Hermes - Towards an Optimal High-Performance Algorithm for Cosmic Statistics of Large Data Sets

Jiawei Li; Long-long Feng; Shiyu Yue; Tengpeng Xu; Tian-Cheng Luan; Weishan Zhu; Wenjie Ju; Xin Sun; Yan-Chuan Cai; Zhuoyang Li

Hermes - Towards an Optimal High-Performance Algorithm for Cosmic Statistics of Large Data Sets

T0 review · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read One reconstructed density field plus window kernels replaces separate tuple-counting algorithms for cosmic clustering statistics.

desk verdict Solid methods/software paper: unified window-on-MRA-field language plus a real PyHermes stack, externally cross-checked; survey-geometry demos are missing but the core math holds in the regime they actually test. read the letter →

arxiv 2607.23494 v1 pith:EN77MUQA submitted 2026-07-26 astro-ph.CO

Long-long Feng , Tengpeng Xu , Tian-Cheng Luan , Jiawei Li , Xin Sun , Wenjie Ju , Zhuoyang Li , Shiyu Yue

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Weishan Zhu Yan-Chuan Cai

This is my paper · ORCID

classification astro-ph.CO

keywords large-scalestructurecorrelationfunctionsmultiresolutionanalysiswindowcounts-in-cellsthree-pointfunctionN-bodycatalogueshigh-performancecomputing

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

Hermes claims that cosmic large-scale-structure statistics need not each be a separate particle-counting program. A galaxy or halo catalogue is projected once onto a continuous multiresolution density field built from compact scaling functions. Counts-in-cells, two-point and three-point correlations, multipoles, marked statistics, and even derived fields such as potential and velocity divergence are then obtained by convolving that field with chosen window kernels and forming products of the filtered fields. Because the expensive step is the field reconstruction and FFT convolutions on a grid, cost tracks grid resolution and the number of windows rather than the combinatorial explosion of N-tuples. A single field can therefore be reused for many standard and custom measurements, which matters for the huge catalogues expected from current and future galaxy surveys.

What carries the argument

The field–window architecture: catalogue → multiresolution scaling-function coefficients → FFT convolution with a chosen window (shell, ring, multipole, mark, derivative, inverse Laplacian, etc.) → inner products of filtered fields. Different statistics are different kernels, not different counting codes.

What would settle it

Measure the same isotropic 2PCF and multipole 3PCF on a large halo catalogue with Hermes at high resolution and with a standard pair/triplet counter on identical bins; if small-scale amplitudes or multipoles disagree beyond resolution-controlled residuals, or if adding a realistic mask and random catalogue breaks the field–window reuse without new cost blow-ups, the central practical claim fails.

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Extended reading notes

Core claim

The paper establishes that conventional cosmic statistics are special cases of window operations on one multiresolution density field: reconstructing a discrete catalogue in a compact scaling-function basis and replacing explicit pair/triplet counting with algebraic products of window-filtered fields yields a unified, reusable estimator whose computational cost is controlled by field resolution and window count rather than by the number of particle tuples.

Load-bearing premise

A finite-resolution multiresolution projection of a periodic box is taken as good enough to stand in for the true continuous field and for real survey geometry, even though the paper’s demos omit masks, selection functions, and varying random catalogues and show small-scale correlation loss at low resolution.

Editorial extensions

If this is right

One catalogue projection can feed CIC PDFs, isotropic and redshift-space 2PCF, marked 2PCF, standard and multipole 3PCF, and operator-built potential/acceleration fields without rewriting estimators.
New statistics can be introduced by designing or composing window kernels rather than by inventing new N-tuple counters.
Runtime for high-order and multi-probe analyses of large surveys scales primarily with grid size and number of windows, not with N_g^N tuple enumeration.
Emulator and covariance pipelines that need many statistics on many mocks become cheaper once fields are cached and reused.

Reading between the lines

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

The same separation of representation from measurement could extend cleanly to four-point and density-split pipelines if higher-fold field products stay memory-feasible.
Survey-realistic randoms and masks will likely force either masked-field reconstructions or hybrid particle corrections; how much of the reuse advantage survives that step is an open stress test.
Fourier differential and inverse-Laplacian windows suggest a path to consistent density–velocity–potential diagnostics inside one codebase, useful for RSD and environment studies beyond pure clustering.

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Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Punchline: this is a usable computational-methods paper, not a cosmology result. They take MRACS (Feng 2007), the in-situ filtered-field view (Yue et al. 2024), and the multipole 3PCF work (Ju et al. 2026), cast a broad class of LSS stats as window operators on one multiresolution reconstruction, and ship PyHermes with MPI/threads/GPU and public code.

