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REVIEW 3 major objections 3 minor 47 references

PhaseJumps: fast computation of zeros from planar grid samples

T0 review · 3 major / 3 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read PhaseJumps recovers zeros of complex-valued functions from planar grid samples by tracking phase changes between neighbors, with √δ Wasserstein accuracy and vanishing failure probability even for non-analytic inputs.

desk verdict Materials mismatch: only the PhaseJumps abstract is present; the full text is an unrelated clustering paper, so the claimed algorithm and smoothed-analysis bound cannot be checked. read the letter →

arxiv 2603.13158 v2 pith:PJHYVPRG submitted 2026-03-13 math.NA cs.NAmath.CV

Antti Haimi , Günther Koliander , José Luis Romero This is my paper · ORCID
classification math.NAcs.NAmath.CV MSC 65H0565T5094A12
keywords PhaseJumpszerofindinggridsamplescomplexshort-timeFouriertransformWassersteinmetricsmoothedanalysisnon-analyticfunctions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces PhaseJumps, an algorithm that locates the zeros of a complex-valued function given only its samples on a finite planar grid. It works by comparing how the complex phase jumps and how the function oscillates from one grid point to its neighbors, and it also returns the sense of phase winding around each detected zero. Because the method never assumes analyticity, it applies directly to the short-time Fourier transform of an analog signal under a completely general analysis window—the first practical way to extract those zeros. Under a stochastic model drawn from signal-processing noise, a variant of the algorithm is proved to return a zero set accurate to order √δ in the Wasserstein metric, with failure probability only O(log²(1/δ)·δ); the rare bad instances are fragile and are regularized by additive noise (smoothed analysis). Numerical experiments indicate still better practical rates. The result therefore removes an analyticity barrier that had limited several recent time-frequency techniques.

What carries the argument

The PhaseJumps algorithm: it detects zeros by comparing phase changes and local oscillations among neighboring grid points, simultaneously recovering the direction of phase winding.

What would settle it

Draw many independent realizations from the paper’s stochastic input model on grids of spacing δ, run the analyzed variant of PhaseJumps, and check whether the empirical Wasserstein error stays order √δ and the observed failure rate remains bounded by a constant times log²(1/δ)·δ; systematic violation of either bound falsifies the claim.

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Extended reading notes

Core claim

Given samples of a complex-valued function on a grid of spacing δ, PhaseJumps (or a stated variant) computes the zeros to accuracy √δ in the Wasserstein metric with failure probability O(log²(1/δ)·δ) under a signal-processing-motivated stochastic model, while also returning phase-winding directions and applying to non-analytic functions such as general-window short-time Fourier transforms.

Load-bearing premise

The accuracy and failure-probability guarantees rest on a specific random model of the input function (motivated by signal processing) together with the claim that ordinary additive noise will smooth away the fragile bad cases.

Editorial extensions

If this is right

  • Zeros of the short-time Fourier transform become computable for arbitrary analysis windows, not only analytic ones.
  • Recent signal-processing insights that previously required analyticity can now be applied more broadly.
  • Pathological inputs that defeat zero recovery are shown to be fragile under additive noise, so practical spectrograms are typically well-behaved.
  • Both location and orientation (winding direction) of zeros are recovered from discrete samples alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same local phase-jump test may extend, with suitable modifications, to higher-order zeros or to branch points of multi-valued functions sampled on grids.
  • If the √δ Wasserstein rate is essentially sharp under the model, it quantifies a fundamental resolution limit for zero recovery without derivative information.
  • Spectrogram pipelines that currently discard non-Gaussian windows can now extract and exploit zero sets that were previously inaccessible.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The submission is labeled as PhaseJumps (arXiv 2603.13158, math.NA): an algorithm that recovers zeros of complex-valued planar functions from finite grid samples by comparing phase changes and local oscillations among neighbors, works for non-analytic inputs (including general-window STFT zeros), returns phase-winding direction, and—under a stochastic model motivated by signal processing—achieves Wasserstein accuracy √δ with failure probability O(log²(1/δ)·δ) via smoothed analysis. The body of the supplied manuscript, however, is an entirely different paper (Catanzaro–van der Hofstad–Garlaschelli, clustering without geometry in sparse networks with independent edges, arXiv 2603.13159): it proves that the multi-scale / Norros–Reittu model with infinite-mean Pareto (or α-stable) fitness yields positive local clustering without geometry or higher-order edge dependence, derives the annealed clustering function, leaf/hub asymptotics, and non-self-averaging of r_{0/1} and of the average clustering coefficient.

