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arxiv: 2605.21828 · v1 · pith:ZJZC6DZHnew · submitted 2026-05-20 · 🧮 math.NA · cs.NA

A Butterfly-Accelerated Manifold Harmonic Transform

Paul G. Beckman , Samuel F. Potter , Michael O'Neil This is my paper

Pith reviewed 2026-05-22 07:54 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords manifold harmonicsbutterfly factorizationLaplace-Beltrami eigenfunctionsfast transformslow-rank approximationssurface discretizationnumerical linear algebra
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The pith

Butterfly factorization enables fast computation of manifold harmonic transforms on arbitrary surfaces.

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

The paper aims to extend fast analysis and synthesis transforms from simple domains like the circle and sphere to eigenfunctions of the Laplace-Beltrami operator on general surfaces. It does so by treating the manifold harmonic transform as a dense matrix and replacing direct multiplication with a compressed representation. The compression comes from a butterfly factorization that builds a hierarchy of nested low-rank approximations to selected submatrices. Numerical tests on varied geometries and discretizations show concrete gains in speed and memory, while a separate analysis treats the flat periodic square in detail.

Core claim

The central claim is that a butterfly factorization, constructed by hierarchically compressing the manifold harmonic transform matrix through nested low-rank approximations of carefully selected submatrices, yields a fast algorithm for computing linear combinations of Laplace-Beltrami eigenfunctions on arbitrary surfaces. This generalizes the classical fast transforms for complex exponentials and spherical harmonics. The resulting method reduces both runtime and storage compared with direct evaluation, as confirmed by examples across multiple geometries and discretizations.

What carries the argument

Butterfly factorization of the manifold harmonic transform matrix, which builds a hierarchy of nested low-rank approximations for selected submatrices.

If this is right

  • Linear combinations of manifold harmonics become practical to evaluate on complex surfaces without quadratic cost.
  • Memory footprint for the transform operator drops substantially through the hierarchical compression.
  • The same framework applies across different surface discretizations and application domains.
  • Error and complexity can be bounded explicitly when the surface is a flat periodic square.

Where Pith is reading between the lines

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

  • The same low-rank hierarchical structure might appear in other integral operators on manifolds, allowing similar accelerations.
  • Adaptive choice of the low-rank blocks could further reduce error for surfaces with sharp features.
  • The approach supplies a concrete route toward real-time harmonic analysis in geometry-processing pipelines.

Load-bearing premise

The underlying surface discretization produces a transform matrix whose selected submatrices possess sufficient hierarchical low-rank structure for the butterfly compression to succeed with controlled error.

What would settle it

Running the algorithm on a surface discretization whose submatrices lack the expected low-rank structure and measuring whether the approximation error grows beyond the claimed tolerance or the runtime fails to improve.

Figures

Figures reproduced from arXiv: 2605.21828 by Michael O'Neil, Paul G. Beckman, Samuel F. Potter.

