Response

In their comment, Schaeben et al.1 present a critique questioning the validity of our TACS methodology, suggesting that our approach is fundamentally flawed based on their theoretical examples. They argue that our use of pole figures to assess the performance of TACS is inappropriate. However, visual inspection of pole figure maps remains a widely accepted and standard method for texture validation in materials science2,3,4,5,6,7,8. Moreover, preserving the crystallographic texture in pole figures is sufficient for many practical applications, such as crystal plasticity simulations. The TACS method was primarily developed with these applications in mind, which is why we used pole figure maps as part of our validation process. In fact, similar approaches have been used by others in the field. For example, Marki et al.8 developed a nonlinear optimization method to compact orientation distribution functions (ODFs) for crystal plasticity simulations. They reduced the number of crystal orientations without sacrificing accuracy by comparing compacted and full-size ODFs using the texture difference index (TDI) and visually validating their method through stereographic pole figures. Their results further affirm that pole figure maps can serve as an appropriate tool for validation.

The authors suggest that we are mistaking examples for proof without providing a detailed mathematical interpretation. Proposing TACS methodology does not require mathematical proof as its use must require the user of the tool to assess the suitability of representation in the same way that Schaeben et al.1 have demonstrated the rotational difference in textures1. Moreover, as TACS is based on the K-means machine learning (ML) algorithm, it is widely accepted that ML algorithms are best validated by testing on a large number of datasets. Attempting to develop a mathematical proof would defeat the purpose of employing an ML-based approach, which is inherently designed to handle complex, real-world data. While TACS might not perform perfectly for every dataset, this is a common characteristic of ML-based approaches—they are not a universal solution for all scenarios. However, the TACS technique outperformed kernel density estimation (KDE) in 20 examples of our manuscript. The use of examples of representation using pole figure comparisons is exactly what users of TACS will perform for themselves to assess whether or not results are beneficial for their work.

In addition, one key issue with the comment is the assumption that our method treats the three Euler angles (φ1, Φ, φ2) as independent variables. This is a fundamental misunderstanding of TACS. Our method treats these angles as a single entity, thereby preserving the full distribution of crystallographic orientations. The three angles provide information to evaluate an orthogonal rotation tensor between the crystal frame and the lab/sample frame so they must be taken together. The matching of pole figure maps is a consequence of this preservation.

To demonstrate that TACS preserves the full distribution of crystallographic orientations, we provide sigma sections of the orientation distribution function (ODF) generated by TACS using reduced datasets of 350 and 100 crystal orientations, compared with the ODF from the raw dataset. These comparisons for three example datasets (cubic, hcp, and orthorhombic) are illustrated in Fig. 1. The close similarity of the ODFs demonstrates that TACS is capable of preserving the distributions of crystallographic orientations. Please note that these three examples were arbitrarily chosen from the 20 datasets that were used in the manuscript. All raw datasets and reduced-order datasets are available in the repository, along with a code to plot the sigma sections.

Fig. 1
figure 1

Comparison of sigma sections from three example datasets showing the distribution of crystallographic orientations in the raw datasets and the reduced-order datasets generated by TACS.

Full size image

The comment also critiques our use of the Kolmogorov–Smirnov (K–S) test, arguing that it is not an ideal statistical measure. While we agree that the K–S test alone is insufficient, we employed it as a supplementary check to demonstrate that TACS quantitatively outperforms KDE in preserving the distribution of crystallographic orientations. We used a combined approach of pole figure comparison and the K–S test to evaluate the similarity of the full distribution of crystallographic orientations and agreed upon during the review. The significant difference in the K–S test scores demonstrated that TACS outperforms KDE in preserving the distribution of crystallographic orientations. We have shown that the KDE method cannot adequately capture crystallographic texture in pole figures without extensive parameter tuning, rendering it unsuitable for robust dataset size reduction. We addressed this limitation in detail in our manuscript. In addition, we demonstrate how the ML-based approach avoids human bias in reducing data sets with other techniques.

Conclusion

In conclusion, the comprehensive approach employed by TACS—preserving the full distribution of crystallographic orientations and maintaining accuracy even with reduced datasets—demonstrates its effectiveness as a robust and practical tool for dataset size reduction for broad applications. TACS consistently outperforms conventional methods such as KDE in key areas, proving its value for a wide range of applications. It is misguided to suggest that conventional methods such as kernel density estimation (KDE) would be more suitable, as we have demonstrated in our manuscript that KDE fails to capture crystallographic texture accurately without extensive parameter tuning. This limitation is a clear reason why KDE is impractical for robust dataset reduction, which we addressed in detail. Finally, we expect any rigorous use of our methodology to be done so with full recognition of any inherent assumptions in Machine Learning techniques, as well as recognizing the inherent enhancement of the elimination of potential human bias in reducing data sets with other techniques.