Significance. If the PhaseJumps claims were substantiated, the work would be significant for numerical analysis and time-frequency analysis: a practical, non-analytic method for STFT zeros with general windows, plus a concrete smoothed-analysis rate, would remove a long-standing analytic-window restriction and make recent signal-processing insights more widely usable. Separately, the clustering manuscript that actually appears in the full text is itself a substantial contribution to network science: it rigorously shows that sparsity + strong local clustering + power-law degrees can arise from edge-independent models with infinite-mean fitness (node-aggregation invariance), without latent geometry, and it carefully characterizes non-self-averaging—directly addressing a debated implication that “clustering implies geometry.” Neither contribution can be properly credited or stress-tested under the present mismatched materials.

major comments (3)
  1. Title/abstract vs. full text mismatch: the abstract and paper_id describe PhaseJumps (zeros from grid samples, STFT application, smoothed-analysis Wasserstein bound √δ with failure O(log²(1/δ)·δ)), but the entire FULL MANUSCRIPT TEXT is the unrelated clustering paper (MSM/Norros–Reittu with Pareto/α-stable fitness, annealed clustering function (6), leaf/hub asymptotics (7)–(8), average clustering (10), non-self-averaging of r_{0/1}). No definition of PhaseJumps, no stochastic input model, no proof of the claimed rate, and no STFT experiments appear. The central PhaseJumps claim is therefore unverifiable from the supplied package.
  2. Because the load-bearing objects for PhaseJumps (algorithm pseudocode, precise variant used in the bound, definition of the stochastic model and additive noise, Wasserstein distance on zero sets, and the smoothed-analysis argument) are absent, the performance guarantee and the “first effective means” claim for general-window STFT zeros cannot be checked for correctness, restrictiveness of assumptions, or applicability to the stated STFT setting. This is not a presentation issue; it blocks assessment of the paper as submitted under the PhaseJumps title.
  3. If the intended submission were instead the clustering manuscript that occupies the full text, its main theorems (annealed clustering function, lim C̄ = 1 or 1−r_{0/1}, hub contribution vanishing, non-self-averaging via α-stable total weight S_n) appear carefully derived in the SI and supported by numerics (Figs. 1–2, S1–S8). Even then, the package would need a matching title/abstract and a clear statement that the PhaseJumps abstract is erroneous; as currently assembled, the submission is not reviewable as either paper.
minor comments (3)
  1. Abstract (PhaseJumps): the phrase “(a variant of) PhaseJumps” already flags that the proved bound may not apply to the exact algorithm used for STFT; that distinction must be made explicit once the correct manuscript is supplied.
  2. Clustering full text: figure captions and some SI labels contain garbled/placeholder characters (e.g., sequences of black squares in Fig. 1 and SI figure titles), which should be cleaned for production.
  3. Clustering full text: the claim that a second-moment argument would show C(k) and C̄(k) coincide w.h.p. is left unproved (footnote [38]); either prove it or clearly mark it as a conjecture.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity can be assessed for PhaseJumps: the supplied full manuscript is a different paper (clustering without geometry); the PhaseJumps abstract alone shows no self-definitional or fitted-as-prediction loop.

full rationale

The requested paper is PhaseJumps (arXiv:2603.13158). Only its abstract is present. The CACHEABLE full text is an unrelated manuscript (Catanzaro–van der Hofstad–Garlaschelli on clustering without geometry, arXiv:2603.13159). From the PhaseJumps abstract alone, the central claim is a smoothed-analysis guarantee: under a stochastic input model, a variant of the algorithm recovers zeros to Wasserstein accuracy √δ with failure probability O(log²(1/δ)·δ). That claim is framed as a theorem under an external model plus additive-noise regularization of fragile instances; nothing in the abstract equates the claimed accuracy or failure rate to a quantity defined by the same fit or by a self-citation that forces the result. No self-definitional step, fitted-input-called-prediction, uniqueness import, or ansatz smuggling is quotable. Per the hard rules, absence of the derivation chain precludes manufacturing circularity; the honest finding is score 0 (no significant circularity identifiable). Residual risk that the inaccessible stochastic model or free constants might later prove load-bearing is a completeness issue, not circularity.