Figure 1
Figure 1. Figure 1: Eigenfunctions φk of the Laplace-Beltrami operator on a cow manifold for k = 2, 4, 8, 78, 340, 1350. eigenfunctions of the Laplacian ∆ on [−π, π] 2 under periodic boundary conditions, it is natural to consider a generalization of the Fourier transform which uses Laplacian eigenfunctions to perform harmonic analysis on domains other than the flat periodic square. For a general compact d-manifold M with boun… view at source ↗
Figure 2
Figure 2. Figure 2: Row tree, column tree, and resulting low-rank blocks of Φ at levels ℓ = 0, . . . , 3. up the column tree (doubling the number of columns). In doing so, the number of entries in each block is held constant, and the complementary low-rank property can be exploited in a hierarchical fashion at each level [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Local node labels for relevant subtrees in the space and frequency trees Tx and Tλ. Since, by assumption, the matrix admits the complementary low-rank property, we can then compute a low-rank factorization of the first matrix above [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: A subset of the nodes of a cut-cell-free Neumann Fiedler tree on the cow mesh, with the second Neumann eigenfunction '2 plotted on each submanifold. quadrilateral mesh, plotting the eigenfunctions '2 used on various submanifolds at each level. 4. The butterfly-accelerated manifold harmonic transform. Having de￾veloped a tree structure on M, we can now apply the butterfly factorization to the MHT. However, … view at source ↗
Figure 5
Figure 5. Figure 5: A schematic depiction of the streaming butterfly factorization. The nodes of the column (frequency) tree are traversed using a post-order traversal. When leaf nodes are visited, new bands of eigenvectors are streamed. When an internal node is visited, the partial factors corresponding to that node’s children are merged and split. Eight steps of the algorithm are shown in sequence, with the block boxes indi… view at source ↗
Figure 6
Figure 6. Figure 6: Diagram of the two-dimensional Fourier transform as a MHT. The annulus A √ √ ρk ρk−1 (left) contains eigenfrequencies ω = [k1, k2] for k1, k2 ∈ Z, denoted by black dots. These eigenfrequencies are mapped by ∥ω∥ 2 = λ to the corresponding eigenvalues λk of ∆[−π,π] 2 (center). Each block Φ(τ, ν) of the MHT maps coefficients of eigenfunctions with λ ∈ [ρk−1, ρk] to values in a disk x ∈ BR(cj ). for any (b − a… view at source ↗
Figure 7
Figure 7. Figure 7: Numerical experiments for the flat torus M = [−π, π] 2 . Scaling with n for m = ⌈n/25⌉ for various tolerances ε (left). Matvec relative error as a function of compression tolerance ε for n = 160,000 and m = 6,400 (center). ε-ranks of circle-to-circle Fourier transforms and of compressed blocks of the BF-MHT corresponding to annulus-to-disk Fourier transforms with n = 106 , m = 40,000, and ε = 10−3 (right).… view at source ↗
Figure 8
Figure 8. Figure 8: Numerical experiments for a deformed torus. Scaling with n for m = ⌈n/32⌉ for various tolerances ε (left). Matvec relative error as a function of compression tolerance ε for n = 51,200 and m = 1,600 (center). Mesh of the geometry (right). numerically. For this purpose we use a fast direct solver for the Laplace-Beltrami problem based on a spectral collocation scheme [20]. We then employ a Krylov eigensolve… view at source ↗
Figure 9
Figure 9. Figure 9: Filtered mesh with low-pass filter F(λ) = 1{λ<1000} (left). Recovered mesh with node coordinates (x˜1)i, . . . ,(x˜n)i for i = 1, 2, 3 (center). Filtered mesh with a Gaussian bump for moderate eigenvalues F(λ) = 1 + 6400 exp(−(λ − 5500)2 /50002 ) (right). for i = 1, 2, 3. The coefficient vectors c (i) are then multiplied pointwise by a filter function c (i) k 7→ F(λk)c (i) k which amplifies or attenuates g… view at source ↗
Figure 10
Figure 10. Figure 10: Samples from GRFs for Mat´ern spectral densities with ν = 3 (left) and ν = 1 (center), and for a random surface wave model Sθ(λ) = exp{−(λ − λ0) 2 /η2 } (right) on the deformed torus quadrilateral mesh and the Lake Superior triangularization with Dirichlet boundary conditions. Laplacian are used as approximations to the Laplace-Beltrami eigenfunctions on the underlying manifold. If the data lie exactly on… view at source ↗
Figure 11
Figure 11. Figure 11: Three eigenfunctions of the weighted graph Laplacian for noisy data on the surface of a hand (left) and scaling of BF-MHT to compress the first m =  n 50  eigenfunctions to tolerance ε = 10−3 (right). butterfly compress the eigenvector matrix. We repeat this experiment for a number of data sizes n [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
read the original abstract

The eigenfunctions of the Laplacian are a natural basis of functions for many tasks in computational mathematics. On the circle and sphere, the eigenfunctions are given by complex periodic exponentials and spherical harmonics, respectively, and much work has been done to develop fast algorithms for analyzing and synthesizing data in these bases. In this work, we generalize these special-case transforms to Laplace-Beltrami eigenfunctions of arbitrary surfaces, referred to as manifold harmonics. The resulting fast algorithm for computing linear combinations of the manifold harmonics is based on a butterfly factorization, which hierarchically compresses the transform matrix by constructing nested low-rank approximations of carefully selected submatrices. Several numerical examples are provided which demonstrate the speedups and reduction in memory requirements achieved by our algorithm for a variety of geometries, discretizations, and applications. In addition, a detailed analysis of the algorithm is given in the case that the underlying manifold is the flat periodic square.

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 generalizes fast harmonic transforms from circles and spheres to Laplace-Beltrami eigenfunctions (manifold harmonics) on arbitrary surfaces. The fast algorithm for linear combinations of these harmonics is obtained by applying a butterfly factorization that hierarchically compresses the transform matrix via nested low-rank approximations of selected submatrices. Numerical examples on a variety of geometries and discretizations demonstrate speedups and memory savings; a detailed analysis of complexity and accuracy is supplied only when the manifold is the flat periodic square.

