Estimating the importance of each atom in a molecule is one of the most appealing and challenging problems in chemistry, physics, and material engineering. The most common way to estimate the atomic importance is to compute the electronic structure using density-functional theory (DFT), and then to interpret it using domain knowledge of human experts. However, this conventional approach is impractical to the large molecular database because DFT calculation requires huge computation, specifically, O(n^4) time complexity w.r.t. the number of electrons in a molecule. Furthermore, the calculation results should be interpreted by the human experts to estimate the atomic importance in terms of the target molecular property. To tackle this problem, we first exploit machine learning-based approach for the atomic importance estimation. To this end, we propose reverse self-attention on graph neural networks and integrate it with graph-based molecular description. Our method provides an efficiently-automated and target-directed way to estimate the atomic importance without any domain knowledge on chemistry and physics. In molecules, each atom has the importance in manifesting the entire molecular properties, and estimating such atomic importance plays a key role in interpreting molecular systems. For these reasons, the atomic importance estimation has been consistently studied in the scientific communities (Yen & Winefordner, 1976; Tang et al., 2016; Pan et al., 2018) . However, estimating the atomic importance is one of the most challenging tasks in chemistry and quantum mechanics because the importance of each atom is comprehensively determined based on atomic properties, neighbor atoms, bonding types, and target molecular property. The most common approach for estimating the atomic importance is to interpret the electronic structure using density-function theory (DFT) (Sholl & Steckel, 2009) . In this approach, the atomic importance is estimated through three steps: 1) A human expert selects appropriate functional and basis sets for a given molecule to apply DFT; 2) The electronic structure of the molecule is calculated based on DFT calculation; 3) The human expert estimates the atomic importance by interpreting the calculated electronic structure in terms of target molecular property. Although some methods are developed to estimate relative contributions of atoms in molecules, their generality is typically limited to the molecular properties (Marenich et al., 2012; Glendening et al., 2019) . For this reason, DFT that can generate a general description of the molecule has been most widely used to interpret the molecular systems and to reveal important atoms for target molecular property (Crimme et al., 2010; Lee et al., 2018b; Chibani et al., 2018) . However, the conventional approach based on DFT has three fundamental limitations in efficiency, automation, and generality. • Efficiency: As an example of the electronic structure computations, DFT calculation requires O(n 4 ) time complexity to compute the electronic structure, where n is the number of basis functions that describe electrons in the molecule. Generally, molecules have more electrons than atoms. • Automation: DFT cannot automatically generate all target-specified physical properties in principle, so human expert should manually select additional calculation method to com-pute target molecular property from the electronic distributions. That is, domain knowledge of the human experts is necessarily required to estimate the atomic importance in terms of the target molecular property. • Generality: For some molecular properties, the relationship between them and the electronic distributions is not clear. Moreover, sometimes the estimation is impossible because the relationships between molecular property and molecular structure are not interpretable. For these limitations, estimating the atomic importance is remaining as one of the most challenging problems in both science and engineering such as physics, chemistry, pharmacy, and material engineering. To overcome the limitations of the conventional approach in estimating the atomic importance, we first exploit machine learning-based approach. To this end, we propose a new concept of reverse self-attention and integrate it with the graph neural networks. The self-attention mechanism was originally designed to determine important elements within the input data to accurately predict its corresponding target or label in natural language processing (Vaswani et al., 2017) . Similarly, in graph neural networks, self-attention is used to determine important neighbor nodes within the input graph to generate more accurate node or graph embeddings (Velickovic et al., 2018) . Our reverse self-attention is defined as the inverse of the self-attention to calculate how important a selected node is considered in the graph. For a given molecule and target property, the proposed estimation method selects the atom that has the largest reverse self-attention score as the most important atom. The proposed method estimates the target-directed atomic importance through two steps: 1) For the given molecular graphs and their corresponding target properties, self-attention-based graph neural network is trained; 2) After the training, the reverse self-attention scores are calculated, and then the atomic importance is estimated according to the reverse self-attention scores. As shown in this estimation process, neither huge computation nor human experts in chemistry and physics is required. Thus, the proposed method provides an efficient and fully-automated way to estimate the atomic importance in terms of the target molecular property via target-aware training of the graph self-attention. To validate the effectiveness of the proposed method, we conducted comprehensive experiments and evaluated the estimation performance using both quantitative and qualitative analyses. The contributions of this paper are summarized as: • This paper first proposes a machine learning-based approach to estimate the atomic importance in the molecule. • The proposed method drastically reduced the computational cost for the atomic importance estimation from O(n 4 ) time complexity to the practical time complexity of the graph-based deep learning. • The proposed method provides a fully-automated and target-directed way to estimate the atomic importance. • We comprehensively validated the effectiveness of the proposed method using both quantitative and qualitative evaluations. However, since none of a labeled dataset for the atomic importance estimation and a systematic way to quantitatively evaluate the estimation accuracy, we devised a systematic quantitative evaluation method and validated the effectiveness of the proposed method using it. This paper first exploited machine learning approach to estimate the atomic importance in molecules. To this end, the reverse self-attention was proposed and integrated with graph attention network. The proposed method is efficient and fully-automated. Furthermore, it can estimate the atomic importance in terms of the given target molecular property without human experts. However, the proposed method can estimate the importance of the group of atoms that consists of k-hop neighbor atoms only, even though some important group of atoms may have an arbitrary shape such as ring and bar. As the future work, it is necessary to modify the proposed method to estimate the importance of the group of atoms with arbitrary shape. Fig. 7 shows the original molecules and their selected sub-molecules with the extremely small error. As shown in the results, even though the molecules have carbon rings, the important group of atoms was correctly captured because the molecules are characterized by nitrogen or oxygen. On the other hand, Fig. 8 shows the original molecules and their sub-molecules with the extremely large error. Chemically, double bond plays an important role in determining HOMO-LUMO gap. However, as shown in the results of Fig. 8 , sub-structures that do not contain the double bonds are selected as an important group of atoms. Thus, we need to develop a descriptor for the molecules that can emphasis the bond-features more strongly.
We first propose a fully-automated and target-directed atomic importance estimator based on the graph neural networks and a new concept of reverse self-attention.
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