Empowering Active Learning for 3D Molecular Graphs with Geometric Graph Isomorphism
Authors: Ronast Subedi
Lu Wei
Wenhan Gao
Shayok Chakraborty
Yi Liu
Venue: NeurIPS 2024 poster (2024)
Keywords: Active learning
3D molecular graphs
graph neural networks
molecular diversity
Primary area: machine_learning_for_other_sciences_and_fields

Abstract: Molecular learning is pivotal in many real-world applications, such as drug discovery. Supervised learning requires heavy human annotation, which is particularly challenging for molecular data, e.g., the commonly used density functional theory (DFT) is highly computationally expensive. Active learning (AL) automatically queries labels for most informative samples, thereby remarkably alleviating the annotation hurdle. In this paper, we present a principled AL paradigm for molecular learning, where we treat molecules as 3D molecular graphs. Specifically, we propose a new diversity sampling method to eliminate mutual redundancy built on distributions of 3D geometries. We first propose a set of new 3D graph isometries for 3D graph isomorphism analysis. Our method is provably at least as expressive as the Geometric Weisfeiler-Lehman (GWL) test. The moments of the distributions of the associated geometries are then extracted for efficient diversity computing. To ensure our AL paradigm selects samples with maximal uncertainties, we carefully design a Bayesian geometric graph neural network to compute uncertainties specifically for 3D molecular graphs. We pose active sampling as a quadratic programming (QP) problem using the proposed components. Experimental results demonstrate the effectiveness of our AL paradigm, as well as the proposed diversity and uncertainty methods.

Empowering Active Learning for 3D Molecular Graphs with Geometric Graph Isomorphism
Ronast Subedi" Florida State University rs22ce@fsu.
Lu Wei Stony Brook University lu wei. 16stonybrook. edu
Wenhan Gao" Stony Brook University
venhan. edu
Shayok Chakraborty Florida State University shayok@ca. fsu. edu
Liu Stony Brook University
yi.1in.48stonybroo. edu
Abstract
Molecular learning is pivotal many real-world applications, such as drug discov- ery. Supervised learning requires heavy human annotation, which is particularly challenging for molecular data; e.g., the commonly used density functional theory (DFT) is highly computationally expensive. Active learning (AL) automatically queries labels for the most informative samples, thereby remarkably alleviating the annotation hurdle. In this paper, we present principled AL paradigm for molecular learning, where we treat molecules as 3D molecular graphs. Specifically, we propose
diversity sampling method eliminate mutual redundancy built on distributions of 3D geometries. We first propose a set of new 3D graph isometries for 3D graph isomorphism analysis. Our method is provably least as expressive as the Geometric Weisfeiler-Lehman (GWL) test. The moments of the distributions of the associated geometries then extracted for efficient diversity computing. To ensure our AL paradigm selects samples with maximal uncertainties, we carefully design Bayesian geometric graph neural network compute uncertainties specifically for 3D molecular graphs. We pose active sam- pling as a quadratic programming (QP) problem using the proposed components. Experimental results demonstrate the effectiveness of our AL paradigm, as well as the proposed diversity and uncertainty methods. The code is publicly available https: htt///tttttbbt/////////// cww haamm//ronast//te aaa//al__daaa al_3dgrpph
Introduction
Molecular representation learning is essential for various real-world applications, such as molecular design, drug discovery, material design, etc.. In recent studies, molecules have been formulated as 3D graphs, based on the evidence that 3D spatial information is crucial to determine the properties of molecules [Liu et al., 2019, Townshend et al. 2019 Axelrod and
2020).
