Boosting decision trees for Main Belt Asteroid selection in planetary ephemerides: an alternative model

Agn\`es Fienga; Jacques Laskar; Micka\"el Gastineau; Vincenzo Mariani; Zachary Murray

REVIEW 3 major objections 7 minor 31 references

Boosting decision trees for Main Belt Asteroid selection in planetary ephemerides: an alternative model

T0 review · 3 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Boosted decision trees rank the 343 main-belt asteroids by effect on Mars residuals, and removing the 147 lowest-ranked ones yields a 196-asteroid model with no significant degradation of the fit.

desk verdict A credible reduced-asteroid ephemeris built with a BDT ranking, but the ranking's causal role is under-tested because there is no control ordering. read the letter →

arxiv 2505.10487 v1 pith:HJG6BMMG submitted 2025-05-15 astro-ph.EP astro-ph.IM

Vincenzo Mariani , Agnès Fienga , Zachary Murray , Mickaël Gastineau , Jacques Laskar This is my paper · ORCID

classification astro-ph.EPastro-ph.IM

keywords boosteddecisiontreesasteroidrankingMainBeltasteroidsplanetaryephemeridesMarsExpressresidualspoint-massmodelmassuncertaintiesINPOP

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 claims that a boosted decision tree can rank the 343 Main Belt asteroids currently treated as point masses by how much each one changes Mars Express ranging residuals, and that removing the 147 lowest-ranked asteroids leaves a 196-asteroid model, called INPOP25c, whose postfit residual quality is essentially unchanged. This matters because unknown asteroid masses are a main limitation on Mars orbit prediction, and cutting the number of fitted masses by more than 140 makes the least-squares fit better conditioned, reduces average mass uncertainties by about 15%, and roughly halves integration time. The ranking is trained on random mass variations of ±10% around previous postfit values using a linearized approximation of the residual change, then validated by full planetary refits in which asteroids are removed cumulatively from least to most important. The central practical claim is that a much smaller point-mass model can replace the 343-asteroid model without degrading the ephemeris.

What carries the argument

The load-bearing object is a supervised ranking of asteroid masses by their marginal effect on Mars ranging residuals, learned by boosted regression trees. A gradient-boosted decision tree is trained on a dataset pairing random asteroid mass perturbations, uniform within ±10% of INPOP21a postfit masses, with the implied change in the Mars Express residual χ2, computed through the linearized residual formula $\Delta \widehat{(O-C)}|_O \equiv \frac{\partial (O-C)|_O}{\partial m}\Delta m$. The tree ensemble assigns a relative importance to each of the 343 masses, and that importance order is used as a removal sequence. The validation machinery is the residual indicator $RI = \sqrt{\sigma^2_{\rm MEX,fit} + \sigma^2_{\rm MEX,ext}}$, combining in-fit and extrapolated Mars Express residuals, whose minimum selects the 147-removal solution.

What would settle it

Recompute the BDT ranking with training perturbations spanning the full removal range, for example setting individual masses to zero or using ±100% mass changes, and refit the ephemerides; if the 147-removal set no longer keeps σMEX oscillations below 20 cm, the linearized ±10% training assumption is the breaking point. A second test would be to refit with a non-MEX observable, such as InSight or Mars orbiter ranging, and check whether residuals remain within 20 cm.

Watch

Extended reading notes

Core claim

The central discovery is that the relative importance of asteroid masses for the Mars Express residual fit, learned by gradient-boosted regression trees, is a reliable guide for deleting asteroids from the dynamical model altogether. Removing objects in increasing order of importance keeps the oscillation in the postfit σMEX below 20 cm up to about 200 removals, and the chosen solution INPOP25c removes 147 asteroids and keeps 196. The reduced model has a conditioning number about 70% lower, average mass uncertainties about 15% smaller, and integration time about 52% lower in user time, while the fitted masses remain consistent with an independent albedo-based posterior and with literature mass estimates. The paper presents this as evidence that the BDT ranking identifies which asteroids can be omitted from the point-mass model without significant degradation of the planetary ephemeris.

Load-bearing premise

The ranking is learned from small, ±10% mass perturbations and a linearized (straight-line) approximation of how the residuals respond, but it is then used to justify deleting asteroids entirely, a 100% change, and the validation refits use the same Mars-Express-based objective, so the order and size of the optimal reduced set could change if the linear ordering is not valid outside its training range.