Significance. If the low-rank structure and error control extend beyond the flat case, the work would supply a practical O(N log N) method for manifold-harmonic analysis and synthesis, generalizing classical FFT-type algorithms to surfaces arising in graphics, geometry processing, and PDEs. The numerical demonstrations already indicate concrete gains in runtime and storage; the rigorous treatment of the flat square provides a useful benchmark and proof-of-concept for the compression technique.

major comments (2)
  1. [Abstract and general-algorithm section] Abstract and the section describing the general algorithm: the claim that the butterfly factorization yields a fast, accurate algorithm for arbitrary manifolds rests on the assumption that carefully chosen submatrices of the manifold-harmonic transform admit nested low-rank approximations whose ranks and errors remain controlled across the hierarchy. Rigorous singular-value decay estimates and error bounds are derived only for the flat periodic square; for curved or irregular surfaces the manuscript supplies only numerical evidence without rank-growth bounds or approximation guarantees. This gap is load-bearing for the central claim of general applicability.
  2. [Numerical-examples section] Numerical-examples section: the reported speedups and accuracy are convincing for the tested discretizations, yet without theoretical control on the effective rank or the constant in the O(N log N) complexity it remains unclear whether the observed performance extrapolates to finer meshes or more complex geometries, or whether it depends on favorable properties of the particular surface discretizations chosen.
minor comments (2)
  1. [Introduction] The notation distinguishing the continuous manifold-harmonic operator from its discrete matrix representation could be introduced earlier and used consistently to improve readability.
  2. [Numerical-examples section] A short table summarizing the observed compression ratios and wall-clock times across all examples would make the empirical results easier to compare at a glance.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the distinction between the rigorous analysis available for the flat periodic square and the numerical support for general manifolds. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract and general-algorithm section] Abstract and the section describing the general algorithm: the claim that the butterfly factorization yields a fast, accurate algorithm for arbitrary manifolds rests on the assumption that carefully chosen submatrices of the manifold-harmonic transform admit nested low-rank approximations whose ranks and errors remain controlled across the hierarchy. Rigorous singular-value decay estimates and error bounds are derived only for the flat periodic square; for curved or irregular surfaces the manuscript supplies only numerical evidence without rank-growth bounds or approximation guarantees. This gap is load-bearing for the central claim of general applicability.

    Authors: We agree that the manuscript derives rigorous singular-value decay estimates and error bounds exclusively for the flat periodic square, as stated in the abstract and the analysis section. For arbitrary manifolds the algorithm is presented as a general procedure whose practical performance rests on the empirical observation that selected submatrices admit useful low-rank approximations; this observation is documented through numerical examples on curved and irregular surfaces. The flat-square case is supplied as a benchmark that proves the O(N log N) complexity under controlled rank growth. We will revise the abstract and the general-algorithm section to state more explicitly that theoretical guarantees are currently limited to the flat case while the method is shown to be effective more broadly via numerical validation. This clarification will be made without altering the algorithmic description or the reported experiments. revision: partial

  2. Referee: [Numerical-examples section] Numerical-examples section: the reported speedups and accuracy are convincing for the tested discretizations, yet without theoretical control on the effective rank or the constant in the O(N log N) complexity it remains unclear whether the observed performance extrapolates to finer meshes or more complex geometries, or whether it depends on favorable properties of the particular surface discretizations chosen.

    Authors: The numerical section already includes results on several distinct geometries and discretization types, all exhibiting comparable speedups and memory reductions. In the flat periodic square we supply the full complexity analysis with explicit rank bounds. For the remaining examples we report the observed effective ranks and runtimes, which remain modest. To address the extrapolation concern we will add a short discussion subsection that tabulates the measured rank growth with respect to mesh size and surface complexity, together with a brief statement on the empirical evidence for scalability. We acknowledge that a general theoretical bound on the rank constant is not yet available. revision: yes

standing simulated objections not resolved
  • Deriving rigorous singular-value decay estimates and approximation guarantees that hold uniformly for arbitrary curved or irregular manifolds would require substantial additional theoretical work that lies outside the scope of the present manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation applies external butterfly compression to manifold-harmonic matrix

full rationale

The manuscript constructs the fast transform by applying a standard butterfly factorization (hierarchical low-rank compression of selected submatrices) to the manifold-harmonic operator. The only analytic error bounds and rank estimates are supplied for the flat periodic square; all other geometries are supported by numerical demonstration. No equation reduces a claimed prediction or complexity result to a fitted parameter or self-citation by construction, and the load-bearing step (existence of controlled nested low-rank structure) is an assumption about the discretization rather than a definitional tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on the abstract alone, no explicit free parameters, axioms, or invented entities are stated; the method appears to rest on standard numerical linear algebra assumptions about low-rank approximability of submatrices.