Generally, a 3D graph, atoms are represented as nodes, each associated with Cartesian coordinates in 3D space. A predefined cut-off distance be used as threshold determine if there is edge between two nodes the 3D graph. With the advance of deep learning, 3D graph neural networks (GNNs) have been developed to learn from 3D molecular graph data Thomas et al.) 2018 Schüt
et al. 2017 Satorras et al.) 2021 Gasteiger et al., 2000cc Liu et al., 2021 2022 Wang et al. 2022
Liao and Smidi 2022 Zhou al. 2022 Yan al. 2022 Wang al. 2023 Lin al. 2023 Zhang
et al.]a0 2023 These models are data- data-hungry and necessitate a large amount of annotated training data
Eqqual contribution.
t *Coresponding author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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to attain good performance. However, annotation usually consumes excessive manpower, which is particularly challenging for molecules, e.g., the commonly used density functional theory (DFT) for molecular energy computing [Hohenberg and Kohn 1964] is very expensive, inducing complexity of (m)), where ne the number of electrons. As concrete example, DFT can be hundreds of thousands of times slower than a reasonably good GNN for inference [Gilmer et al. 2017].
Active Learning (AL) algorithms automatically identify the salient and exemplar samples from large amounts of unlabeled data [Settles] 2009, Ren et al. 202T]. This tremendously reduces the human annotation effort, as only the few samples identified by the algorithm need to be labeled manually. Further, since the deep network gets trained on the representative samples from the underlying data population, typically depicts better generalization capability than a passive learner, where the training data are selected at random. Deep AL has been used with remarkable success in various applications, such as computer vision (Yoo and Kweon 2019, Sinha et al., 2019], natural language processing (Zhang et al., 2022], medical diagnosis [Blanch et al. 2017]. chemistry (Smith et al. 2018). and anomaly detection PPimentel et al. 2020) among others. There are a few AL. applications for 31>
GNNs (Smith et al.) 202] van der Oord et al. 2023]; however, these works do not specifically account for 3D geometric information. The 3D geometry of molecules is crucial for determining molecular properties, but it introduces unique challenges in designing effective AL schemes. Currently, principled AL algorithm for 3D molecular graphs is still lacking.
In this paper, we propose a principled AL paradigm for 3D molecular graphs. We formulate criterion based on uncertainty and diversity, which ensures that the queried molecules are those where the graph learning model has maximal uncertainty about the labels, and that are also mutually diverse to avoid duplicate sample queries. In particular, diversity computing for 3D graphs is challenging and the difficulties are twofold. Firstly, the AL pipeline requires computing the difference between any two 3D molecular graphs, which could have different planar (2D) molecules (entangling different atom numbers, etc.), in most cases. Secondly, the 3D shape (geometry) of 3D graph should be captured completely for expressive geometric representations and accurate diversity computation.
To tackle these challenges, we propose a novel diversity sampling method for 3D molecular graphs based on distributions of important 3D geometries. We propose a set of new 3D graph isometries for geometric modeling. which produces geometric representations that are at least as powerful as the Geometric Weisfeiler-Lemme geometries. This indicates our approach sets an upper bound on the expressive power of any existing 3D GNN models. Hence, the geometries derived from our geometric modeling method (e.g., reference distances, triangles) can be used for accurate diversity computing. To compare any two 3D molecules (with different planar graphs), the moments of the distributions of the derived geometries are extracted for final diversity computing of 3D graphs. In addition, to ensure our AL paradigm selects samples with maximal uncertainties, we carefully design a Bayesian geometric GNN specifically for 3D graph uncertainty computing. Our method is shown to be effective and efficient based set of ground approximations. With our novel components, we pose the sample selection as a quadratic programming (QP) problem and implement fast QP solver to identify exemplar molecules to be annotated. Our method is easy to implement and can be applied in conjunction with any 3D GNN architecture.
(GWL.) test [Joshi et al.) 2023] distinguishing 3D graph
Overall, our proposed AL paradigm incorporates both diversity and uncertainty for 3D molecular graphs. The diversity component, driven by proposed geometric isometries, captures diverse chemical properties from geometries. The uncertainty component leverages chemical contexts, such as atom types, as node features, enhancing the model's ability to identify and learn from uncertain chemical interactions. By considering both our method represents a powerful AL paradigm for 3D molecular graphs. We conduct extensive experiments, and the results demonstrate the effectiveness of the proposed diversity and uncertainty methods as well as the overall AL paradigm.