Editorial extensions

If this is right

A 196-asteroid point-mass model can replace the 343-asteroid model without significant degradation of postfit Mars Express residuals.
The conditioning number of the least-squares fit drops by about 70%, and average fitted mass uncertainties improve by about 15%.
Integration of the planetary ephemeris takes about 52% less user time and 30% less real time.
The mass estimates from the reduced model are statistically consistent with an independent posterior built from neural-network albedo predictions.
The close match between σMEX and global χ2 trends indicates that Mars Express residuals are a reliable proxy for global ephemeris quality.

Reading between the lines

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

Extending beyond the tested case, the same ranking procedure could likely be applied to trans-Neptunian objects or other perturbing populations; the authors mention this as future work but do not test it, and the computational cost of building a comparable training set would need to be addressed first.
Because the tree-based importance reflects correlations among asteroids, the method could be extended to identify groups of asteroids that can be replaced by a single effective mass, potentially reducing the model further than the 196-asteroid solution.
If the ranking proves stable under larger training perturbations, the reduced model could guide which asteroid masses most need independent determination, for example by spacecraft flybys or occultations, to keep Mars ephemerides accurate.

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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

The thing to know: this is a useful engineering result, not a deep scientific breakthrough. The authors train XGBoost to rank the 343 INPOP main-belt asteroids by importance derived from linearized MEX residuals, then sequentially remove asteroids and do full ephemeris refits. The resulting 196-asteroid model INPOP25c holds postfit MEX residuals within about 20 cm, cuts integration time roughly in half, and tightens mass uncertainties by about 15% because the Jacobian is better conditioned. Those numbers are credible because they come from full refits, not just from the surrogate model.

What I like: the validation loop is honest. Each removal step is a full fit with the complete observation set, not just the ranking metric. The physical plausibility checks against albedo-based posteriors and the Kretlow/Carry mass catalogs are reasonable and show no red flags. The paper also tells you where the assumptions are weak, which I appreciate.

The soft spots, in order of weight. First, there is no control ordering. The paper interprets the slow rise of sigma_MEX over the first 200 removals as validation of the BDT ranking. But any ordering that removes unimportant asteroids first will show a plateau; you need a random or reverse-ranking trajectory to show the BDT actually identified the right set. Without that, the claim that the BDT identifies which asteroids can be omitted is not established, even though the reduced model itself clearly works. Second, the ranking is trained on plus-or-minus 10% linearized mass perturbations and applied to 100% removals. The authors say the results prove the assumption, but the refit metric overlaps with the training objective, so that proof is partly circular. Third, the choice of 147 removals is the minimum of a residual indicator with no error bars; the differences between adjacent points look small. Minor: no code, data, or hyperparameters are released, so independent reproduction is impossible.

Bottom line: the reduced ephemeris is a real product and worth having. The paper deserves a serious referee and probably acceptance after a control experiment. I would ask for a random-removal and maybe a mass-cut comparison, plus artifact release. For someone working on planetary ephemerides, this is a useful reference; I would cite the INPOP25c result, with a caveat about the ranking claim.

Referee Report

3 major / 7 minor

Summary. The paper proposes a boosted decision tree (BDT) ranking of the 343 Main Belt asteroids used as point masses in the INPOP planetary ephemerides. The BDT is trained on a synthetic dataset of ±10% random mass perturbations, with the target being the linearized change in Mars Express (MEX) residual chi-square. Starting from the INPOP25b ephemeris, the authors remove asteroids one by one in increasing BDT-importance order, performing a full planetary fit after each removal. They find that removing up to 200 asteroids keeps the postfit MEX sigma within 20 cm of the reference, and they select INPOP25c, a solution with 147 asteroids removed (196 remaining), as the minimum of a residual indicator that combines fit and extrapolation dispersions. They report that INPOP25c improves the conditioning of the fit, reduces average mass uncertainties by about 15%, and cuts integration time by about half. The fitted masses are then compared with independent mass estimates from albedo-based neural networks, the literature, and Wasserstein barycenters.

