Unsupervised Discovery of Formulas for Mathematical Constants

Summary

The authors propose an algorithm to filter, cluster and label polynomial continued fractions. It is based on several features linked to the asymptotic behaviour of their rational approximations. The authors detail how these are computed. They apply this algorithm to a large set of formulae ; they discuss the properties of the obtained clusters and highlight some novel PCFs linked to known mathematical constants.

Strengths

This work seems to be an interesting application of unsupervised data analysis to fundamental science. It proposes a way to build relevant metrics from sequences of numbers, and to numerically estimate them. More generally, the structures revealed on fig. 3 and 4 are intriguing and the article is well written.

Weaknesses

The claim that "we connect the challenge of formula creation to modern approaches in AI for Science" seems a bit bold. To my understanding this work does not involve modern AI in the sense that it requires manual and careful feature extraction and that nothing is learnt.

Questions

- Part 3.4 was a bit confusing. The Blind-δ Algorithm is presented as a proxy for delta eq. 4. Is it not rather to estimate the approximation error epsilon ? - There is an inconsistency : part 3.2 it is n^beta vs table 1 P(n) and no beta ; is it because beta is hard to estimate ? is it usefull for clustering ? - Could the authors comment on the fact that the predicted delta eq. 5 can be quite far from the actual delta (fig. 2a, fig. 2b) ? Yet it seems it provides relevant information for clustering ; what does it capture ? - More generally, did the authors try to extract more features, other metrics, not necessary having a mathematical interpretation, in an automated way ? l. 100 "equval" l. 152 "but how is it related to the actual series delta?" is not clear l. 197 "We’ll start" informal

Rating

5

Confidence

3

Soundness

3

Presentation

3

Contribution

2

Limitations

-

Summary

The paper presents a classification of 1.5 million polynomial continued fractions (PCF), continued fractions having as coefficients the integer values of two polynomials, $A(n)$ and $B(n)$, with $A$ $B$ of degree two, with integer coefficients in $[-5,5]$. PCF are classified according to the asymptotic properties of the sequence of approximation errors (difference between the convergents and the limit), the asymptotic properties of the denominator of the convergents (in simplest terms), and the measure of irrationality, which is the difference between the logs of the two previous metrics. Along these metrics, the authors observe that groups of PCF with the same limit tend to cluster together, and that by "anchoring" these clusters on known formulas, one can discover new PCF decompositions of mathematical constants.

Strengths

The paper is well written, and interesting to read. It is an original work, and both theoretical and experimental results are adequately supported. The results are intriguing, and seem to point to scientific discovery.

Weaknesses

The link with AI or machine learning is not completely clear to me. Whereas the clustering techniques used in the paper, and the t-SNE projections used to represent the results, are commonly used by ML practitionners, most of the analyses conducted in this paper amount to descriptive statistical analysis of a specific mathematical dataset. To demonstrate a possible link with Machine Learning, it would be useful that the authors discuss (and perhaps demonstrate) how their approach scales to large sets of PCF, by letting the coefficients and degrees of $A$ and $B$ grow larger. I lean towards acceptance because of the potential interest of such approaches in AI for Science.

Questions

* Figure 1 caption is very long, in the interest of clarity, it might be worth describing your methodology in a specific section (3.2?) * l.96: couldn't we assume that the convergents $p_n/q_n$ are always in simplest terms? this would avoid having to introduce of $\tilde q_n$. Besides, I believe it is assumed in the usual definition of the measure of irrationality. * in section 3.1, you explain that the irrationality measure is either 0 or larger than 1. You then claim that the blind-$\delta$ method provides a good estimator of $\delta$, and figure 2.a seems to support this, yet in figure 2.b most estimates of $\delta$ are below $1$, and some are even negative. What happens? Doesn't this compromise the use of blind-$\delta$ as an estimator of the irrationality measure? * Section 3.5, can you elaborate on "representation power", what it is? why is it important? * Table 1: The exponential factor of the growth coefficient seems useless, why is it? Also, could the larger value of the Davies Boulding Index the irrationality measure be a sign of the problem with its estimation? * Figure 4: can you provide a description of the axes in the tSNE graph? (maybe switch to PCA for more explainability) * l.29: shouldn't Lambert's original paper be quoted, instead of a modern compilation? * l. 100 "equal" (typo) * l.135: the error rate $\epsilon$ is used before it is defined (in section 3.4) * Figure 4, constant $C1$ is not defined anywhere, lemniscate?