Our contributions are summarized below. (i) We propose principled AL paradigm to alleviate the annotation hurdle of 3D molecular graphs. We employ diversity and uncertainty measures to select the most informative subset for AL. (ii) We introduce novel diversity component for 3D molecular graphs. Investigating geometric graph isomorphism, we introduce a model-agnostic geometric modeling method, which is provably at least as expressive the GWL test. Our method can significantly enhance the accuracy of diversity computing for 3D molecular graphs. (iii) Our proposed graph isometries set the theoretical upper bound to the expressive power of all existing 3D GNNs, and thus can serve as the new gold standard to test the expressiveness of various 3D

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GNNs. (iv) Rooted in Bayesian inference, we develop an effective and efficient pipeline to compute uncertainties for 3D molecular graphs. (v) Our framework significantly outperforms mainstream AL baselines, achieving remarkable efficiency owing to the cheap complexity of D(22) as well as the implementation of fast QP solver.
2 Methods
2.1 Diversity Computing for 3D Molecular Graphs
In molecular AL tasks, diversity sampling is important for eliminating redundancy, thereby wisely leveraging the annotation budget. The model's capability of capturing the 3D shape diversity among molecules is crucial for informed sampling. A particular challenge that a diversity measure for two 3D molecules with different planar graphs is indispensable. Methods for diversity measures for 3D molecules with the same planar graph have been developed [Kumar and Zhang 2018 Kearnes et al. 2016 Gfeller et al. 2013]. but diversity method for two 3D molecules with different planar graphs (entailing different atoms, etc) is demanding. Inspired by the USR method (Ballester and Richards 2007]. propose novel solution to achieve the goal from the distribution perspective. Generally, we develop a set of new isometries for expressive representations of 3D molecular graphs, after which the distributions of geometries associated with the isometries are obtained for diversity computing.
2.1.1 Isometries of 3D Molecular Graphs
As the first step, we introduce a set of new isometries as a basis, aiming at expressive representations of 3D graphs. As we focus on 3D geometry of molecules in this section, for simplicity, we use 3D point clouds to illustrate our ideas. Let A {al, 02, .... an and B {f((a) f be two sets representing 3D point clouds. Here, each in A is associated with a positional vector a, = (Ia Ma. Za;) in 3D space. f denotes a bijective mapping between and B. Then, similarly, each point [a;; in B is associated with positional vector f(;) = (T))) ()(a))) + =f(a)))
.2,...
faa), ..., f(a)))
Two 3D point clouds, A and B, are said to be E(3) ((3)-isomorpphc. if there exists 7 E E(3) such that A = B. We further choose compute consistent reference point (e.g., centroid) for each point cloud, denoted as rı and r2, respectively. Without loss of generality, we use aar to denote the farthest point from the reference point in point cloud A. Below, will define three levels of isometries, each of which fulfills an isometric mapping between A and B. To satisfy any isometry, there needs to exist bijective function
and - B, such that ()) ha for any node A. Here, h (a) ha denote the node feature vectors for (a) and a, respectively.
Reference Distance Isometry: there exists collection of global group elements % E E(3). such that fa:)) = (7T1, ma.) for each point a, E A, A is reference distance isometric to B.
Reference distance isometry involves the Euclidean distance between any atom in the molecule and the predefined reference point.
If there exists collection of global group elements 7/ E E(3), such that (r2, f (dma) {a:)) (MT11 7arr 7/,0,) for each point a, E A. A is triangular isometric to B.
