance of the method on sparse computational tomography reconstruction, where it achieves competitive reconstruction quality while significantly reducing computational time relative to standard total variation--based approaches.

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 manuscript proposes a data-driven interpolation method for real-valued functions on smooth manifolds. It constructs a continuous extension via a Nadaraya-Watson kernel regressor whose bandwidths are set by Voronoi cell volumes on the point cloud, combined with a diffusion step that approximates the Laplace-Beltrami operator. The central claims are that the resulting interpolant has vanishing gradient at the sample points, suppresses high-frequency components with high probability as the number of samples grows, and admits a variational interpretation as minimizing a total-variation-type energy, thereby furnishing a closed-form proxy for compressed sensing with the identity operator. The method is illustrated on sparse computational tomography reconstruction, where it is reported to achieve competitive quality at linear inference cost.

Significance. If the claimed regularity and spectral properties can be rigorously established, the approach would supply a scalable, training-free interpolant with built-in smoothness and frequency-control features that are attractive for manifold-valued inverse problems. The linear complexity and absence of preprocessing are concrete practical strengths. The TV-energy reading, if substantiated, would also link the construction to compressed-sensing literature in a novel way.

major comments (2)
  1. [Abstract and §4] Abstract and §4 (theorems on vanishing gradient and high-probability frequency suppression): the high-probability statement that high-frequency components are suppressed as n→∞ rests on the unverified claim that the Voronoi-weighted diffusion kernel converges to the manifold heat kernel at a sufficient rate. No explicit error bound, discretization analysis, or sampling-density assumption is supplied that would justify the claimed spectral decay; without this, the frequency-suppression and TV-minimization interpretations remain conditional rather than proven.
  2. [§5] §5 (variational interpretation): the assertion that the interpolant minimizes a total-variation-type energy and thereby approximates compressed sensing with identity forward operator is presented without showing that the discrete energy induced by the Voronoi measure converges to the continuous TV functional on the manifold. This equivalence is load-bearing for the compressed-sensing claim yet is left at the level of interpretation.
minor comments (2)
  1. [§3.1] Clarify in §3.1 how the Voronoi cell volumes enter the kernel normalization; the current notation leaves ambiguous whether the estimator remains a partition of unity for non-uniform sampling.
  2. [Experiments] In the CT experiment (Figure 4), state the assumed manifold dimension and the distribution of the point samples; without this, it is difficult to assess whether the Voronoi tessellation faithfully captures intrinsic geometry.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. The comments correctly identify areas where the theoretical analysis requires additional rigor to support the main claims. We will revise the manuscript accordingly by adding the necessary discretization analysis and convergence results.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4 (theorems on vanishing gradient and high-probability frequency suppression): the high-probability statement that high-frequency components are suppressed as n→∞ rests on the unverified claim that the Voronoi-weighted diffusion kernel converges to the manifold heat kernel at a sufficient rate. No explicit error bound, discretization analysis, or sampling-density assumption is supplied that would justify the claimed spectral decay; without this, the frequency-suppression and TV-minimization interpretations remain conditional rather than proven.

    Authors: We agree that the high-probability frequency suppression and related claims depend on establishing a sufficiently rapid convergence of the Voronoi-weighted diffusion kernel to the manifold heat kernel. In the revised version we will add an appendix containing a discretization analysis. Under standard assumptions on the sampling density (e.g., the point cloud being an ε-net with ε = O(n^{-1/d}) on a d-dimensional manifold), we will derive explicit bounds on the kernel approximation error and show that these bounds imply the desired spectral decay with high probability as n → ∞. The vanishing-gradient property at sample points will also be strengthened by the same analysis. revision: yes

  2. Referee: [§5] §5 (variational interpretation): the assertion that the interpolant minimizes a total-variation-type energy and thereby approximates compressed sensing with identity forward operator is presented without showing that the discrete energy induced by the Voronoi measure converges to the continuous TV functional on the manifold. This equivalence is load-bearing for the compressed-sensing claim yet is left at the level of interpretation.

    Authors: We acknowledge that the variational interpretation as a total-variation minimizer requires a rigorous convergence statement. We will revise §5 to include a theorem establishing that the discrete energy functional, constructed from the Voronoi measure and the diffusion kernel, Γ-converges to the continuous total-variation semi-norm on the manifold in the dense-sampling limit. This result will directly support the claim that the method furnishes a closed-form proxy for compressed sensing with the identity operator. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on the standard Nadaraya-Watson kernel regression estimator whose bandwidth is set by Voronoi cell volumes on the manifold, combined with diffusion processes approximating the Laplace-Beltrami operator. Claims of vanishing gradients at sample points follow directly from the interpolant reproducing the data values, while high-probability high-frequency suppression and the total-variation energy interpretation are presented as consequences of the asymptotic behavior of the discrete kernel under increasing sample density. No step reduces a target result to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness theorem, or renames a known empirical pattern as a new derivation. The construction remains self-contained against external manifold-learning benchmarks and does not require the target claims to be presupposed in the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard differential-geometric assumptions about smooth manifolds and the Laplace-Beltrami operator; no free parameters or new entities are declared in the abstract.