What is actually new is the packaging and the breadth: one SFC field reused for CIC, isotropic/anisotropic 2PCF, marked and wavelet stats, standard and multipole 3PCF, plus differential/potential operators, with systematic timing/memory numbers. That is real engineering value for mock suites and multi-statistic pipelines.

What they do well: external checks. 2PCF converges to pycorr/Corrfunc/TreeCorr as J rises (Fig. 13). CIC PDFs match pyLDT. Angular vs multipole 3PCF consistency is shown. Marked/wavelet/3PCF monopoles track Kun emulators as comparison targets, not circular proofs. Operator windows recover the expected large-scale continuity/Poisson patterns. The math (Toeplitz kernels → FFT convolution, difference-field LS, connection-coefficient contractions) is coherent and cited to prior work honestly.

Soft spots, in proportion. Finite-J small-scale suppression is real and they show it; it is a resolution choice, not a hidden bug. The stress-test point lands but is not fatal: all body demos are periodic boxes with analytic uniform randoms; masks, selection, and varying randoms are asserted as “same framework” without a masked-mock figure. Padding a survey footprint will inflate N_grid and can eat the memory advantage (Table 3 already ~60 GiB at J=9). That limits the “future surveys” claim until someone runs it; it does not invalidate the periodic-box identities or the code as shipped. Title’s “optimal” is marketing; the content is “unified and scalable under stated assumptions.”

Who it is for: people building multi-probe LSS pipelines, mock factories, or custom windows (marks, wavelets, operators). Not required reading if you only need a standard pair counter on a small catalogue.

I would send this to peer review. Ask referees for a masked/random-catalogue demo or a clear cost model with padding, and dial back “optimal,” but the work is serious and citeable as methods infrastructure.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: methods paper reformulates known estimators as windowed field products and validates externally.

full rationale

Hermes/PyHermes is a computational-methods paper. Its load-bearing chain is algebraic reformulation (discrete tuple counting → MRA coefficients + window convolution + field products), not a first-principles physical prediction. Conventional 2PCF/3PCF/CIC estimators are recovered as special window choices (Eqs. 17–21, 54–58, 70–75) and checked against independent codes (pycorr/Corrfunc, TreeCorr) and theory (pyLDT-cosmo). Self-citations (Feng 2007; Yue et al. 2024; Ju et al. 2026) supply algorithmic ancestry for the in-situ/MRA viewpoint but are not used as uniqueness theorems that force the result; numerical claims stand on external benchmarks and public simulation suites (Quijote, Kun). Emulator curves trained on Kun are demonstration targets for marked/3PCF workflows, not self-defining proofs of estimator identity. Periodic-box and uniform-random limitations affect applicability, not circularity of the derivation.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

Load-bearing content is standard LSS statistics plus MRA functional analysis and engineering choices (basis, resolution, parallelism). No new physical entities. Free parameters are numerical/analysis choices (J, wavelet family, bin widths, marks, ℓ_max, Monte Carlo rotations) that control accuracy–cost tradeoffs rather than fitted cosmological constants in the central claim.

free parameters (5)

MRA resolution level J = typically 8 (benchmarks); convergence tests to 9
Controls grid size N=2^{3J} and small-scale fidelity; demos use J=7–9 and show resolution-dependent 2PCF bias at low J.
Scaling-function / wavelet family = db2 default
Default Daubechies D4 (db2); changes compact support and connection-coefficient stencils.
Binning and smoothing window parameters (R, ΔR, σ, ring/cylinder geometry) = example-dependent (e.g. shells, σ_⊥=5 h^{-1} Mpc)
Define the statistic being measured; user-chosen, not cosmologically fitted in the method claim.
Mark function hyperparameters (α, p, ε, ρ*, R_smooth) = e.g. R=8 h^{-1} Mpc, ε=0.1, ρ*=1
Choose environmental weighting for marked 2PCF demos (Eqs. 66–67).
3PCF multipole cutoff ℓ_max and Monte Carlo n_rot / n_cen = ℓ_max up to 20 in demos; n_rot=200–1000
Truncate angular expansion or set sampling noise in particle-based 3PCF path.