Triangular Isometry:
With reference point r, we define the refer- ence vector vn as pointing to the farthest Figure 1:
The illustrations of encoding the molecular point Ojr in a 3D molecule. Based on ref- triangular and cross-and -ooslla isometries erence distance isometry, triangular isometry further involves the angle between vn and other vectors
pointing from r to any other point in the molecule, computed as 0g = cos -1
where ik
|
denotes vectors originating from and directed towards hth atoms in the molecule. The process is illustrated in part A of Fig. [ For a molecule with N nodes, compute N angles. Essentially, such angles provide insights into the spatial arrangement of atoms with respect to the pre-assigned reference vector.
Cross-angle Isometry: there exists a collection of global group elements Yi E E(3), such that (r2, I (az), (a:)) (19771, Tijay, wyaa), Va,, aj E A (i + j), A is cross-angle isometric to B.
Beyond the angles triangular isometry as well as based on reference distance isometry, cross- angular isometry further considers angles formed by any two atoms in the molecule with respect to

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the reference vector as above. Specifically, for every pair of atoms and j, vector Vij is formed from i to j. With the reference vector vo, the cross angle is computed as aij cos This approach, as depicted in part B of Fig. essentially reflects cross-angle information globally. For molecule with N nodes, we compute N(N 1)/2 cross angles with the complexity of (
P ||
Next, we propose Theorem to indi- cate the relationship between these three isometries as below.
Theorem 1. If A and B are triangular isometric, then A and B are reference dis- tance isometric; If A and B are cross- angle isometric, then A and B are trian- gular isometric.
fidi
Figure 2: A and B are triangular isometric but not cross- isometric. The angles afrr L and Zdr far in structure A are equal to the angles
The proof of Theorem D can be found in Appendix A.I Generally, we define (f
(((c)r=f(aaar)) and (d)r2 farr) three levels of isometries for graph iso- in structure B, respectively. However, the cross angle morphism. Reference distance isometry Ldr |c not equal to the cross angle (d)))
ensures that the Euclidean distance between each point and predefined reference point is consistent in two different point clouds. Triangular isometry further manifests the spatial arrangement of atoms referring the pre-assigned pivot. Built on triangular isometry, cross-angular isometry then reflects the pair-wise wise global information. An illustrative example for Iriangular isometry and cross-angular isometry is also given Fig. [ Clearly, cross-angular isometry represents the strictest isometry among the three. In the following Sec. 2.1.2 we show that a designed geometric representation based on cross- aaooslr isometry can exhibit great expressive power.
2.1.2 Expressive Power of Our Geometric Representations
In this section, we aim to formally elucidate the expressive power of geometric representation (GR) based on our developed isometries in Sec. 2.1.1 Naturally, we formulate GRoars as a set containing all reference distances, triangles, and eross angles 3D graph.
We explore the Geometric Weisfeiler-Leman (GWL) test [Joshi et al. 2023], and then leverage GWL to illustrate the expressiveness power of our model. GWL test is extension of the classic WL Test, enhancing its capabilities by incorporating both the topological structure of the graph and the geometric attributes of its vertices. Such integration allows the GWL test especially apt for evaluating all 3D graph representation methods. Similar to the regular WL test, GWL test imposes an upper bound to the expressive power of 3D GNNs, i.e., if GWL test fails distinguish two 3D graphs, then all existing 3D GNNs would also fail. See details of the GWL test in Appendix [A.2
GRear is at least as expressive as the GWL test. In other words, Goorr suffices to distinguish any non-isomorphic molecular structures that are distinguishuble by any 3D GNN.
Proposition 1.