Rating

7

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

yes

Summary

They generate continued fraction formulas and test if they evaluate to mathematical constants. They introduce a distance metric to compare formulas. They discover novel formulas for known constants.

Strengths

Mathematical constants are always used, so it is important to have formulas to calculate them well-written throughout experiments

Weaknesses

It generates formula hypotheses, so we do not always know if the formulas are actually correct Rather limited structure of the formulas

Questions

all formulas are continued fraction formula of quadratic polynomials? is that not rather limited? in (1) you give the example tan(x), but in the definition of a/b (p4 118), there is only n and no x. So you cannot actually get a formula for tan(x), can you? is there a list of all 1,543,926 formulas? how many are equal to pi or e? is there pi^pi or e^e among them? or ln(pi)? do you know how many are irrational? or even how many are transcendent? how many are proven to be correct and how many remain hypotheses? have you consulted mathematicians if they are going to use these formulas for anything? >(4) is it necessary to introduce q̃_n and not use q_n without tilde there?

Rating

4

Confidence

3

Soundness

2

Presentation

3

Contribution

2

Limitations

yes

Summary

The paper addresses a long-standing challenge in number theory by proposing a new methodology for the categorization, characterization, and pattern identification of mathematical formulas, specifically Polynomial Continued Fraction (PCF) formulas. The authors introduce metrics based on the convergence dynamics of these formulas, enabling the first automated clustering of mathematical formulas. The methodology is demonstrated on a dataset of 1.7M PCF formulas, leading to the identification of both known and previously unknown formulas for significant mathematical constants such as π, ln(2), Gauss, and Lemniscate constants. The uncovered patterns allow for the generalization of individual formulas to infinite families, revealing rich mathematical structures. This work sets the stage for a generative model capable of creating continued fractions with specified mathematical properties, potentially accelerating the discovery of useful formulas.

Strengths

The work introduces a novel methodology for the automated investigation of mathematical formulas, specifically focusing on Polynomial Continued Fractions. This approach is new in its use of convergence dynamics as metrics for clustering and categorization. The methodology is rigorously tested on a large dataset, resulting in the discovery of both known and previously unknown formulas for important mathematical constants. This demonstrates the robustness and potential of the approach. The paper clearly outlines the problem, the new methodology proposed, and the important findings. The inclusion of detailed explanations and relevant figures helps in understanding the approach and its implications. By automating the discovery of mathematical formulas, this work has the potential to impact the field of number theory and mathematical discovery. The ability to generalize formulas into infinite families could lead to new insights and advancements in mathematics.

Weaknesses

The paper has some weaknesses that should be addressed to improve its quality: While the addressed task and the methodology is unique, it may not be the best fit for a machine learning conference like NeurIPS. The focus on mathematical discovery might be better suited for a specialized conference or journal in mathematics or computational mathematics. The paper’s writing can be further improved for clarity and conciseness. The abstract could be more concise, and the figures are currently blurry and not well-organized, detracting from the overall presentation. The study is based on a limited-size dataset and a small set of metrics. Expanding the dataset and incorporating a broader range of metrics could enhance the robustness and applicability of the findings. The newly identified formulas for significant constants need to be further verified. Ensuring their correctness and utility is crucial for the validity of the contributions.

Questions

How do the authors plan to further verify the newly discovered formulas for significant constants such as π and ln(2)? Can the authors provide more details on how their methodology can be scaled to larger datasets and more complex formulas? How can the approach be adapted or extended to other types of mathematical formulas beyond Polynomial Continued Fractions?

Rating

5

Confidence

3

Soundness

2

Presentation

2

Contribution

3

Limitations

The paper could benefit from a more detailed discussion on the impact of the limited dataset size and the potential benefits of using larger and more diverse datasets. The choice of metrics is crucial for the clustering and characterization of formulas. Providing a rationale for the selected metrics and discussing potential additional metrics would strengthen the paper. Further elaboration on the methods for verifying the newly discovered formulas would enhance the credibility of the findings.