Interpolating amplitudes

- Self-contained review of interpolation methods for the approximation of higher-order loops scattering amplitudes with a detailed discussion of the benefits and pitfalls of all the interpolation techniques;
- Extensive exploration of the configurations used to define the interpolation functions and the quantity of interest, either $a$ or $f$;
- The work is timely and addresses important questions for future precision measurements;
- The manuscript could be a common ground for the collaboration between physicists and mathematicians.

- The methods themselves are not novel. The splines and the polynomial methods seem to be already well-established, and the machine learning approaches have also been applied to NLO and high-multiplicity processes;
- Evaluation speed is a recurrent theme throughout the paper, however, no numerical comparison between the methods has been made;
- The comparison of machine learning methods is rather limited;
- The datasets of all the test functions could be published. It will improve the reproducibility and future developments.

The work proposed in the paper is valid, and the authors show a great understanding of the numerical tools. It is a timely study, which clarifies the direction for future research on amplitude regression. Some clarifications are needed, in particular, the introduction should be supported by references. The authors refer to two-loop amplitudes as problematic but only test the numerical tool up to one-loop contributions. The studies in the machine learning section are minimal compared to the other methods. Considering that the authors conclude that machine learning is the most promising tool, the studies could be extended.
Before recommending publication, these points should be addressed. Please refer to the requested changes for a detailed list.

1. The introduction should be extended with references to support the authors' claims. For instance, in the first two paragraphs, the authors refer to the need for precision for future BSM searches and the difficulties of performing higher-order analytic calculations.
Similarly, "... the test functions used in mathematics often have different properties than the functions related to loop amplitudes" should be clarified and referenced;
2. In Sec.2.3 the authors introduce a target $\epsilon \sim 1\%$. How do the authors decide on this target? How does it would change with different evaluation metrics?
3. In the same section, I think the symbol $\sigma_R$ is not introduced;
4a. The purpose of the slices in $x$ shown in Sec.2.5 is unclear since it is limited to only a few variables and does not cover the full space. The authors could consider reducing the number of slices and highlight important aspects of the amplitudes, e.g. divergent regions or symmetries;
4b. The formatting could be improved by including the slices as figures, which should also ease the discussion;
5a. The distinction between sections in Sec.3 seems overdone. Is there a motivation for dividing the discussion on Chebyshev polynomials in the three short subsections 3.1, 3.2, and 3.3? The authors could consider merging them into a single subsection;
5b. Similar argument applies to 3.8,3.9;
6. Eq.(36), $\text{d}x$ is missing;
7. What is the origin of the checkerboard pattern in Fig.3?
8. The order of Fig.4 and Fig.5 should be swapped since the noise is discussed first.
9. The motivation and the discussion on the effects of noise should be extended. The noise indeed acts as a lower bound for $\epsilon$ but it also worsens the scaling of the methods with the number of amplitudes, as can be seen from Fig.5. This potentially undermines the main scope of the numerical technique and should be discussed.
9. At the end of page 17 there is a typo: was -> way;
10. At the end of Sec.4.3 the authors introduce the advantageous evaluation time of the B-splines. However, no numerical evidence is shown.
This is marginally important for this specific evaluation since the scaling is different but it is relevant given one of the main motivations of the paper, e.g. is the evaluation with a machine learning model faster than B-splines? The authors should consider introducing the evaluation time for all the methods studied.
11. In the machine learning section, the authors move from the parameterization introduced in Sec.2.5 to the 4-momenta of the particles. Could the authors comment on this choice or show evidence of the superior performance of the proposed methods?
12. The LGATR network is seven times bigger than the MLP. How was the choice of hyperparameters done for the two networks? Is this a "fair" comparison?
The authors could extend the discussion with the following:
a. Results with varying number of layers;
b. Evaluation time on CPU and GPU;
c. Comparison at fixed budget, e.g. the number of parameters, training time, or number of computations.

Ask for minor revision

1- The authors demonstrate the ability to leverage interpolation techniques to speed up scattering amplitude calculations
2- The authors perform a detailed analysis of various methods
3- The authors provide their code, improving the reproducibility of their work

1- The authors neglect a large swath of literature that focuses on improving computational costs for event generation.
2- The authors have written the paper trying to balance between explaining the physics to mathematicians and the math to physicists. However, this balance does not work well in the introduction. They over simplify both sides of the discussion too far to make it clear what is going on.

The work of the authors is interesting, and an important development towards making efficient high-precision predictions including higher order corrections. While the work is interesting and an important contribution to the field, it is not suitable for publication in its current form. The authors neglect a large swath of other research on improving the computational efficiency of event generators, do not discuss where the most significant computational bottlenecks arise from in the calculations, nor do they test out any two-loop processes in which this technique would prove to be of most use. Below are a list of requested changes to improve the overall quality and presentation of the work.