The proof of Proposition| can be found in Appendix A. In conclusion, the molecular geometric representation GR developed in this work has the greater expressive power than the GWL test, which indicates our diversity sampling method is accurate enough to capture the 3D shape diversity among different molecules. Notably, as mentioned before, GWL test sets the upper bound to the expresiveness of any existing 3D GNNs. Apparently, our geometric representation GRowr is provably at least as powerful as any existing 3D GNN for learning geometric features. Essentially, the three isometries associated with GRuus define expressiveness at different levels. For example, as only considering distance information, a well pretrained SchNet is upper bounded by reference distance isometry (but not triangular isometry or cross- aoossla isometry); than SchNet, a well pretrained DimeNet is upper bounded by triangular isometry (but not cross- angular isometry). Additionally, learning accurate geometric representations requires a perfectly pretrained 3D GNN model, which is hard to guarantee in practice. Our isomorphy study provides deterministic and model-agnostic diversity component for 3D graphs, avoiding the need of "perfectly' pretrained 3D GNN model, as well as achieving theoretically guaranteed upper bound of the expressiveness of all existing 3D GNN models.
more powerful model
2.1.3 Final Distributional Representations
Based on the isomorphy study in Sec. 2.1.1 we obtain our geometric representation method G Rours and prove G.ours possesses greater expressive power than any existing 3D GNN models in Sec.

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2.1.2 In this section, we aim to extract the distributions of the eniangled three geometries in Goouss including reference distances, triangles, and cross angles, for diversity computing. Fortunately, we have the theorem [Hall 1983] implying that the sequence of translated moments can be used to determine the original distribution. Following the USK work [Ballester and Richards [007], for each of the three aforementioned geometries, we also use four reference points to reflect the ""ranslated" geometries; those are, the centroid (denoted as ctd) computed by the mean position of all the atoms in the 3D molecule, the point closest to the centroid (denoted as cst), the point farthest from the centroid (denoted as fct), and the point farthest from fet (denoted as ftf). For each reference point, we use set of moments, including mean, variance, skewness, and kurtosis, which describe distribution from different angles, e.g., skewness indicates the asymmetry and kurtosis describes the tailedness of distribution. Detailed formulae for these moments can be found in the Appendix A Notably, we compute these translated moments for all three entangled geometries as above. Eventually, we obtain summarized representations of distributions over geometries of 3D graphs, capturing essential characteristics of a molecule's shape.
We use cross angles as example to describe the final distributional vector. For a molecule with N atoms, as shown in Fig. we can obtain a set of cross angles [a (uN-))(N11//3 (,5,,,0.... for a reference point (e.g., ctd). After applying statistical moments as an approximation, we can obtain a 4-dimensional vector ref = [m tof: ieet "if: ker)). where the four elements denote the mean, variance, skewness, and kurtosis for this reference point, respectively. We perform similar process for all four reference
points mentioned above. By doing this, we can obtain four 4-dimensional vectors including M
M ME. and MM which are then concatenated together, resulting in the final 16-dimensional vector to represent the distribution of cross angles. We repeat the similar process for reference distances and triangles, and then all three corresponding 16-dimensional vectors are further concatenated as 48- 18-dimensional distributional vector to represent the geometric information of the input molecule. The 48 48-dimensional distributional vectors are then used to compute the diversity matrix. For any two molecules 11 and ny in the dataset with N molecules, we perform the inner product on their distributional vectors to achieve the similarity, and then use 1- similarity to obtain the final value D Dring as the diversity measure between them. Finally, matrix D E R x is obtained, which contains the diversity between every pair of molecules.
Comparing Our Method to Traditional Structural Descriptors. Our method generates a 48- dimensional vector that encodes the geometric structure of molecule. This representation is both cquivariant to roto-translations and invariant to atomic permutations as the statistical quantities remain unchanged under such transformations. In contrast, Smooth Overlap of Atomic Positions (SOAP) Barttkk et al. [013 De et al. 6 Jäger et al., 2018] generates atom-wise vectors that capture local atomic environments by employing spherical harmonics and radial basis functions. While SOAP is also equivariant to roto-translations it is not invariant to atomic permutations. On the other hand, Atomic Cluster Expansion (ACE) [Drautz 2019 uses a systematic expansion to describe interactions of varying orders (e.g., two-body, three-body interactions). However, ACE is less of a traditional descriptor compared to our method and SOAP; it is designed provide a complete and systematic representation of atomic interactions by focusing on higher-order expansions (e.g., two-body, three- body interactions). This makes ACE more comprehensive in capturing the physical interactions within system, but less suited for producing a fixed-dimensional, flexible descriptor. Unlike our method and SOAP, which generate more compact and adaptable descriptors, ACE emphasizes thorough expansions, making it less ideal for tasks requiring flexible, low-dimensional representations that can adapt casily to the active learning scheme. An empirical comparison between our method and the approach that uses SOAP will be provided, highlighting the effectiveness of our method in capturing molecular geometries.
