SISSO: A compressed-sensing method for identifying the best low-dimensional descriptor in an immensity of offered candidates

The lack of reliable methods for identifying descriptors—the sets of parameters capturing the underlying mechanisms of a material's property—is one of the key factors hindering efficient materials development. Here, we propose a systematic approach for discovering descriptors for materials' properties, within the framework of compressed-sensing-based dimensionality reduction. The sure independence screening and sparsifying operator (SISSO) tackles immense and correlated features spaces, and converges to the optimal solution from a combination of features relevant to the materials' property of interest. In addition, SISSO gives stable results also with small training sets. The methodology is benchmarked with the quantitative prediction of the ground-state enthalpies of octet binary materials (using ab initio data) and applied to the showcase example of predicting the metal/insulator classification of binaries (with experimental data). Accurate, predictive models are found in both cases. For the metal-insulator classification model, the predictive capability is tested beyond the training data: It rediscovers the available pressure-induced insulator-to-metal transitions and it allows for the prediction of yet unknown transition candidates, ripe for experimental validation. As a step forward with respect to previous model-identification methods, SISSO can become an effective tool for automatic materials development.

Figure 1

The method SISSO combines unified subspaces having the largest correlation with residual errors $\mathbf{\Delta }$ (or $\mathit{P})$ generated by sure independence screening (SIS) with sparsifying operator (SO) to further extract the best descriptor.

Figure 2

Benchmark of algorithms. (a) Training error: RMSE versus descriptor dimension for different SOs operating on the smallest ${\mathbf{\Phi }}_1$. (b) Training error: RMSE versus subspace size in the SIS step to find a three-dimensional (3D) descriptor by OMP or SISSO with the same large features space ${\mathbf{\Phi }}_2$ (see Supplemental Material [40] for a similar picture for a 2D descriptor). (c) Training error: RMSE by $\text{SISSO}\left({\ell }_0\right)$ with ${\mathbf{\Phi }}_2$ and ${\mathbf{\Phi }}_3$ compared with previous work [7] (features space size $\sim 4500)$ and with the eureqa software [41] (evaluated functions ${10}^{12}$, larger than $\#{\mathbf{\Phi }}_3)$.

Figure 3

Benchmark of algorithms. (a) Cross validation: LTOCV and LOOCV results for the features space ${\mathrm{\Phi }}_3$ with OMP and SISSO $\left({\ell }_0\right)$. (b) Cross validation: Box plots of the absolute errors for the $\text{SISSO}\left({\ell }_0\right)$-LTOCV results with features space ${\mathrm{\Phi }}_3$. The upper and lower limits of the rectangles mark the 75th and 25th percentiles of the distribution, the internal horizontal line indicates the median (50th percentile), and the upper and lower limits of the error bars depict the 99th and 1st percentiles. The crosses represent the maximum absolute errors.

Figure 4

SISSO for classification. (a) An almost perfect classification (99%) of metal/nonmetal for 299 materials. Symbols: $\chi$, Pauling electronegativity; $IE$, ionization energy; $x$, atomic composition; $\sum V_{\mathrm{atom}}/V_{\mathrm{cell}}$, packing fraction. Red circles, blue squares, and open blue squares represent metals, nonmetals, and the three erroneously characterized nonmetals, respectively. (c) Reproduction of pressure-induced insulatormetals transitions (red arrows), of materials that remain insulators upon compression (blue arrows), and computational predictions at step of 1 GPa (green bars).

Figure 5

SISSO for classification. Correlation between the band gap of the nonmetals and the scaled coordinate from the dividing line.