Assumptions & free parameters 2 free parameters · 3 assumptions · 1 invented entities

With only the PhaseJumps abstract available, the ledger is limited to assumptions stated there. The performance claim depends on a stochastic input model, grid spacing δ, additive-noise regularization (smoothed analysis), and the informal algorithmic idea of comparing phase changes and local oscillations. No free parameters are numerically fitted in the abstract. No new physical entities are introduced; PhaseJumps is an algorithm name.

free parameters (2)
  • grid spacing δ
    Accuracy √δ and failure probability scale with δ; δ is a design/input parameter of the sampling grid, not fitted, but the rates are stated in terms of it.
  • stochastic input model (unspecified details)
    Failure probability and smoothed-analysis claim are relative to a model ‘motivated by signal processing applications’; distributional parameters are not given in the abstract.
assumptions (3)
  • ad hoc to paper Zeros of the target complex function can be recovered from finite planar grid samples by comparing phase changes and local oscillations among neighbors.
    Core algorithmic hypothesis of PhaseJumps; stated in the abstract without proof text available here.
  • domain assumption Additive noise regularizes the fragile failure instances (smoothed analysis).
    Used to justify that bad inputs have small measure under the stochastic model; abstract performance paragraph.
  • domain assumption The method applies to possibly non-analytic functions, including STFTs with general analysis windows.
    Central application claim; overcomes prior analytic-only constraints according to the abstract.
invented entities (1)
  • PhaseJumps algorithm
    purpose: Compute zeros (and phase-winding direction) of complex planar functions from grid samples.
    Named method introduced in the paper; independent evidence would be code, benchmarks, and proofs, which are not in the supplied text.

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Cite this review

Pith. "Pith review of PhaseJumps: fast computation of zeros from planar grid samples." pith.science (2026). https://pith.science/paper/PJHYVPRG

@misc{pith2026260313158,
  author       = {Pith},
  title        = {Pith review of: PhaseJumps: fast computation of zeros from planar grid samples},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PJHYVPRG}},
  note         = {Machine review of arXiv:2603.13158}
}
abstract

We consider complex-valued functions on the complex plane and the task of computing their zeros from samples taken along a finite grid. We introduce PhaseJumps, an algorithm based on comparing changes in the complex phase and local oscillations among neighboring grid points. The algorithm is applicable to possibly non-analytic input functions, and also computes the direction of phase winding around zeros. PhaseJumps provides a first effective means to compute the zeros of the short-time Fourier transform of an analog signal with respect to a general analysis window, and makes certain recent signal processing insights more widely applicable, overcoming previous constraints to analytic transformations. We study the performance of (a variant of) PhaseJumps under a stochastic input model motivated by signal processing applications and show that the input instances that may cause the algorithm to fail are fragile, in the sense that they are regularized by additive noise (smoothed analysis). Precisely, given samples of a function on a grid with spacing $\delta$, we show that our algorithm computes zeros with accuracy $\sqrt{\delta}$ in the Wasserstein metric with failure probability $O\big(\log^2(\tfrac{1}{\delta}) \delta\big)$, while numerical experiments suggest even better performance.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 2 linked inside Pith

  1. [1]

    Caldarelli,Scale-free networks: complex webs in na- ture and technology(Oxford University Press, 2007)

    G. Caldarelli,Scale-free networks: complex webs in na- ture and technology(Oxford University Press, 2007)

    2007

  2. [2]

    Newman,Networks(Oxford university press, 2018)

    M. Newman,Networks(Oxford university press, 2018)

    2018

  3. [3]

    Van Der Hofstad, Random graphs and complex net- works: Volume 1 (2016)

    R. Van Der Hofstad, Random graphs and complex net- works: Volume 1 (2016)

    2016

  4. [4]

    Erd˝ os, A

    P. Erd˝ os, A. R´ enyi,et al., On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. (1960)

    1960

  5. [5]

    M. E. Newman, Properties of highly clustered networks, Physical Review E��, 026121 (2003)

    2003

  6. [6]

    M. E. Newman, Random graphs with clustering, Physical review letters���, 058701 (2009)

    2009

  7. [7]

    Bollob´ as, S

    B. Bollob´ as, S. Janson, and O. Riordan, Sparse random graphs with clustering, Random Structures & Algorithms ��, 269 (2011)

    2011

  8. [8]

    In monopartite projections of bipartite networks, the original nodes are first initially connected (possibly in- dependently of each other) to ‘auxiliary’ nodes of a dif- ferent type (representing for instance the possibile affil- iations or memberships of the original nodes), and the auxiliary nodes are then eliminated while connecting the original nodes...