1- In the abstract, the authors talk about the dimensionality of the interpolation problems. However, a reader may confuse this with techniques such as dimensional regularization used for evaluating loop integrals, since the discussion of multi-loop amplitudes features prominently in the previous sentence. The authors should make the language more precise.
2- Also in the abstract, the authors state that "This work aims to fill this gap...", but it is unclear what they mean by the gap. The gap in efficient interpolation frameworks? The gap in numerical evaluations of amplitudes?
3- The authors state in the abstract that they are reviewing interpolation methods and "assessing their performance for concrete examples drawn from particle physics." However, the whole first paragraph focuses solely on problems related to particle physics, so here mentioning this seems out of place since the authors have only talked about particle physics thus far.
4- In the introduction, there are exactly 3 references which are just reviews of the progress in higher loop calculations. There are many claims throughout the introduction that should be supported with appropriate references:
4a- The first sentence claims about the precision needs of the experiments for BSM searches. There should be some general references to current collider papers from CMS and ATLAS highlighting their needs.
4b- The authors point out that "two-loop amplitudes become rapidly unfeasible as the number of external legs..." but provide no reference to any of the work that demonstrates this.
4c- The authors claim that numerical methods are more tractable, but again without any reference to works proving this.
4d- The authors talk about the traditional pipeline for Monte Carlo event generation in which millions of phase-space points need to be evaluated, but neglect a large collection of work in this area investigating computational costs, where the bottlenecks are, and attempts to address them. These would include papers such as: 2004.13687 (and references within), 2209.00843, 2203.11110 (and references within), 2112.09588, 1905.05120, etc.
4e- The authors also don't seem to recognize that in modern event generators, most of the computational time is spent on phase space generation, tree-level diagrams, real corrections, and double real corrections (see 2209.00843, talk by Simone Alioli at the "Event generators' and N(n)LO codes' acceleration" workshop held at CERN in November 2023, 2407.02194 (abstract mentions that double-reals are the most computationally intensive part), etc).
4f- To the end above, the authors neglect all efforts in improving the computational efficiencies of the base calculations, such as working on implementing the evaluation of amplitudes on GPUs (2106.06507,1305.0708,0908.4403,0909.5257,2303.18244, 2105.10529,2106.10279,2106.12631,2311.06198,2502.07060,2503.07439), improving phase space handling (1810.11509, 2001.05478, 2001.05486, 2001.10028, 2212.06172, 2211.02834, 2205,01697, 2302.10449, 2311.01548, 2401.09069, 2408.01486, etc.), reducing negative weights (2109.07851,2303.15246,2005.09375,2007.11586,2002.12716,2411.11651,etc.), etc.
5- In the section "What is an amplitude," the authors introduce the calculations of the perturbative series, but focus solely on the loop corrections and completely neglect any discussion of the real corrections that are also required.
6- The authors set the target goal to be at 1%, but provide no justification for this number, and if it is sufficient for the HL-LHC. This is at least an order of magnitude too small for the processes they consider in their tests.
7- The authors discuss how their work would provide a low precision result in the "very-high-energy region" where the contribution to the total cross section is small. However, in the introduction the authors state that precision is needed for BSM, and the effects of BSM will mostly show up in the tails of the distributions and not in the peaks. The precision in the peaks will be needed for electroweak precision tests, Higgs measurements, etc. but not general BSM searches.
8- The authors point out that normalizing flows can be used for find the variable transformation, and simply cite a review article on normalizing flows and neglect the work done by the particle physics community to use this technique.
9- In the test function section, the authors use the language "Leading-order (tree-level) amplitudes", but do not clearly define this. If the goal is to obtain help from mathematicians, these terms should be more well defined.
10- The authors show projections of the test functions, but these are not labeled as figures. These should be figures with captions and not inline plots.
11- The authors use the phrase "integrable Coulomb-type singularity" which should be better defined for the intended mathematician audience.
12- In the introduction, the authors discuss how the two-loop amplitudes really are a challenge as motivation for their study. It would be very useful for them to include some simple two-loop example as a test function to demonstrate that their approach will work for two loops and not limited to tree and one-loop amplitudes.
13- The authors claim that the RAMBO sampling technique is "widely" used. This is not the case in most generators any more. The authors should remove the "widely" and supply references to more modern phase space techniques such as the multichannel approaches, diagram based approaches (MadGraph), recursion-based approaches (COMIX), etc.
14- In all the figures, the y-axes have no labels. The y-axis should be labeled to make it more clear what is shown.
15- In figure 2, the authors should add x- and y-axes labels to all plots.
16- In figure 4, the authors mention that n is the number of evaluations, but do not mention this in the other figures. The authors should add what n is in the caption of the other figures for clarity.
17- Section 3.8 is dedicated to discussing the impact of noise on the interpolation, but do not motivate why there would be such low precision values of a. A discussion of this should be included.
18- Figure 5 is mentioned before figure 4. The order should be switched as is standard requirements for publications
19- At the end of section 5.3, the authors discuss the difference between a greedy approach and a balanced approach as show in Figure 10, but do not explain what function was used in this demonstration. It is mentioned in the caption, but should be mentioned in the main text as well.
20- In section 5.5, the authors discuss that greedy refinement is "not very robust against numerical artefacts and noise." What do the authors mean here by numerical artefacts?
21- The sentence "We see in each test function that the scaling is better, however, and that at high amounts of training data the improvements are more significant" is not clear at all. The authors should rework this sentence.
22- In the discussion of machine learning techniques, the authors talk about the importance of enforcing Lorentz invariance in their network, but do not discuss the importance of enforcing the correct QCD soft and collinear structure. They cite the paper that demonstrated the importance of this as Ref. 57, but do not discuss the novelty of this work.
23- In the sentence "... we can train the MLP on either uniform or unweighted samples and the get similar performance...", the "the" should be dropped after the "and".
24- Overall, the paper could use a read through for clarity and grammar.

Ask for major revision