Significance. If the central claims hold, the work would provide a practical and inexpensive way to reduce the dimension of the main-belt model in planetary ephemerides, with concrete benefits for conditioning, mass-parameter uncertainties, and computational cost. The paper's strongest evidence is the repeated full ephemeris refit after each cumulative removal, which is a much more demanding validation than a simple proxy test. The comparison of the INPOP25c masses with independent albedo-based and literature estimates is also a valuable sanity check. However, the central methodological claim that the BDT ranking identifies which asteroids can be omitted is not yet supported, because no alternative removal order is tested; and the training/application mismatch between ±10% linearized perturbations and 100% removal leaves open the possibility that the reduced model works for reasons unrelated to the specific ranking. The absence of public code or data also limits reproducibility of the machine-learning component. The contribution is potentially significant for the ephemerides community, but it needs stronger validation before the ranking claim can be accepted.

major comments (3)

[Sec. 2.4, Fig. 3] The validation of the ranking is not controlled. The statement that 'the ranking is validated by the increase of the differences to the reference solution with the increase of the ranking' describes a property that any cumulative removal order necessarily possesses: residuals grow as more asteroids are removed. The plateau below 20 cm for up to 200 removals shows that the removed set as a whole is collectively unimportant, but it does not show that this particular 196-asteroid subset is the one identified by the BDT, nor that a random ordering, a reverse ranking, or a simple size/mass-based cut would not perform equally well. Without such control experiments, the causal role of the BDT in producing INPOP25c is not established, and the paper's central claim about the ranking is unsupported.
[Sec. 2.1.3, Eqs. (3)-(5)] The training set is built from random mass variations of only ±10% around the INPOP21a postfit values, with a target defined by linearized residual changes, while the actual operation is complete removal of the asteroid, i.e., a 100% change. The paper states that the 'validity of such assumptions is going to be proved by the results presented in Sect. 2.4', but those results use the same MEX-based objective and the same full global chi-square for validation; they do not independently probe nonlinearities or interactions among the many simultaneously removed asteroids. This is a load-bearing gap because the reduced model's success might be insensitive to the ordering even if the BDT ranking is unreliable outside the training range. A focused nonlinear test, such as direct integration of a few removal scenarios or an ordering based on full-removal chi-square changes for a subset, would materially strengthen the claim.
[Sec. 2.4.1, Fig. 4] The selection of INPOP25c as the model with 147 asteroids removed relies on the minimum of the residual indicator RI, which is computed on the same fitted quantities and the same extrapolation interval used to define 'best'. No uncertainty on the RI values, no error bars, and no stability analysis of the minimum across the training-set sizes and hyperparameters are reported. Because the same index is used both to evaluate candidates and to select the final solution, the exact number of removals may be noise-driven. The authors should report the RI values at the minimum, the sensitivity of the minimum to the chosen training set, and ideally a statement of whether solutions between, say, 100 and 200 removals are statistically indistinguishable.

minor comments (7)

[Fig. 5 and Fig. 6 captions; Sec. 3.1] The captions of Fig. 5 and Fig. 6 refer to 'INPOP26c' while the text uses 'INPOP25c'; also Sec. 3.1 states that 196 masses are fitted by both INPOP25b and INPOP25c, whereas the Fig. 5 caption says 193 masses. Please reconcile these numbers.
[Sec. 3.2.1 and Sec. 3] There are several typos: 'preform' should be 'perform', 'wether' should be 'whether', and the Table 1 header 'noise-to-signal ration' should be 'ratio'.
[Sec. 2.1.3, Eqs. (4)-(5)] The notation ∆^(O−C)|O is not defined; please explain the hat operator and the restriction to O (MEX) explicitly, since Eqs. (4)-(5) are central to the training-set construction.
[Sec. 2.4] The sentence 'For this study, we used the INPOP25a datasets' is ambiguous after the paper states that the ranking is implemented starting from INPOP25b; clarify whether 'datasets' refers to the observation data set or to the ephemeris version used as the starting model.
[Sec. 3.2.1] The text mentions p-values for the Kolmogorov-Smirnov tests, but Fig. 7 does not show them; please report the p-values in the text or in the figure.
[Sec. 2.4.1, RI definition] The residual indicator RI combines σ_MEX,fit and σ_MEX,ext in quadrature without an explicit justification for this particular combination; please state the units and whether the two dispersions are weighted equally by construction or by choice.
[Sec. 2.3 and Sec. 3.2.2] Minor inconsistencies in formatting include 'several trainingsets' (missing space) and inconsistent capitalization of 'INPOP25C' versus 'INPOP25c' in Sec. 3.2.2.