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Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [1]

    Alsnayyan and B

    A. Alsnayyan and B. Shanker,Iso-geometric integral equation solvers and their compression via manifold harmonics, IEEE Trans. Antennas Propag., 70 (2022), pp. 6893–6904. A BUTTERFLY-ACCELERATED MANIFOLD HARMONIC TRANSFORM23

    work page 2022

  2. [2]

    A. M. Alsnayyan, L. Kempel, and S. Balasubramaniam,Manifold harmonics and their application to computational electromagnetics, IEEE J. Multiscale Multiphysics Comput. Tech., 7 (2022), pp. 336–345

    work page 2022

  3. [3]

    P. G. Beckman and M. O’Neil,A nonuniform fast hankel transform, SIAM Journal on Scientific Computing, 48 (2026), pp. A784–A803

    work page 2026

  4. [4]

    Belkin and P

    M. Belkin and P. Niyogi,Laplacian eigenmaps for dimensionality reduction and data repre- sentation, Neural computation, 15 (2003), pp. 1373–1396

    work page 2003

  5. [5]

    J. K. Bennighof and R. B. Lehoucq,An automated multilevel substructuring method for eigenspace computation in linear elastodynamics, SIAM Journal on Scientific Computing, 25 (2004), pp. 2084–2106

    work page 2004

  6. [6]

    Berger, L

    M. Berger, L. G. Nonato, V. Pascucci, and C. T. Silva,Fiedler trees for multiscale surface analysis, Computers & Graphics, 34 (2010), pp. 272–281

    work page 2010

  7. [7]

    Bonito, J

    A. Bonito, J. M. Casc ´on, P. Morin, and R. H. Nochetto,AFEM for geometric PDE: the Laplace-Beltrami operator, Analysis and numerics of partial differential equations, (2013), pp. 257–306

    work page 2013

  8. [8]

    Borovitskiy, A

    V. Borovitskiy, A. Terenin, P. Mostowsky, et al.,Mat´ ern gaussian processes on riemannian manifolds, Advances in Neural Information Processing Systems, 33 (2020), pp. 12426–12437

    work page 2020

  9. [9]

    Candes, L

    E. Candes, L. Demanet, and L. Ying,A fast butterfly algorithm for the computation of Fourier integral operators, Multiscale Model. Simul., 7 (2009), pp. 1727–1750. [10]I. Chavel,Eigenvalues in Riemannian geometry, vol. 115, Academic press, 1984

    work page 2009

  10. [10]

    Cheeger,A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis, 625 (1970), p

    J. Cheeger,A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis, 625 (1970), p. 110

    work page 1970

  11. [11]

    Chiu and L

    J. Chiu and L. Demanet,Sublinear randomized algorithms for skeleton decompositions, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 1361–1383

    work page 2013

  12. [12]

    R. R. Coifman and M. Maggioni,Diffusion wavelets, Applied and computational harmonic analysis, 21 (2006), pp. 53–94

    work page 2006

  13. [13]

    Cortinovis and L

    A. Cortinovis and L. Ying,A sublinear-time randomized algorithm for column and row subset selection based on strong rank-revealing qr factorizations, SIAM Journal on Matrix Analysis and Applications, 46 (2025), pp. 22–44

    work page 2025

  14. [14]

    Demanet, M

    L. Demanet, M. Ferrara, N. Maxwell, J. Poulson, and L. Ying,A butterfly algorithm for synthetic aperture radar imaging, SIAM J. Imag. Sci., 5 (2012), pp. 203–243

    work page 2012

  15. [15]

    Dutt and V

    A. Dutt and V. Rokhlin,Fast Fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14 (1993), pp. 1368–1393

    work page 1993

  16. [16]

    Dziuk and C

    G. Dziuk and C. M. Elliott,Finite element methods for surface PDEs, Acta Numerica, 22 (2013), pp. 289–396

    work page 2013

  17. [17]

