GeoBench: A new benchmark on Symbolic Regression with Geometric Expressions

Authors: Yilong Xu, Kai Ruan, Mengzhen Liu, Hao Sun
Primary area: datasets and benchmarks
Keywords: Symbolic Regression, Geometry

Abstract:
Symbolic regression (SR) is a powerful technique for deriving mathematical expressions from data. With the emergence of numerous SR methods, SRBench made a significant contribution by providing a standardized testing platform that includes 130 SR datasets and evaluates 14 SR methods. However, the methods included in SRBench are outdated, and the dataset does not feature results from more recent approaches such as TPSR. Additionally, the metrics used in SRBench do not adequately capture the full capabilities of symbolic regression methods, and the benchmark data has scientific problems. Although Matsubara et al. (2022) address some of these issues,
their approach remains incomplete.  In response, we propose a new benchmark consisting of 71 expressions derived from geometric contexts, categorized into three difficulty levels: easy, medium, and hard. We evaluate 20 SR methods on these expressions, focusing exclusively on the symbolic regression capability of each model, assessed through recovery rates across the different levels and overall. We provide a detailed methodology for reproducing the experiments and include results for newly developed SR methods within this updated benchmark. The results demonstrate significant variation in symbolic regression ability across models.

Decision: reject
Reviewer scores: 6, 3, 3, 5

Reviews and discussion:
<thread>
creation date:
1730381031071

thread_type:
official_review

<thread_object>
creator: reviewer_bdjg

<content>
summary:
This paper presents a new benchmark for symbolic regression, constructing 71 datasets on which the SR methods are tested. These datasets focus on 2D and 3D geometric problems and are categorized into three difficulty levels, in which recovery rate is used as a metric to evaluate the SR models.

strengths:
The proposed benchmark integrates a variety of the newest SR methods as baselines, allowing the comprehensive comparison of their recovery performance for the following work.
The authors construct new datasets based on 2D and 3D geometric problems, complementing existing physical symbolic regression datasets.
Overall, the presentation of the paper is clear.

weaknesses:
The paper lacks a description of the advantages of the proposed geometric datasets over the existing datasets like the feynman dataset in evaluating SR methods, in which I think the proposed geometric datasets should be one of the main contributions. 
The paper uses only recovery rate as the evaluation metric, missing assessments of other important factors, such as runtime and the robustness to noise for the SR methods.

questions:
From the hyperparameter settings, it seems the runtime of the presented methods may appear to vary significantly. Should runtime comparisons be included to better promote fairness, especially when different types of SR models are involved? For instance, in some algorithms, like gplearn, there are only 20 generations, which looks far away from the runtime constraints. The presentation of the runtime can also promote fairness for the comparisons of the follwing work, avoiding the simple trade-off of increased time costs for improved recovery performance.
Since the search space grows exponentially with the increase in input dimension, is the same 500 dataset size for each geometric problem sufficient for SR models to recover the geometric equations with high input dimensions, particularly for those classified as hard?
It is not exactly clear to me how the runtime constraints of 1 hour, 5 hours, and 24 hours are derived for different difficulty levels.
Given that the purpose of the datasets is to provide a comprehensive evaluation of symbolic regression models, it would be beneficial to strengthen the description on how the proposed geometric problem-based dataset offers unique advantages over the existing datasets like the Feynman datasets in terms of model evaluation.
</content>
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</thread>
<thread>
creation date:
1730664952329

thread_type:
official_review

<thread_object>
creator: reviewer_eak1

<content>
summary:
The paper proposes a new benchmark for symbolic regression (SR) based on 71 geometric expressions and 20 SR methods. The chosen expressions are used to describe properties of 2D and 3D geometric figures, for instance, the area of a triangle given the lengths of its three sides. These expressions are supposed to be meaningful for real-life applications and complement the current SR datasets. Moreover, the authors categorize the expressions into three difficulty levels.

strengths:
- The problem addressed by the authors - designing a new benchmark for SR - is important.
- The authors provide a good overview of currently used SR methods and benchmarks, discuss their advantages and disadvantages.
- The proposed benchmark is extensive. It includes 71 equations as well as many contemporary SR methods, including recent ones.
- Geometric equations have not been extensively used previously for evaluating SR methods.