2.2 Uncertainty Computing for 3D Molecular Graphs
In Sec. 2.1 we develop an effective method for diversity computing among different 3D molecular graphs. In addition to selecting diverse molecules, it is important to select molecules where the model has maximal prediction uncertainty about the labels, so as to append maximal information to the model. Uncertainty qualification is well-studied in planar graph analysis (Hirschfeld et al. [220]. but an effective paradigm for 3D molecular graphs is currently lacking. Additionally, existing methods, such as Bayesian neural networks (BNNs) [Lampinen and Vehtari [00T] [Tttteington, 2004 Goan and Fookes 2020) and deep model ensemble methods [Lakshminarayaaan et al. [017] Huang et al.,
2017]. excessively computationally expensive, limiting their capacity in 3D graph analyses. In
5

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concurrent work (Thaler et al. frameworks, a dropout Monte Carlo scheme has been proposed to lessen these issues.
2024] on active learning for partial charge prediction of metal-organic
In this work, we develop an effective and efficient method, known as Bayesian geometric graph neural network (BGGNN), that takes 3D graph as input and produces the demanding properties well as uncertainty values, e.g. mean and variance. Formally, a 3D graph is represented as G = (V, E, P). where V denotes the set of vertices (atoms), E' denotes the set of edges (bonds), and P denotes the set of Cartesian coordinates for all atoms. A 3D molecular graph is associated with a set of properties, denoted as O. Recently, researchers have developed 3D GNNs, such as SchNet (Schütt et al. 2017]. DimeNet [Gasteiger et al.) (020b), SphereNet [Liu et al. 2022 and GemNet [Gasteiger al. 2021]. for 3D graph representation learning. The likelihood
3D GNN can be represented
as ( G. G.w). where 3DGNN indicates any existing 3D GNN and w denotes the set of
parameters of the used 3D GNN. We also use (EDGNN(W) to represent the prior distribution for the
parameters. Assume we collect new input and output pair, denoted as g" and o'. Then based on the conventional Bayesian theorem, Bayesian inference for this new output o" is given by
PJDGNN (o" g.' G,
PEDGNN (o" Ig" w) PHDGNN G, G,O)dw, R
(1)
where IR" is the whole space of n parameters in 3DGNN. It's infeasible to perform the above integration on R" due to prohibitive computational cost. To tackle this, the variational inference method is introduced to approximate PSIXINN O G, w) with the parameterized qo go(w) through minimizing the Kullback-Leibler (KI.) divergence between these two distributions. After applying Bayesian theorem once more, the minimization objective becomes
Lv) = - go ()) (o(w) log PADGNNN (O G. d)dw + KL (go ( (ww]lp/llwww)))
(2)
To completely avoid the integration over the whole parameter space, the MC-dropout method Gaa
and Ghahramani] 2016 [Srivastava et al., [014] is further used in our BGGNN. Specifically, it employes the Monte-Carde estimator [Gal et al. [016 Gal and Ghahramani 2016 to approximate the integration by performing summation over the sampled models. In practice, researchers implement an MC-dropout network by using dropout as the network's regularization (Gai and Ghahramani 2016). Following this, we propose to insert dropout layers after the linear layers in our used 3DGNN as an effective yet efficient estimation of Bayesian inference.