assumptions (5)

domain assumption Discrete catalogues are Poisson samples of an underlying continuous density; multi-point statistics are spatial averages of products of density contrasts (ergodicity in a large volume).
Standard LSS point-process setup in §2.1–2.4; underpins replacing ensemble averages by survey/box averages.
standard math Compactly supported orthonormal scaling functions form a complete multiresolution basis that approximates δ_D as J→∞ (completeness relation Eq. 49).
MRA theory (Daubechies); used to justify finite-J field reconstruction in §3.1.
domain assumption Finite-bin and multipole NPCF estimators equal inner products of window-filtered MRA coefficient fields (in situ formulation), including Landy–Szalay-style data−random combinations.
Core equivalence claimed in §2–3; validated numerically against pair counters for 2PCF, less exhaustively for all higher-order variants.
ad hoc to paper Periodic boxes and analytic uniform randoms suffice to demonstrate estimator correctness and scaling for the paper's claims.
Explicitly stated limitation in §4; real masks/randoms deferred to documentation/future work.
standard math FFT-based convolution on the MRA grid with basis autocorrelation Φ correctly implements continuous window convolution up to resolution limits.
Toeplitz/FFT argument in §3.1–3.3; standard numerical harmonic analysis under periodic BC.

invented entities (1)

Hermes / PyHermes field–window software architecture independent evidence
purpose: Name the unified multiresolution framework and its Python implementation separating SFC fields from statistical windows/tasks.
Engineering/system entity, not a new physical particle or force; independent evidence is the public code and benchmarks.

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Cite this review

Pith. "Pith review of Hermes - Towards an Optimal High-Performance Algorithm for Cosmic Statistics of Large Data Sets." pith.science (2026). https://pith.science/paper/EN77MUQA

@misc{pith2026260723494,
  author       = {Pith},
  title        = {Pith review of: Hermes - Towards an Optimal High-Performance Algorithm for Cosmic Statistics of Large Data Sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EN77MUQA}},
  note         = {Machine review of arXiv:2607.23494}
}

read the original abstract

We present Hermes, an in situ multiresolution framework for efficient and flexible measurements of cosmic large-scale-structure statistics from discrete catalogues. Hermes reconstructs a catalogue as a continuous density field in a compact scaling-function basis and replaces explicit counting of particle tuples with algebraic operations among window-filtered fields. Standard binning schemes for counts-in-cells, two-point and higher-order correlation functions are thereby expressed through choices of window functions, while new statistics can be constructed by modifying the kernels without redesigning the estimator. We introduce PyHermes, an open-source Python implementation combining multiresolution reconstruction, FFT-based convolution, MPI/thread parallelism, and GPU acceleration. It supports isotropic and anisotropic two-point statistics, marked correlations, standard and multipole three-point functions, filtered statistics, and differential operators for derived physical fields. Tests with cosmological N-body halo catalogues demonstrate a range of clustering measurements and quantify the computational efficiency and scalability of the approach. By separating field representation from statistical windows, a single reconstructed field can be reused for many standard and customised measurements, making Hermes well suited to large data sets from current and future galaxy surveys.

Figures

Figures reproduced from arXiv: 2607.23494 by the authors.

**Figure 1.** Real-space construction of a spherical-shell window as the subtraction of two spherical top-hat windows. The outer top-hat has radius 𝑅 + Δ𝑅/2, while the inner top-hat has radius 𝑅 − Δ𝑅/2. Their difference leaves a shell of mean radius 𝑅 and width Δ𝑅, corresponding to the normalised shell window 𝑊𝑅,Δ𝑅 (𝑟 ). Since the Dirac delta selects whether the integration domain contains the origin, the binning kernel can be wr… view at source ↗

**Figure 2.** Real-space construction of a Gaussian-blurred spherical-shell window. An infinitesimally thin spherical shell of radius 𝑅 is convolved with a threedimensional Gaussian smoothing kernel of width 𝜎. The convolution preserves spherical symmetry and normalisation while broadening the delta-function shell into a smooth shell of finite radial width around 𝑟 = 𝑅. and thus 𝜉Δ𝑅 (𝑅) = 1 𝑉𝑅,Δ𝑅 ∫ 𝑉𝑅,Δ𝑅 𝜉 (𝑟) 𝑑𝑉𝑟 = 1 𝑉𝑅,Δ𝑅 ∫ 𝑅+… view at source ↗

**Figure 3.** Real-space construction of the Gaussian-blurred double-ring window via convolution. The transverse ring window 𝑊ring,⊥ (𝜌) is first combined with the line-of-sight double-delta window 𝑊los(𝑧), producing two thin rings located at 𝑧 = ±𝐻. Convolution with the transverse Gaussian smoothing kernel 𝑊G,⊥ (𝜌, 𝜎⊥ ) broadens each ring in the transverse direction, yielding a three-dimensional Gaussian double-ring window with … view at source ↗