Show all 47 references
  1. [9]

    M. E. Newman and J. Park, Why social networks are different from other types of networks, Physical review E ��, 036122 (2003)

    2003

  2. [10]

    Battiston and G

    F. Battiston and G. Petri,Higher-order systems (Springer, 2022)

    2022

  3. [11]

    Boccaletti, P

    S. Boccaletti, P. De Lellis, C. Del Genio, K. Alfaro- Bittner, R. Criado, S. Jalan, and M. Romance, The struc- ture and dynamics of networks with higher order inter- actions, Physics Reports����, 1 (2023)

    2023

  4. [12]

    Caldarelli, A

    G. Caldarelli, A. Capocci, P. De Los Rios, and M. A. Munoz, Scale-free networks from varying vertex intrinsic fitness, Physical review letters��, 258702 (2002)

    2002

  5. [13]

    Bogun´ a and R

    M. Bogun´ a and R. Pastor-Satorras, Class of correlated random networks with hidden variables, Physical Review E��, 036112 (2003)

    2003

  6. [14]

    Squartini and D

    T. Squartini and D. Garlaschelli,Maximum-entropy net- works: Pattern detection, network reconstruction and graph combinatorics(Springer, 2017)

    2017

  7. [15]

    Some popular examples are�(� �� ��) =�� �� �, �(� �� ��) = Θ(� � +� � ��),�(� �� ��) =�� �� ��(1 + �� �� �), and�(� �� ��) = 1�e ��� �� � , where� �0 is a global parameter controlling for the overall link density and� � �0��

  8. [16]

    For instance, if� � and� � are scalar (one-dimensional) coordinates and their distance is defined as� �� =�� � � � ��, then� � and� � can simultaneously increase while keeping� �� (and hence� ��) unchanged

  9. [17]

    Krioukov, Clustering implies geometry in networks, Physical review letters���, 208302 (2016)

    D. Krioukov, Clustering implies geometry in networks, Physical review letters���, 208302 (2016)

    2016

  10. [18]

    Aliakbarisani, M

    R. Aliakbarisani, M. Bogu˜ n´ a, and M.´A. Serrano, Clus- tering does not always imply latent geometry, Physical Review Letters���, 197402 (2025)

    2025

  11. [19]

    M. ´A. Serrano and M. Boguna, Clustering in com- plex networks. i. general formalism, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics��, 056114 (2006). 6

    2006

  12. [20]

    M. ´A. Serrano and M. Bogun´ a, Clustering in com- plex networks. ii. percolation properties, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics��, 056115 (2006)

    2006

  13. [21]

    Boguna, I

    M. Boguna, I. Bonamassa, M. De Domenico, S. Havlin, D. Krioukov, and M. ´A. Serrano, Network geometry, Na- ture Reviews Physics�, 114 (2021)

    2021

  14. [22]

    Krioukov, F

    D. Krioukov, F. Papadopoulos, M. Kitsak, A. Vahdat, and M. Bogun´ a, Hyperbolic geometry of complex net- works, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics��, 036106 (2010)

    2010

  15. [23]

    Michielan, N

    R. Michielan, N. Litvak, and C. Stegehuis, Detecting hy- perbolic geometry in networks: Why triangles are not enough, Physical Review E���, 054303 (2022)

    2022

  16. [24]

    Allard, M

    A. Allard, M. ´A. Serrano, and M. Bogu˜ n´ a, Geometric description of clustering in directed networks, Nature Physics��, 150 (2024)

    2024

  17. [25]

    Candellero and N

    E. Candellero and N. Fountoulakis, Clustering and the hyperbolic geometry of complex networks, inAlgorithms and Models for the Web Graph: 11th International Work- shop, WAW 2014, Beijing, China, December 17-18, 2014, Proceedings 11(Springer, 2014) pp. 1–12

    2014

  18. [26]

    Boguna, D

    M. Boguna, D. Krioukov, and K. C. Claffy, Navigability of complex networks, Nature Physics�, 74 (2009)

    2009

  19. [27]

    Bogun´ a, F

    M. Bogun´ a, F. Papadopoulos, and D. Krioukov, Sustain- ing the internet with hyperbolic mapping, Nature com- munications�, 62 (2010)

    2010

  20. [28]

    Garuccio, M

    E. Garuccio, M. Lalli, and D. Garlaschelli, Multiscale network renormalization: Scale-invariance without geom- etry, Physical Review Research�, 043101 (2023)

    2023

  21. [29]