Circularity Check

1 steps flagged · score 4.0 of 10

BDT ranking and its validation share the MEX residual metric, so the main degradation curve is partly a restatement of the training objective; independent nonlinear refits and external mass checks keep the central result from being fully circular.

fitted input called prediction [Sec. 2.1.2-2.1.3 (Eqs. 2-5) and Sec. 2.4 (Fig. 3)]
"f∗ : ∆mAST 7→ ∆˜χ2 OMEX(∆mAST) (2) ... ∆˜χ2 O(∆m)≡ [∆ ^(O−C)|O] T [∆ ^(O−C)|O] . (4) ... The ranking is validated by the increase of the differences to the reference solution (INPOP25b) with the increase of the ranking."

The BDT importance is trained to reproduce Δ~χ2_OMEX, a linearized approximation of the MEX-residual χ2 (Eqs. 2-5). The validation in Sec. 2.4 then reports σMEX of the postfit MEX residuals and the RI based on σMEX,fit and σMEX,ext. To first order, removing asteroids with the smallest linearized effect on MEX residuals must least increase the linearized MEX χ2; the <20 cm plateau up to 200 removals is therefore partly a consequence of the training objective rather than an independent confirmation. The paper explicitly relies on Sec. 2.4 to 'prove' the linearity assumption, but Sec. 2.4 uses the same MEX-based observable, so that proof is circular. The full nonlinear refits, global χ2, and external mass comparisons add independent content, making the circularity partial.

full rationale

The paper's claimed reduction chain is: (i) generate a training set of ±10% mass perturbations with target Δ~χ2_OMEX computed from linearized MEX residuals (Eqs. 2-5); (ii) train a BDT to rank asteroids by importance for that target; (iii) remove asteroids cumulatively from lowest importance, refit the full planetary model, and show σMEX stays below 20 cm for up to 200 removals; (iv) select INPOP25c by minimizing an RI that again combines σMEX,fit and σMEX,ext. Step (iii) is the load-bearing validation of the ranking, but it uses the same MEX residual metric that defined the training target. Under the linearity assumption of Eq. (5), the smallest-importance asteroids are, by construction, the ones whose removal least increases the linearized MEX χ2; the observed plateau is therefore partly encoded in the ranking itself. The paper's statement that the linearity assumption 'is going to be proved by the results presented in Sect. 2.4' is circular in that the same observable is used both to define importance and to test it. This does not make the work wholly circular: the cumulative removals are tested through full nonlinear least-squares refits of all available planetary observations, the global χ2 follows the same trend, and the resulting masses are compared with independent albedo-based posteriors and literature mass determinations (Sec. 3.2). These provide outside support. The absence of a control ordering (e.g., random or reverse ranking) is a substantive limitation of the evidence, but it is a correctness/design issue rather than a circularity per se. Overall, the central reduction claim is partially self-confirming in its validation metric but retains independent content, so the score is 4.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The Wasserstein barycenter is a statistical aggregation tool, not an entity. The central assumptions are domain assumptions about linearization, the MEX proxy, feature importance reliability, and the albedo-based mass posterior; none reduce the result to a definitional identity.

free parameters (3)

196 INPOP25c asteroid masses = given in Appendix Table 1
The masses of retained asteroids are fitted by BVLS to the full observation set; uncertainty improvement is one of the main results and depends on these fitted values.
XGBoost hyperparameters = not reported
Tree depth, learning rate, regularization, and number of trees are not specified; only the training set size M varies (5e5, 2e6, 6e6), yet the ranking and the chosen model depend on these choices.
Mass boundary relaxation factor = 1.25 times the Fienga et al. (2019) boundaries
INPOP25b is built by relaxing the prior mass boundaries by 25% to improve extrapolation; INPOP25c inherits this condition through the reference ephemeris.

assumptions (5)

domain assumption Linearized residual approximation of Eq. (5) is accurate for ±10% mass changes
The BDT training set and target function are built from this linearization; the paper defers proof of validity to the refit results in Sec. 2.4 instead of an independent nonlinear test.
domain assumption MEX tracking residuals are a sufficient proxy for global planetary ephemeris quality
The BDT ranking uses only Mars Express data (Eq. 2); the paper argues Fig. 3 shows similar trends between σMEX and global χ2, but the ranking itself does not use the full dataset.
domain assumption XGBoost split-based feature importance correctly ranks strongly correlated asteroid masses
The paper notes the widespread correlation among asteroids in the fit (Sec. 2.3); tree importance measures are not guaranteed to be reliable for correlated inputs, though the full refits partially mitigate this.
domain assumption Albedo-based mass posterior, with density uniform in 0.5 to 4.5 g/cm3 and albedo from Murray (2023), represents independent physical mass constraints
Used in Sec. 3.2.1 to assess whether fitted masses are physically reasonable; the wide density range makes the check weak.
domain assumption INPOP25c and Kretlow (2020) mass determinations are treated as normal distributions for Wasserstein barycenters
Sec. 3.3 computes barycenters of normal distributions; the barycenter can become non-normal, as the Bellona example shows, but normality is assumed for the inputs.