weaknesses:
### Real-world applicability
I am not convinced that geometric expressions are appropriate for evaluating symbolic regression. Authors criticize other benchmarks for "lacking real-world applicability" (lines 80-81) and claim that their equations are "meaningful in real-world applications" (lines 103-104). However, I do not think people are going to use symbolic regression to discover simple mathematical expressions that are either known or can be derived in a few steps. I suspect SR to be much more useful when applied to real datasets to uncover *unknown* relationships. However, I doubt the proposed equations are representative of equations describing the real world, because *they lack constants*. Only several equations proposed contain constants and all of them are either small integers or $\pi$. However, in reality, we may need many different constants at least to take care of non-matching units. It seems unlikely that $x_1 x_2 \sin(x_3)$ (rectangle-6) fits the data well. A good fit is more likely to look like $(3.3 x_1 + 0.5)(5 x_2)\sin(1.2 + 13.1 x_3)$.

From what I understand, the dataset also does not contain any noise, which is quite unrealistic. 

In short, why should I evaluate a method on datasets that look very different from the ones I will apply it to? 

### Difficulty levels
The definitions of different difficulty levels seem a bit arbitrary. I understand that "easy" equations are mainly polynomials. But what is the defining feature of "medium" equations? They are supposed to involve "non-linear terms", but higher-order polynomials are already non-linear. "Hard" equations are defined as having a "longer and deeper mathematical structure". But this is very arbitrary. For each category, it is stated how much time is needed to find them but it is not stated using which algorithm. I imagine this may vary a lot for different methods. I do not understand why "vector3d-3" is classified as "medium" whereas "vector3d-4" is classified as "easy" and they are both just polynomials. Moreover, "sphere-4" is classified as "easy" even though it is not a polynomial.

### Algorithm 1
I would like to understand the overall intention behind this algorithm. What was the goal? When would you say two expressions are the same?

### Clarifications needed
- line 71: What is a "non-prime function"?
- line 72: What are "more fundamental expressions"?
- lines 108-109: What does "extended period" mean here?
- line 316: What is a "normal model" compared to "machine learning model"?
- line 443: KAN's output is not a symbolic expression by default. How is the expression extracted from the model?
- line 992: What does "in most models" mean here? Why would it be different for different models? The token set impacts the performance.
- SINDy is used to discover ODEs, how do you adapt it to static symbolic regression?
- How do you make sure the results are comparable in between methods like gplearn and pysr where the population size, the number of generations, or the number of iterations may have a big impact on the performance?

### Possible errors
-  lines 85-86: Although Feynman Symbolic Regression Database contains 120 equations, I think only 100 of them are from Feynman's lectures. The rest are from different sources.
- line 409: EQL has been proposed by Martius & Lampert (2017)

### Minor
- line 81: why is "log" bold?

questions:
1. Why is the performance of an SR method at discovering these geometric equations significant? It is unlikely my dataset will be governed by such clear and nice equations that do not contain any constants or noise.
2. How are the difficulty levels defined exactly?
3. What was the overall goal of Algorithm 1? When would you say two expressions are the same?

In addition, please answer my questions in Weaknesses#Clarifications needed.
</content>
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</thread>
<thread>
creation date:
1730677456232

thread_type:
official_review

<thread_object>
creator: reviewer_4kbm

<content>
summary:
This paper attempts to refine the existing Symbolic Regression (SR) benchmark, SRBench. The paper does this by introducing equation recovery datasets based on geometry and evaluating 20 different SR algorithms on these datasets.

strengths:
The organization of the paper is clear.

weaknesses:
This paper positions itself as one that addresses the weaknesses of SRBench [1] and SRSD [2]. However, the new datasets introduced lack important characteristics as elaborated below:

i). Lack of real-world datasets, which is important for benchmarking SR algorithms. SR is an important explainable machine learning method in which its performance on real-world datasets, which may not have an underlying ground truth equation, is important. The benchmark proposed which the paper claims to be “a refined version of the SRBench dataset”, seems to be a step backwards when compared to SRBench’s real-world datasets.

ii). No results on noise added to datasets, in contrast to existing benchmarks available such as SRBench and SRSD. SRBench even has varying noise levels as acknowledged in the paper itself.