    Lalli and D

    M. Lalli and D. Garlaschelli, Geometry-free renormal- ization of directed networks: scale-invariance and reci- procity, arXiv preprint arXiv:2403.00235 (2024)

    2024 arXiv

  22. [30]

    Avena, D

    L. Avena, D. Garlaschelli, R. S. Hazra, and M. Lalli, Inhomogeneous random graphs with infinite-mean fitness variables, Journal of Applied Probability , 1–26 (2025)

    2025

  23. [31]

    Catanzaro, R

    A. Catanzaro, R. S. Hazra, and D. Garlaschelli, Spectra of random graphs with discrete scale invariance, arXiv preprint arXiv:2509.12407 (2025)

    2025

  24. [32]

    Norros and H

    I. Norros and H. Reittu, On a conditionally poissonian graph process, Advances in Applied Probability��, 59 (2006)

    2006

  25. [33]

    G. J. Rodgers, K. Austin, B. Kahng, and D. Kim, Eigen- value spectra of complex networks, Journal of Physics A: Mathematical and General��, 9431 (2005)

    2005

  26. [34]

    Bhamidi, R

    S. Bhamidi, R. van der Hofstad, and J. van Leeuwaarden, Scaling limits for critical inhomogeneous random graphs with finite third moments, Electronic Journal of Proba- bility��, 1682 (2010)

    2010

  27. [35]

    Gabrielli, D

    A. Gabrielli, D. Garlaschelli, S. P. Patil, and M. Serrano, Network renormalization, Nature Reviews Physics�, 203 (2025)

    2025

  28. [36]

    Samorodnitsky and M

    G. Samorodnitsky and M. S. Taqqu,Stable non-Gaussian random processes: stochastic models with infinite vari- ance, Vol. 1 (CRC press, 1994)

    1994

  29. [37]

    Nolan,Univariate stable distributions: models for heavy tailed data, Springer Series in Operations Re- search and Financial Engineering (Springer, Cham, [2020]�2020) pp

    J. Nolan,Univariate stable distributions: models for heavy tailed data, Springer Series in Operations Re- search and Financial Engineering (Springer, Cham, [2020]�2020) pp. xv+333

    2020

  30. [38]

    We will however not prove this claim in the present work

    We also believe that, through a second-moment analy- sis, it could be proven that in the large�limit, with high probability, the two clustering functions become the same. We will however not prove this claim in the present work

  31. [39]

    Leipus, J

    R. Leipus, J. ˇ Siaulys, and D. Konstantinides,Closure properties for heavy-tailed and related distributions—an overview, SpringerBriefs in Statistics (Springer, Cham,

  32. [40]

    Bollob´ as, W

    B. Bollob´ as, W. Fulton, A. Katok,et al., Cambridge studies in advanced mathematics, inRandom graphs (Cambridge University Press Cambridge, UK, 2001)

    2001

  33. [41]

    Janson, T

    S. Janson, T. Luczak, and A. Ruci´ nski, Wiley- interscience series in discrete mathematics and optimiza- tion, inRandom Graphs(2000)

    2000

  34. [42]

    Van Der Hofstad,Random graphs and complex net- works, : Volume 2(Cambridge university press, 2024)

    R. Van Der Hofstad,Random graphs and complex net- works, : Volume 2(Cambridge university press, 2024)

    2024

  35. [43]

    Van Den Esker, R

    H. Van Den Esker, R. Van Der Hofstad, G. Hooghiem- stra, and D. Znamenski, Distances in random graphs with infinite mean degrees, Extremes�, 111 (2005)

    2005

  36. [44]

    Baptista, R

    A. Baptista, R. J. S´ anchez-Garc´ ıa, A. Baudot, and G. Bianconi, Zoo guide to network embedding, Journal of Physics: Complexity�, 042001 (2023)

    2023

  37. [45]

    Ballerini, N

    M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini,et al., Interaction ruling an- imal collective behavior depends on topological rather than metric distance: Evidence from a field study, Pro- ceedings ...

    2008

  38. [46]

    Van Der Hofstad, G

    R. Van Der Hofstad, G. Hooghiemstra, and D. Znamen- ski, Random graphs with arbitrary iid degrees, arXiv preprint math/0502580 (2005)

    2005 arXiv

  39. [47]

    Clustering without geometry in sparse networks with independent edges

    D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world’networks, nature���, 440 (1998). 1 ������������� ����������� accompanying the paper “Clustering without geometry in sparse networks with independent edges” by A. Catanzaro, R. van der Hofstad and D. Garlaschel...

    1998

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