Figures from the paper (15 more)

**Figure 4.** Figure 4: Schematic reconstruction of the same point distribution in the multiresolution scaling-function basis at three resolution levels. At low resolution, each particle is represented by a broad compact-support basis response, producing a smooth coarse-grained field. As the …

**Figure 5.** Figure 5: Demonstration of continuous density-field reconstruction from a discrete Quijote halo catalogue using the MRA representation in PyHermes. The projected density fields are reconstructed at three resolution levels, 𝐽 = 7, 𝐽 = 8, and 𝐽 = 9 (panels A–C), and compared with …

**Figure 6.** Figure 6: PDFs of the normalised matter density field smoothed with spherical windows of radii 𝑅 = 10, 15, and 20 ℎ −1Mpc at 𝑧 = 0. Symbols show the Hermes measurements from the Quijote dark-matter field, while solid curves give the theoretical predictions from pyLDT-cosmo (Cata…

**Figure 7.** Figure 7: Joint PDFs of the matter densities measured with Hermes from the same data set as in [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Examples of marked two-point correlation functions measured with two different density-dependent marks. The left panel shows the power-law density mark defined in Eq. (66), with 𝛼 = 0, +0.3, and −0.3. The case 𝛼 = 0 corresponds to the standard unmarked two-point correl…

**Figure 9.** Figure 9: Examples of the GDW and CWS windowed statistics for three different cosmologies. The left and right panels show 𝜉GDW (𝑟 ) and 𝜉CWS (𝑟 ), respectively. Solid lines show the Hermes results, while filled circles show the emulator predictions. The cosmological parameters a…

**Figure 10.** Figure 10: Examples of the 3PCF monopole for three different cosmologies. The first side length is fixed at 𝑟12 = 20 ℎ −1Mpc, while the second side length varies over 40 ≤ 𝑟13 ≤ 140 ℎ −1Mpc. From left to right, the panels show shell thicknesses Δ𝑅 = 0, 6, and 12 ℎ −1Mpc, respec…

**Figure 11.** Figure 11: Visual connection between the standard angular 3PCF and its multipole representation at 𝑟12 = 20 ℎ −1Mpc and 𝑟13 = 40 ℎ −1Mpc, computed with PyHermes for the Quijote halo example. The particle-centred curve uses 𝑛part = 406728 halo centres and 𝑛rot = 1000, while the b…

**Figure 12.** Figure 12: Schematic overview of the field–window workflow in PyHermes. The framework is organised into four layers: catalogue, window, MRA, and task. The catalogue layer supplies input data, including data and random catalogues, gridded fields, and physical weights or marks. Th…

**Figure 13.** Figure 13: Two complementary checks of the 2PCF window language. Left: one-dimensional binning-window choices measured from the same reconstructed halo field. The thin shell is compared with finite-thickness shells and Gaussian-shell binning windows, showing how the binning defi…

**Figure 14.** Figure 14: Redshift-space 2PCF measured with two different transverse window choices. The left panel uses the standard thin ring window in the (𝑠⊥, 𝑠∥ ) plane. The right panel replaces the transverse ring selection with a Gaussian-smoothed ring window, where the transverse smoot…

**Figure 15.** Figure 15: 3PCF multipole spectra measured from the Quijote halo field at fixed field resolution 𝐽 = 8 with ℓmax = 20. Left: fixed side lengths 𝑟12 = 20 ℎ −1Mpc and 𝑟13 = 40 ℎ −1Mpc, comparing spherical top-hat smoothing windows (solid lines) and Gaussian smoothing windows (dash…

**Figure 16.** Figure 16: Weighted-field and Newtonian-potential diagnostics on the same two-dimensional slice. Left: halo velocity divergence ∇ · v reconstructed from velocity-weighted fields, with transverse halo velocity arrows overlaid. Middle: smoothed dark-matter density contrast 𝛿m, wit…

**Figure 17.** Figure 17: Linear-theory checks for the weighted-field and Newtonian-potential construction. Left: point-by-point comparison between the halo velocity divergence and the dark-matter prediction in Eq. (114), with the diagonal showing equality and the top axis giving the correspon…

**Figure 18.** Figure 18: CPU and GPU execution of the same five 3PCF multipole benchmarks. Left: time spent in the multipole calculation after excluding input, setup, and output overhead. The solid part is the effective time dominated by CPU-side window convolution and MPI communication, and …

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