Lack of R2 score, with a focus on only exact recovery rate, which again is a step backward from existing SR benchmarks. Both metrics are important because in the context of using SR for machine learning, the use-case may not always have an underlying ground-truth equation, so R2 score is an important metric that complements recovery rate.

Novelty of the paper is low, especially in comparison to [2], which does similar benchmarking on Physics equations. [2] also introduced the categories of easy, medium and hard equations. In the reviewer’s opinion, changing the domain from Physics to Geometry, while not introducing anything substantially new has low novelty.

Lack of strong justification for why a new dataset is required. The paper does not give a comparison of the conclusions drawn from the new datasets and the conclusions drawn from SRBench and SRSD. If there are similarities in the conclusions, then why is the new dataset required? If there are differences in the conclusions, then the paper should discuss why that is so. Currently, the paper does not even compare against the conclusions that SRBench and SRSD provide.

[1] La Cava, William, et al. "Contemporary symbolic regression methods and their relative performance." Advances in neural information processing systems 2021. 

[2] Matsubara, Yoshitomo, et al. "Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discovery." Journal of Data-centric Machine Learning Research.

questions:
How does this benchmark help SR researchers better evaluate SR algorithms compared to SRBench and SRSD? A comparison of the conclusions drawn from the new datasets and the conclusions drawn from SRBench and SRSD would make the paper more meaningful.

The error referred to in the paper is not clear. This is especially important given that fixed thresholds for this error is given in Algorithm 1. Does the paper use normalized error (e.g., R2 score, normalized MSE) or not (e.g., MSE)?

Would simply updating the set of SR algorithms evaluated for SRBench and SRSD be sufficient to address all the issues that this paper attempts to address?
</content>
</thread_object>

</thread>
<thread>
creation date:
1730687750250

thread_type:
official_review

<thread_object>
creator: reviewer_jkuh

<content>
summary:
This paper contributes a benchmark of symbolic regression methods using synthetic problems derived from 2D and 3d geometry. It includes a description of these problems, a review of other benchmarking work in SR, and a benchmark of several SR methods on the proposed benchmark.

strengths:
The paper is easy to follow and covers a lot of existing literature. The description of the problems is clear and for the most part the benchmarked methods cover a wide swatch of proposed algorithms. It may be of interest to those looking to benchmark new symbolic regression techniques.

weaknesses:
While the paper has many strengths, one of the main weaknesses is that it does not make it clear to the reader why this set of synthetic benchmark problems is a good assessment of the efficacy of these methods in any real-world applications. The benchmark datasets are noiseless geometric functions, which do represent some aspect of the real-world, but this reviewer is left wondering how a real-world problem in which SR would be applied would relate to these problems. The authors do describe the significance of the equations mathematically, but do not offer a compelling example of why one would expect the best algorithms from this benchmark to perform well in some significant real-world version of these problems.

As an example, with other benchmarks like Feynman and Strogatz, we can imagine a scientist collecting the observations that comprise the dataset from real-world phenomena and using SR to discover the underlying physical relation governing the system. Even these problems suffer from a lack of realism which is why noise is synthetically added in prior benchmarks (SRBench) or data is collected from physical examplar systems (Schmidt & Lipson 2009). But what is the equivalent real-world setting for these geometry problems? And, how well can we expect this benchmark's assessment of methods to transfer to those tasks, especially when the data is noiseless?

Below are some more specific critiques: 

> SR offers greater interpretability and superior generalization, avoiding the complexity often
associated with opaque models.

Two strong claims made without justification - I suggest softening the language or referencing examples. 

L046-065: I found this to be a confusing and somewhat apocryphal description of SR history. One might argue that polynomial fitting and/or even classification and regression trees are all versions of symbolic regression, but the authors are stretching in refering to those methods as the origin of SR. SR as it is thought of today (structural and parametric equation fitting to data) is typically dated to Koza's 1992 book Genetic Programming. The paragraph on constant optimization is also not representative of this history; constant optimization for expression trees didn't begin with a "linear regression framework", they started as "ephemeral random constants" (ERCs), i.e. additional terminals made of random values that were "tuned" via expression tree manipulation. The integration of constant optimization into expression tree SR is often attributed to Topchy and Punch 2003. The authors also seem to describe stochastic hill climbing of constants which made popular in Lipson's lab's papers (maybe first with Bongard & Lipson PNAS 2005 or Zykov, Bongard & Lipson IEEE 2005) but they don't describe it clearly or cite it appropriately. 