    Engquist and L

    B. Engquist and L. Ying,A fast directional algorithm for high frequency acoustic scattering in two dimensions, Commun. Math. Sci., 7 (2009), pp. 327–345

    work page 2009

  18. [18]

    Fiedler,A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory, Czechoslovak mathematical journal, 25 (1975), pp

    M. Fiedler,A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory, Czechoslovak mathematical journal, 25 (1975), pp. 619–633

    work page 1975

  19. [19]

    Fortunato,A high-order fast direct solver for surface pdes, SIAM Journal on Scientific Computing, 46 (2024), pp

    D. Fortunato,A high-order fast direct solver for surface pdes, SIAM Journal on Scientific Computing, 46 (2024), pp. A2582–A2606

    work page 2024

  20. [20]

    Giannakis,Data-driven spectral decomposition and forecasting of ergodic dynamical systems, Appl

    D. Giannakis,Data-driven spectral decomposition and forecasting of ergodic dynamical systems, Appl. Comput. Harm. Anal., 47 (2019), pp. 338–396

    work page 2019

  21. [21]

    G. H. Golub and C. F. Van Loan,Matrix computations, John Hopkins University Press, 1983

    work page 1983

  22. [22]

    Greengard and J.-Y

    L. Greengard and J.-Y. Lee,Accelerating the nonuniform fast Fourier transform, SIAM Review, 46 (2004), pp. 443–454

    work page 2004

  23. [23]

    Lang and M

    A. Lang and M. Pereira,Galerkin–Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds, BIT Numerical Mathematics, 63 (2023), p. 51

    work page 2023

  24. [24]

    understands

    B. L´evy,Laplace-Beltrami eigenfunctions towards an algorithm that “understands” geometry, in IEEE International Conference on Shape Modeling and Applications 2006 (SMI’06), IEEE, 2006, pp. 13–13

    work page 2006

  25. [25]

    Li and H

    Y. Li and H. Yang,Interpolative butterfly factorization, SIAM J. Sci. Comput., 39 (2017), pp. A503–A531

    work page 2017

  26. [26]

    Y. Li, H. Yang, E. R. Martin, K. L. Ho, and L. Ying,Butterfly factorization, Multiscale Modeling & Simulation, 13 (2015), pp. 714–732

    work page 2015

  27. [27]

    Y. Liu, X. Xing, H. Guo, E. Michielssen, P. Ghysels, and X. S. Li,Butterfly factorization via randomized matrix-vector multiplications, SIAM Journal on Scientific Computing, 43 (2021), pp. A883–A907

    work page 2021

  28. [28]

    Michielssen and A

    E. Michielssen and A. Boag,A multilevel matrix decomposition algorithm for analyzing scattering from large structures, IEEE Trans. Antennas Propag., 44 (1996), pp. 1086–1093

    work page 1996

  29. [29]

    M. J. Mohlenkamp,A fast transform for spherical harmonics, Journal of Fourier analysis and applications, 5 (1999), pp. 159–184. 24BECKMAN, POTTER, AND O’NEIL

    work page 1999

  30. [30]

    Nasikun, C

    A. Nasikun, C. Brandt, and K. Hildebrandt,Fast approximation of Laplace-Beltrami eigenproblems, in Computer Graphics Forum, vol. 37, Wiley Online Library, 2018, pp. 121– 134

    work page 2018

  31. [31]

    F. W. Olver, D. W. Lozier, R. Boisvert, and C. W. Clark,The NIST handbook of mathematical functions, (2010). [33]M. O’Neil,A new class of analysis-based fast transforms, tech. report, Yale University, 2007

    work page 2010

  32. [32]

    O’Neil, F

    M. O’Neil, F. Woolfe, and V. Rokhlin,An algorithm for the rapid evaluation of special function transforms, Appl. Comput. Harm. Anal., 28 (2010), pp. 203–226

    work page 2010

  33. [33]

    C. C. Paige and M. A. Saunders,Lsqr: An algorithm for sparse linear equations and sparse least squares, ACM Transactions on Mathematical Software (TOMS), 8 (1982), pp. 43–71

    work page 1982

  34. [34]

    Q. Pang, K. L. Ho, and H. Yang,Interpolative decomposition butterfly factorization, SIAM J. Sci. Comput., 42 (2020), pp. A1097–A1115

    work page 2020

  35. [35]

    Patan`e,Laplacian spectral basis functions, Computer aided geometric design, 65 (2018), pp