axioms (1)
  • domain assumption The underlying space is a smooth manifold on which the Laplace-Beltrami operator governs diffusion.
    Invoked to justify the diffusion-process construction of the interpolant.

pith-pipeline@v0.9.0 · 5749 in / 1156 out tokens · 39778 ms · 2026-05-18T18:47:06.669646+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

27 extracted references · 27 canonical work pages · 1 internal anchor

  1. [1]

    Theoretical foundations of t-sne for visualizing high-dimensional clustered data

    T Tony Cai and Rong Ma. Theoretical foundations of t-sne for visualizing high-dimensional clustered data. Journal of Machine Learning Research, 23(301):1–54, 2022

    work page 2022

  2. [2]

    Kernel principal component analysis

    Bernhard Sch ¨olkopf, Alexander Smola, and Klaus-Robert M ¨uller. Kernel principal component analysis. In International conference on artificial neural networks, pages 583–588. Springer, 1997

    work page 1997

  3. [3]

    UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

    Leland McInnes, John Healy, and James Melville. Umap: Uniform manifold approximation and projection for dimension reduction.arXiv preprint arXiv:1802.03426, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  4. [4]

    Laplacian eigenmaps for dimensionality reduction and data representation

    Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural computation, 15(6):1373–1396, 2003

    work page 2003

  5. [5]

    Diffusion maps.Applied and computational harmonic analysis, 21(1):5– 30, 2006

    Ronald R Coifman and St´ephane Lafon. Diffusion maps.Applied and computational harmonic analysis, 21(1):5– 30, 2006

    work page 2006

  6. [6]

    Introduction to multi-layer feed-forward neural networks

    Daniel Svozil, Vladimir Kvasnicka, and Jiri Pospichal. Introduction to multi-layer feed-forward neural networks. Chemometrics and intelligent laboratory systems, 39(1):43–62, 1997

    work page 1997

  7. [7]

    Feed-forward neural networks.Ieee Potentials, 13(4):27–31, 2002

    George Bebis and Michael Georgiopoulos. Feed-forward neural networks.Ieee Potentials, 13(4):27–31, 2002

    work page 2002

  8. [8]

    Understanding the difficulty of training deep feedforward neural networks

    Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. InProceedings of the thirteenth international conference on artificial intelligence and statistics, pages 249–256. JMLR Workshop and Conference Proceedings, 2010

    work page 2010

  9. [9]

    A sparse-view ct reconstruc- tion method based on combination of densenet and deconvolution.IEEE transactions on medical imaging, 37(6):1407–1417, 2018

    Zhicheng Zhang, Xiaokun Liang, Xu Dong, Yaoqin Xie, and Guohua Cao. A sparse-view ct reconstruc- tion method based on combination of densenet and deconvolution.IEEE transactions on medical imaging, 37(6):1407–1417, 2018. 17 APREPRINT- SEPTEMBER24, 2025

    work page 2018

  10. [10]

    Sparse-view ct reconstruction via convolutional sparse coding

    Peng Bao, Wenjun Xia, Kang Yang, Jiliu Zhou, and Yi Zhang. Sparse-view ct reconstruction via convolutional sparse coding. In2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI 2019), pages 1446–

    work page 2019

  11. [11]

    Diffusion representation for asymmetric kernels.Applied Numerical Mathematics, 166:208–226, 2021

    Alvaro Almeida Gomez, Ant ˆonio J Silva Neto, and Jorge P Zubelli. Diffusion representation for asymmetric kernels.Applied Numerical Mathematics, 166:208–226, 2021

    work page 2021

  12. [12]

    Diffusion representation for asymmetric kernels via magnetic transform.Advances in Neural Information Processing Systems, 36:53742–53761, 2023

    Mingzhen He, Fan He, Ruikai Yang, and Xiaolin Huang. Diffusion representation for asymmetric kernels via magnetic transform.Advances in Neural Information Processing Systems, 36:53742–53761, 2023

    work page 2023

  13. [13]

    A diffusion-map-based algorithm for gradient computation on manifolds and applications.IEEE Access, 11:90622–90640, 2023

    Alvaro Almeida Gomez, Ant ˆonio J Silva Neto, and Jorge P Zubelli. A diffusion-map-based algorithm for gradient computation on manifolds and applications.IEEE Access, 11:90622–90640, 2023

    work page 2023

  14. [14]

    Numerical implementation of learning functions through diffusion maps in MATLAB

    Alvaro Almeida Gomez. Numerical implementation of learning functions through diffusion maps in MATLAB. https://github.com/alvaroalmeidagomez/LTD/tree/main/SE, 2025. Accessed: September 2, 2025

    work page 2025

  15. [15]

    Iterative image reconstruction for sparse-view ct using normal-dose image induced total variation prior.PloS one, 8(11):e79709, 2013