In their discussion of benchmarking history, the authors should also mention McDermott et al's 2012 paper, GP needs better benchmarks (http://dl.acm.org/citation.cfm?id=2330273), which was a bellwether moment for many in the SR community, who were often benchmarking on overly simplistic datasets. It was a key moment in this history as it incorporated community feedback and black-listed several overly simplistic SR benchmarks problems like the quartic polynomial and many of the Nguyen problems. Incidentally, as the field of SR has crossed to new communities, this step forward is often lost, and results in recent papers benchmarking on datasets that were deemed too easy many years prior. 

The introduction description of SRBench only describes half of the datasets in the benchmark paper, and neglects to mention (importantly) that half of SRBench was dedicated to benchmarking on real-world datasets. These are crucial for benchmarking since otherwise SR methods are being compared only on synthetic data which comes with a load of limitations. 

> Furthermore, the benchmark includes some scientifically unrealistic assumptions, such as treating the gravitational constant as a
variable, and has been criticized for its oversimplified sampling process and inappropriate formulas
(Matsubara et al., 2022).

The gravitational constant treatment is from the Feynman equations, and not specific to SRBench. Matsuraba et al's critique applied to the synthetic datasets in Feynman and ODE Strogatz, but the authors here attribute it to SRBench.

> DEAP (d’Ascoli et al., 2022)

This is the wrong reference; d'Ascoli et al 2022 is not about DEAP. 

Furthermore, it is puzzling that the authors benchmark against DEAP and gplearn, which are both super simple and traditional GP-based SR approaches (they date from 2012 and 2016 but implement the algorithm as it was in the 90s), and then against PySR which is more recent. Why not pick other more recent GP-based SR method that have performed well in other benchmarking works, e.g. Operon (2020) etc?

Figure 4 should label the subplots as easy, medium, hard. If space allows it would be nice for this figure to be in the main text.

questions:
Please see above for questions
</content>
</thread_object>

</thread>
<thread>
creation date:
1732343755832

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Difference from SRbench

comment:
We think we have 5 major different from the SRbench

Purpose of the Dataset: Our dataset is designed to identify symbolic equations that are both simple and explainable to effectively solve problems. We have intentionally composed this dataset of ground truth equations rather than real-world scenarios that lack verifiable explanations for their functions. Consequently, traditional metrics like R-squared or other error measures are not applicable to our goals since they do not align with our focus on explainability.

 

Patterns in Geometric: Unlike the Feynman dataset, which encompasses equations from diverse regions and subjects, our dataset is specifically focused on geometric data within a defined area. We concentrate on discovering patterns in geometric properties such as volume, area, and length. The primary motivation for selecting geometric equations is their inherent potential to unveil these patterns.

 

Structured Learning Progression: The Feynman dataset includes a few sequences that progress from easy to difficult, such as the series from I.6.20 a to I.6.20 b. Our dataset, however, clearly illustrates many such progressions: for instance, from Helen's law to the calculation of circumcircle or incircle radii, which utilize Helen's law, or from Pythagoras' theorem to the cosine law, with the former being a special case of the latter. These process facilitates a deeper and more sequential learning experience.

 

Realistic Constraints in Equations: Our dataset includes equations with generational constraints, such as the triangle constraint where the sum of two edges must exceed the third, and their difference must be less. These constraints make our data more realistic compared to data from SRbench, which is typically generated from uniform distributions. This approach ensures that our dataset not only supports the discovery of geometric relationships but also adheres more closely to real-world scenarios.

 

Complex Equations with Few Inputs: Geometry excels at establishing intricate relationships between variables using a minimal number of tokens, as exemplified by Helen’s law. In symbolic regression, inputs are often chosen or crafted through feature engineering to reduce their number, but this does not necessarily simplify the underlying relationships between them. Therefore, having complex equations with few inputs is crucial because it challenges the models to uncover deep relationships without relying on a large number of variables.