    G. Patan`e,Laplacian spectral basis functions, Computer aided geometric design, 65 (2018), pp. 31–47

    work page 2018

  36. [36]

    Reuter, S

    M. Reuter, S. Biasotti, D. Giorgi, G. Patan `e, and M. Spagnuolo,Discrete Laplace– Beltrami operators for shape analysis and segmentation, Computers & Graphics, 33 (2009), pp. 381–390

    work page 2009

  37. [37]

    Seljebotn,Wavemoth–fast spherical harmonic transforms by butterfly matrix compression, The Astrophysical Journal Supplement Series, 199 (2012), p

    D. Seljebotn,Wavemoth–fast spherical harmonic transforms by butterfly matrix compression, The Astrophysical Journal Supplement Series, 199 (2012), p. 5

    work page 2012

  38. [38]

    S. Seo, M. K. Chung, and H. K. Vorperian,Heat kernel smoothing using Laplace-Beltrami eigenfunctions, in Medical Image Computing and Computer-Assisted Intervention–MICCAI 2010: 13th International Conference, Beijing, China, September 20-24, 2010, Proceedings, Part III 13, Springer, 2010, pp. 505–512

    work page 2010

  39. [39]

    Shi and J

    J. Shi and J. Malik,Normalized cuts and image segmentation, IEEE Transactions on pattern analysis and machine intelligence, 22 (2000), pp. 888–905

    work page 2000

  40. [40]

    Singer,From graph to manifold laplacian: The convergence rate, Applied and Computational Harmonic Analysis, 21 (2006), pp

    A. Singer,From graph to manifold laplacian: The convergence rate, Applied and Computational Harmonic Analysis, 21 (2006), pp. 128–134

    work page 2006

  41. [41]

    R. M. Slevinsky,Fast and backward stable transforms between spherical harmonic expansions and bivariate Fourier series, Appl. Comput. Harm. Anal., 47 (2019), pp. 585–606

    work page 2019

  42. [42]

    Solin and M

    A. Solin and M. Kok,Know your boundaries: Constraining gaussian processes by variational harmonic features, in The 22nd International Conference on Artificial Intelligence and Statistics, PMLR, 2019, pp. 2193–2202

    work page 2019

  43. [43]

    Solin and S

    A. Solin and S. S ¨arkk¨a,Hilbert space methods for reduced-rank gaussian process regression, Statistics and Computing, 30 (2020), pp. 419–446

    work page 2020

  44. [44]

    G. W. Stewart,A Krylov–Schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 601–614

    work page 2002

  45. [45]

    Su and R

    P.-C. Su and R. R. Coifman,Learning the analytic geometry of transformations to achieve efficient computation, arXiv preprint arXiv:2506.11990, (2025)

    work page arXiv 2025

  46. [46]

    A. D. Szlam, M. Maggioni, R. R. Coifman, and J. C. Bremer Jr,Diffusion-driven multiscale analysis on manifolds and graphs: top-down and bottom-up constructions, in Wavelets XI, vol. 5914, SPIE, 2005, pp. 445–455

    work page 2005

  47. [47]

    Townsend,A fast analysis-based discrete Hankel transform using asymptotic expansions, SIAM Journal on Numerical Analysis, 53 (2015), pp

    A. Townsend,A fast analysis-based discrete Hankel transform using asymptotic expansions, SIAM Journal on Numerical Analysis, 53 (2015), pp. 1897–1917

    work page 2015

  48. [48]

    Tygert,Fast algorithms for spherical harmonic expansions, III, J

    M. Tygert,Fast algorithms for spherical harmonic expansions, III, J. Comput. Phys., 229 (2010), pp. 6181–6192

    work page 2010

  49. [49]

    Vallet and B

    B. Vallet and B. L´evy,Spectral geometry processing with manifold harmonics, in Computer Graphics Forum, vol. 27, Wiley Online Library, 2008, pp. 251–260. [52]C. Van Loan,Computational frameworks for the fast Fourier transform, SIAM, 1992

    work page 2008

  50. [50]

    C. K. Williams and C. E. Rasmussen,Gaussian processes for machine learning, vol. 2, MIT press Cambridge, MA, 2006

    work page 2006

  51. [51]

    Zheng, E

    L. Zheng, E. Riccietti, and R. Gribonval,Efficient identification of butterfly sparse matrix factorizations, SIAM J. Math. Data Sci., 5 (2023), pp. 22–49

    work page 2023