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Cite this review

Pith. "Pith review of Boosting decision trees for Main Belt Asteroid selection in planetary ephemerides: an alternative model." pith.science (2026). https://pith.science/paper/HJG6BMMG

@misc{pith2026250510487,
  author       = {Pith},
  title        = {Pith review of: Boosting decision trees for Main Belt Asteroid selection in planetary ephemerides: an alternative model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HJG6BMMG}},
  note         = {Machine review of arXiv:2505.10487}
}

read the original abstract

One of the main bottleneck in assessing the accuracy of Mars orbit is the unknown value of the asteroids in the Main Asteroid Belt. Nowadays a modeling with 343 asteroids as point masses is used, with the relative masses fitted to observational data. In the current work we propose an innovative methodology to reduce the number of asteroids implemented as point masses, thus reducing the number of parameters to be fitted, without a significant degradation of the postfit residuals.

Figures

Figures reproduced from arXiv: 2505.10487 by the authors.

**Figure 1.** Distributions of residuals for the extrapolation period using the original boundary interval (in blue) and the extended boundaries (in red). 2.3. Ranking As explained in Secs. 2.1.3 and 2.1.2, the idea is to use a BDT for the regression of a training set D, as defined in Eq. (3), from which we obtain a ranking of the variables in input to the function f ∗ of Eq. (2). As a byproduct, we get the relative importance (… view at source ↗

**Figure 3.** Standard deviation (σMEX) of the MEX residuals in meters and global χ 2 obtained with the model with X asteroids removed. X is given by the x-axis according to the ranking given by Sec. 2.3. The dotted line represent a reference of 20 cm increase relative to INPOP25b σMEX plotted with the red full line.. ilar trends, meaning that indeed MEX is a good proxy of the quality of the global ephemerides. The ranking is val… view at source ↗

**Figure 4.** Residual Indicator obtained taking into account the σMEX,fit obtained after fit and the σMEX,ext obtained on the extrapolation interval for different sizes of the training sets. Panel (b) is an enlargement of Panel (a). The grey vertical line indicates the minimum reached by the residual indicator RI . The corresponding solution also maximize the number of asteroids removed from the Main belt original selection. The… view at source ↗

Figures from the paper (5 more)

**Figure 7.** Figure 7: Here we show a comparison of the ECDF of the distribution of percentile scores of our fitted masses with respect to the posterior mass distributions. Perfect agreement is denoted by the red diagonal line. We see the distributions found in INPOP25c are comparable with …

**Figure 6.** Figure 6: Histograms of the masses of asteroids common to INPOP25b, INPOP26c, INPOP21a (Fienga et al. 2021), INPOP19a (Fienga et al. 2019), Li et al. (2023), Kretlow (2020) and Carry (2012). fit masses with respect to the corresponding posterior distributions. If the generated p…

**Figure 8.** Figure 8: Comparisons between INPOP25c, Carry (2012) and Kretlow (2020). Panels A and B give the 2D histograms of the radii in kilometers and ratio Kretlow (2020)/INPOP25c (A) and the ratio Carry (2012)/INPOP25c (B). On Panels C and D, one finds the masses of INPOP26c versus the…

**Figure 9.** Figure 9: Histograms of the mass distributions for 20 Massalia estimated from INPOP25c, neural networks and albedo (Sec. 3.2.1 and Murray (2023)) and Kretlow (2020). In yellow the Wasserstein barycenter resulting from the three posteriors cited. several distributions involved…

**Figure 11.** Figure 11: Histograms of the mass distributions for 28 Bellona estimated from INPOP25c, neural networks and albedo (Sec. 3.2.1 and Murray (2023)) and Kretlow (2020). In yellow the Wasserstein barycenter resulting from the three posteriors cited. end, using the Wasserstein bar…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts ...

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write newline

" write newline "" before.all 'output.state := FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix "arXiv" = new.block " " eprint * " " * new.block " " eprint * " " * if if if FUNCTION format.doi doi empty "" " " doi * " " * if FUNCTION format.pid doi empty eprint empty ur...

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thebibliography [1] 20pt to REFERENCES 6pt =0pt -12pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command E...

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