 

The results from our benchmark also differ from those of SRBench. Many baselines in SRBench focus primarily on the R² score, which may suggest they are better at fitting curves. However, their capability to accurately recover true symbolic equations is lacking. Moreover, thanks to the Structured Learning Progression, we are able to categorize these symbolic equations and assess model performance across different levels of difficulty. Additionally, the Patterns in Geometry enable us to evaluate each model's performance within specific patterns. This understanding allows us to select better baselines for future problem-solving involving these patterns.
</content>
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</thread>
<thread>
creation date:
1732343804931

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Extra experiment part1

comment:
We have expanded our dataset to include four additional experiments concerning noise, speed, more baselines, and the introduction of constants.

Noise: We incorporated datasets with two levels of noise—1e-5 and 0.0001—to evaluate how well the models perform under noisy conditions. Unlike typical setups where only the target variable is affected by noise, we introduced noise equally to both the input variables and the target. This simulates a more realistic scenario where the measurement of both features and targets may be impacted by noise.

Speed: To assess the computational efficiency of each baseline, we measured the speed by averaging the results of 100 parallel runs for each category. Understanding the speed of each model is crucial as it allows us to create a Pareto front that balances the recovery rate against the computational time cost, providing a comprehensive view of model performance.

More Baselines: Recognizing the importance of robust comparison, we included additional baselines from the era of genetic programming. Specifically, we added Operon to our benchmarking table to evaluate its performance against other established methods.

Adding Constants: Since our dataset primarily comprises geometric equations where the only constant is π, we tested the ability of the symbolic regression models to handle constants by multiplying each dataset by a uniform constant ranging from 0 to 5. This test aims to assess each baseline's capability in accurately recovering symbolic expressions that incorporate constants.

The outcomes of these experiments are detailed below, illustrating how each model fares across these varied conditions and providing insights into their overall robustness and effectiveness in symbolic regression tasks.

 Table 1: result of data with noise> In each column, the noise level is before the slash and the metric is after that, the discount means the recovery rate discount between no noise with $10^{-3}$ noise.

| baselines | 0-count | 0-recovery | 0.00001-count | 0.00001-recovery | 0.001-count | 0.001-recovery | discount |
|---------|:-------:|:----------:|:-------------:|:----------------:|:-----------:|:--------------:|:--------:|
| PSRN|52 | 68.75% |47|59.65%|40 | 48.32% |70.29%|
| Pysr|48 | 59.31% |44|53.51%|44 | 49.31% |83.13%|
| NGGP|56 | 64.86% |50|52.37%|48 | 50.84% |78.39%|
| UDSR|46 | 47.11% |43|37.56%|43 | 35.56% |75.47%|
| RSRM|57 | 72.92% |52|61.15%|48 | 51.89% |71.17%|
| KAN |4|4.39% |4|3.78%|4|3.11% |70.68%|
| BMS |45 | 54.55% |39|44.74%|37 | 38.74% |71.01%|
|PhySO|25 | 24.44% |25|22.90%|23 | 21.81% |89.23%|
| symindy |26 | 29.18% |20|25.37%|18 | 21.34% |73.11%|
| gplearn |25 | 16.46% |23|13.19%|21 | 11.99% |72.83%|
| deap|37 | 22.65% |30|17.85%|27 | 17.54% |77.43%|
| EQL |3|0.94% |2|0.83%|2|0.67% |70.94%|
|Sindy|16 | 22.54% |15|19.35%|13 | 17.14% |76.05%|
| SPL |22 | 23.99% |20|21.55%|18 | 20.16% |84.03%|
| E2E |28 | 29.39% |23|25.64%|20 | 21.06% |71.64%|
| TPSR|24 | 24.01% |14|19.06%|10 | 14.85% |61.84%|
| AIF |23 | 25.13% |18|21.87%|17 | 15.60% |62.10%|
| SNIP|12 | 11.82% |10|10.23%|6|6.31% |53.36%|
| DGSR|35 | 43.17% |29|34.60%|28 | 30.41% |70.44%|
| NSRS|15 | 18.10% |9|15.70%|5| 10.42% |57.59%|

from the result, all baselines suffer a lot from noise. And baselines with transformer pre-train module like SNIP, NSRS suffers most. And AIF RSRM also does not perform well due to their searching algorithm is not able to displace these noise.

As well, PhySO and pysr still have low discount due to their symbolic ability on physical dimensions. With the dimension , they can cut a lot of useless equation. Also, dimension is not affected through noise.

