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Non-parametric learning critical behavior in Ising partition functions: PCA entropy and intrinsic dimension
by Rajat K. Panda, Roberto Verdel, Alex Rodriguez, Hanlin Sun, Ginestra Bianconi, Marcello Dalmonte
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Submission summary
| Authors (as registered SciPost users): | Marcello Dalmonte · Rajat Panda |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2308.13636v2 (pdf) |
| Date submitted: | 2023-09-12 16:32 |
| Submitted by: | Panda, Rajat |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
We provide and critically analyze a framework to learn critical behavior in classical partition functions through the application of non-parametric methods to data sets of thermal configurations. We illustrate our approach in phase transitions in 2D and 3D Ising models. First, we extend previous studies on the intrinsic dimension of 2D partition function data sets, by exploring the effect of volume in 3D Ising data. We find that as opposed to 2D systems for which this quantity has been successfully used in unsupervised characterizations of critical phenomena, in the 3D case its estimation is far more challenging. To circumvent this limitation, we then use the principal component analysis (PCA) entropy, a "Shannon entropy" of the normalized spectrum of the covariance matrix. We find a striking qualitative similarity to the thermodynamic entropy, which the PCA entropy approaches asymptotically. The latter allows us to extract -- through a conventional finite-size scaling analysis with modest lattice sizes -- the critical temperature with less than $1\%$ error for both 2D and 3D models while being computationally efficient. The PCA entropy can readily be applied to characterize correlations and critical phenomena in a huge variety of many-body problems and suggests a (direct) link between easy-to-compute quantities and entropies.
Current status:
Reports on this Submission
Report 1 by Biagio Lucini on 2023-10-17 (Invited Report)
- Cite as: Biagio Lucini, Report on arXiv:2308.13636v2, delivered 2023-10-16, doi: 10.21468/SciPost.Report.7958
Strengths
(1) The paper provides a robust analysis of phase transitions using two different unsupervised machine learning methods.
(2) The paper contrasts strengths and weaknesses of the two approaches in a sound way.
(3) While the paper uses the Ising model as a test system, conclusions seem generalisable, with the results possibly opening new avenues of investigation in Statistical Mechanics and Quantum Field Theory at finite temperature.
Weaknesses
In my opinion, the paper does not have any weaknesses. There are some technical points that I would like the authors to clarify, but I consider those addressable at this stage. Details are in the report below.
Report
Machine learning methods are gaining adoption also in fundamental physics as tools for getting insides on physical systems. In this work, the Authors explore two unsupervised methods, the intrinsic dimensionality, and the PCA entropy, as tools to gain quantitative information on phase transitions. The intrinsic dimensionality approach is applied to the 3D Ising model, following previous promising explorations in 2D. The authors find that as a tool this method is rather inefficient in the 3D case, suggesting the curse of dimensionality as an explanation. The second method, the PCA entropy, is more promising and provides very good results. The method is physically insightful and generalizable to other systems. As far as I know, this proposal is original. Therefore, in terms of originality and interest, I believe that the paper meets the mandatory criteria of Scipost Physics Core. In addition, in my opinion, the paper also meet the general criteria (e.g., in terms of clear structure, sound conclusions, very good English, clarity etc.).
However, before I can recommend the work for publication, I would like the authors to consider the changes I suggest below.
Requested changes
(1) The value Nb = 10 seems rather small for the central limit theorem to apply, which is a general requirement of resampling methods. The authors should comment on the choice of Nb in appendix B, providing an example of what happens for a larger value (e.g., Nb = 20);
(2) The authors should explain the size of the errors in Fig. 5b, which appears to be of very different magnitudes, and sharply decreasing for larger lattices.
(3) It is not clear to me whether the interpolation error for the computation of the PCA entropy (e.g., Fig. 5a) has been accounted for. This could be done, e.g., by double resampling, but possibly there are simpler ways to estimate it. The authors should provide an estimate of this error with a short explanatory discussion.
Author: Rajat Panda on 2023-11-13 [id 4109]
(in reply to Report 1 by Biagio Lucini on 2023-10-17)We thank the Referee for their time in critically assessing our manuscript and for the high evaluation of our work.
Below we address the requested changes.
(1) We thank the Referee for raising this point. In Appendix B, we have added a discussion on the effect of changing $N_b$ on $S_{PCA}$. In both the 2D and 3D cases, by changing the value of the number of batches $N_b$, from $10$ to $20$, we observe no significant change in $S_{PCA}$ (we expect a similar behavior for our intrinsic dimension calculations). Therefore, we have kept the plots in the main text unchanged.
(2) We appreciate the referee for pointing this out to us. In Fig. 5a, we can observe that for smaller system sizes the numerical derivative doesn't have a prominent peak, affecting the precision on the estimated $T^{\ast}(L)$ (which is the position of the local maximum in the derivative of the entropy). For larger system sizes the peak is more pronounced and therefore one can locate $T^{\ast}(L)$ more precisely. Further, we observe that the interpolation is more affected by fluctuations in the sampling for the smaller system size as compared to larger lattices. We have amended the manuscript to make this point clear.
(3) We thank the referee for raising this point, as it was probably not so clear in our original manuscript. The interpolation error is accounted for in the following way: We generate a smooth spline approximation for each of $N_b=10$ batches of data (generated as described in the main text) and identify the associated position of the local maximum $T^*_i$ of each of the spline curves. Afterward, we calculate the average $\overline{T^*}$ and estimate the standard error according to the subsampling algorithm described in Appendix B. We have added a sentence to our manuscript to clarify this point.