    Jing Huang, Yunwan Zhang, Jianhua Ma, Dong Zeng, Zhaoying Bian, Shanzhou Niu, Qianjin Feng, Zhengrong Liang, and Wufan Chen. Iterative image reconstruction for sparse-view ct using normal-dose image induced total variation prior.PloS one, 8(11):e79709, 2013

    work page 2013

  16. [16]

    Super-sparsely view-sampled cone-beam ct by incorporating prior data.Journal of X-ray science and technology, 21(1):71–83, 2013

    Sajid Abbas, Jonghwan Min, and Seungryong Cho. Super-sparsely view-sampled cone-beam ct by incorporating prior data.Journal of X-ray science and technology, 21(1):71–83, 2013

    work page 2013

  17. [17]

    Learn: Learned experts’ assessment-based reconstruction network for sparse-data ct.IEEE transactions on medical imaging, 37(6):1333–1347, 2018

    Hu Chen, Yi Zhang, Yunjin Chen, Junfeng Zhang, Weihua Zhang, Huaiqiang Sun, Yang Lv, Peixi Liao, Jiliu Zhou, and Ge Wang. Learn: Learned experts’ assessment-based reconstruction network for sparse-data ct.IEEE transactions on medical imaging, 37(6):1333–1347, 2018

    work page 2018

  18. [18]

    Deep-neural-network- based sinogram synthesis for sparse-view ct image reconstruction.IEEE Transactions on Radiation and Plasma Medical Sciences, 3(2):109–119, 2018

    Hoyeon Lee, Jongha Lee, Hyeongseok Kim, Byungchul Cho, and Seungryong Cho. Deep-neural-network- based sinogram synthesis for sparse-view ct image reconstruction.IEEE Transactions on Radiation and Plasma Medical Sciences, 3(2):109–119, 2018

    work page 2018

  19. [19]

    Dual-domain deep prior guided sparse-view ct reconstruction with multi-scale fusion attention.Scientific Reports, 15(1):16894, 2025

    Jia Wu, Jinzhao Lin, Xiaoming Jiang, Wei Zheng, Lisha Zhong, Yu Pang, Hongying Meng, and Zhangyong Li. Dual-domain deep prior guided sparse-view ct reconstruction with multi-scale fusion attention.Scientific Reports, 15(1):16894, 2025

    work page 2025

  20. [20]

    he visible human project.https://www.nlm.nih.gov/research/ visible/mri.html, 2025

    National Library of Medicine NLM. he visible human project.https://www.nlm.nih.gov/research/ visible/mri.html, 2025. Accessed: September 2, 2025

    work page 2025

  21. [21]

    Numerical implementation of sparse ct reconstruction in MATLAB.https://github

    Alvaro Almeida Gomez. Numerical implementation of sparse ct reconstruction in MATLAB.https://github. com/alvaroalmeidagomez/LTD/tree/main/Sparse_CT, 2025. Accessed: September 2, 2025

    work page 2025

  22. [22]

    Optimal dynamic fixed-mix portfolios based on rein- forcement learning with second order stochastic dominance.Engineering Applications of Artificial Intelligence, 133:108599, 2024

    Giorgio Consigli, Alvaro A Gomez, and Jorge P Zubelli. Optimal dynamic fixed-mix portfolios based on rein- forcement learning with second order stochastic dominance.Engineering Applications of Artificial Intelligence, 133:108599, 2024

    work page 2024

  23. [23]

    Reinforcement learning with dynamic convex risk measures.Mathe- matical Finance, 34(2):557–587, 2024

    Anthony Coache and Sebastian Jaimungal. Reinforcement learning with dynamic convex risk measures.Mathe- matical Finance, 34(2):557–587, 2024

    work page 2024

  24. [24]

    Reinforcement learning for algorithmic trading.Machine Learning and Data Sciences for Financial Markets: A Guide to Contemporary Practices

    ´Alvaro Cartea, Sebastian Jaimungal, and Leandro S ´anchez-Betancourt. Reinforcement learning for algorithmic trading.Machine Learning and Data Sciences for Financial Markets: A Guide to Contemporary Practices. Cambridge University Press, 2023

    work page 2023

  25. [25]

    Dgm: A deep learning algorithm for solving partial differential equations.Journal of computational physics, 375:1339–1364, 2018

    Justin Sirignano and Konstantinos Spiliopoulos. Dgm: A deep learning algorithm for solving partial differential equations.Journal of computational physics, 375:1339–1364, 2018

    work page 2018

  26. [26]

    Solving high-dimensional partial differential equations using deep learning.Proceedings of the National Academy of Sciences, 115(34):8505–8510, 2018

    Jiequn Han, Arnulf Jentzen, and Weinan E. Solving high-dimensional partial differential equations using deep learning.Proceedings of the National Academy of Sciences, 115(34):8505–8510, 2018

    work page 2018

  27. [27]

    Deepxde: A deep learning library for solving differential equations.SIAM review, 63(1):208–228, 2021

    Lu Lu, Xuhui Meng, Zhiping Mao, and George Em Karniadakis. Deepxde: A deep learning library for solving differential equations.SIAM review, 63(1):208–228, 2021. 18

    work page 2021