Table 2:the average time cost within each baselines within 71 datasets and 100 parallel runs

| baselines | time cost(s) |
| --------- | :----------: |
| PSRN| 270|
| Pysr| 374|
| NGGP| 2341 |
| UDSR| 3512 |
| RSRM| 2794 |
| KAN | 130|
| BMS | 2371 |
| physo | 2098 |
| symindy | 478|
| gplearn | 523|
| deap| 476|
| EQL | 1209 |
| Sindy |0 |
| SPL | 1438 |
| E2E |1 |
| TPSR| 3602 |
| AIF | 3475 |
| SNIP| 2975 |
| DGSR| 1097 |
| NSRS| 746|

Next comes the speed test. In this test, we test each model's running speed through all 71 datasets. SIndy model, KAN model runs fast due tothey are linear model. Also, end2end transformer model alsoruns fast for it only runs once and optimize its constant.

Then is the new baselines. Operon can reach 50.1% with 45 reachable, which is sightly below PySR within 127s average, but much more better than gplearn or deap.
</content>
</thread_object>

</thread>
<thread>
creation date:
1732343824682

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
extra experiment part 2

comment:
Final is the constant learning:

 Table 3: result of data with constant, the discount means the recovery rate discount between no constantand with constant.

| baselines | nonconstant-recovery | constant-recovery | discount |
| --------- | -------------------- | ----------------- | -------- |
| PSRN| 68.75% | 51.21%| 74.49% |
| Pysr| 59.31% | 56.48%| 95.23% |
| NGGP| 64.86% | 59.78%| 92.17% |
| UDSR| 47.11% | 42.96%| 91.19% |
| RSRM| 72.92% | 63.98%| 87.74% |
| KAN | 4.39%| 3.97% | 90.49% |
| BMS | 54.55% | 47.24%| 86.60% |
| PhySO | 24.44% | 23.58%| 96.49% |
| symindy | 29.18% | 27.16%| 93.07% |
| gplearn | 16.46% | 15.82%| 96.09% |
| deap| 22.65% | 20.23%| 89.32% |
| EQL | 0.94%| 0.97% | 103.00%|
| Sindy | 22.54% | 22.00%| 97.60% |
| SPL | 23.99% | 22.45%| 93.56% |
| E2E | 29.39% | 26.13%| 88.91% |
| TPSR| 24.01% | 22.12%| 92.11% |
| AIF | 25.13% | 21.86%| 86.98% |
| SNIP| 11.82% | 10.85%| 91.81% |
| DGSR| 43.17% | 39.79%| 92.17% |
| NSRS| 18.10% | 16.84%| 93.05% |

We can conclude that PSRN can not good at handle with constant while the others can fit as well as before since the constant is only multiplies outside the equation.
</content>
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</thread>


Metareview:
metareview:
**(a) Summary**

This paper introduces a new benchmark for symbolic regression (SR) to address several limitations of the existing SRBench benchmark. The proposed benchmark includes 71 datasets derived from 2D and 3D geometric objects, with ground truth equations provided. The dataset is categorized into three difficulty levels, enabling a more structured benchmarking of SR methods. A total of 20 SR methods were evaluated on the proposed benchmark, with their performance assessed using the recovery rate metric.

**(b) Strengths**

Benchmarking for SR methods is still in its early stages compared to other well-established machine learning tasks, such as regression and classification. The addition of a new benchmark is a relevant and valuable contribution to the SR research community.

The presentation is clear, and the paper is easy to follow.

**(c) Weaknesses**

As highlighted by the reviewers, existing SR benchmark datasets, such SRSD (formerly known as the AI-Feynman database), already address benchmarking of symbolic regression to some extent. The advantages of the proposed benchmark over these existing datasets are fundamentally unclear. Specifically, the reviewers raised concerns regarding the lack of real-world datasets, the absence of datasets with diverse noise levels, and the lack of R2 score-based evaluation.

**(d) Decision Reasoning**

While I acknowledge the authors' efforts, the identified weaknesses are significant and need to be thoroughly addressed. A key issue with the current submission is the lack of a direct evaluation of the proposed benchmark dataset itself. Without addressing these concerns, the paper does not provide sufficient justification for the introduction of the new benchmark. Therefore, I recommend rejecting the paper.