Now as we have obtained the variational predictive distribution of a new output with (w). we can casily compute the predictive mean and variance of this distribution. For the molecular property prediction tasks, after we sample N outputs from the same input, the heteroscedastic predictive uncertainty is then given by
1 N
t² o0"
g
Σ
(6,))
N Σ 6
Σ T
N
where o is the th sampled output and 62 is the variance that is the same among all the data samples. By doing this, we can obtain an uncertainty value (variance) for each molecule. Additionally, built on a 3D GNN, our BGGNN can faithfully produce a set of molecular properties O.
Practically, any of the existing 3D GNN can be used as the backbone network for property prediction and uncertainty computing. In this study, we employ SphereNet [Liu et al. [022] as our 3DGNN, owing to its great power in incorporating 3D geometric information. We apply dropout layers onto the linear layers of SphereNet for Bayesian inference in our BGGNN. To allow more accurate AL selections, we particularly employ the concrete dropout with learnable dropout rate [Gal et al. 2017] in our BGGNN. Overall, our method is shown to he an effective and efficient paradigm for 3D) graph uncertainty computing, further empirically demonstrated in Sec.
2.3 Active Sampling
A schematic diagram of our active sampling framework is depicted in Fig. aand described in A.4 in Appendix. Specifically, in Sec. 2.1
TT A2 DD
we obtain the matrix D E 3 N'N' containing the mutual diversity between every pair of unlabeled molecules, where N is the number of unlabeled molecules. In Sec.
max
N
2.2 we employ our designed quantifying the prediction uncertainty score of each unlabeled molecule. In the AI. setting, our objective is to select a batch of k unlabeled molecules (Ic is
s.t.
Σ 2 k i=1
BGGNN to achieve the vector E .
4 E (0,1), (4)

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pre-defined query batch size) with high prediction uncertainty and high mutual diversity among them. Let E {0, 11 be a binary vector with N entries which denotes whether the unlabeled molecule T, will be included in the batch (2; 1) or not (z; = 0). The molecule selection can thus be posed as the following optimization problem as in Eq. (4) where À a weight parameter governing the relative importance of the two terms. This is a standard quadratic programming (QP) problem; we relax the integer constraints into continuous constraints and solve the problem using an off-the-shelf QP solver. In this work, we employ the widely used Operator Splitting Quadratic Program (OSQP) (Stellato) et al. 2020] to solve the QP problem in Eq. 4 We then apply a greedy approach to project the continuous solution back to the binary space, where the k highest entries of the continuous solution vector are set to 1 and the remaining to O. Such an approach is commonly used to convert continuous solutions obtained from a QP solver to binary solutions in AL [Chattopadhyay et al.. 2013 Wang and
Ye, 2013 To accelerate the optimization, we implement a solution to execute the problem in the GPU (instead of the CPU) using the parallel implementation of the alternating direction method of multipliers, as detailed in [Scuubieer et al. 2020]. Notably, the predictions in the main tasks (e-g., molecular properties) are produced by our BGGNN built on SphereNet as in Sec. 2.2
3
Related Work
3.1 Active Learning
AL is a well-researched problem in the machine learning community [Settles] 2009]. There exist two commonly used strategies for AL sampling. Uncertainty based sampling queries unlabeled samples with the highest prediction uncertainties for annotation. Divensity/represenaavvvesss based sampling aims select the subset that can well represent the entire data distribution. A full review of the two AL sampling methods is provided in Appendix A.5
3.2 Molecular Shape Similarity
Molecular shape similarity plays pivotal role in drug discovery and virtual screening of compounds
[Kumar and Zhang 2018| Murgueitio et al. 2012
Shang et al.) 2017]. Methods predominantly fall
into several categories [Kumar and Zhang 2018 including descriptor- -asedd methods [Schreyer]
and Blundell 2012, Canmon et 20088] Li et al. 2016 Armstrong et (209 Zhou et al. 2010].
atom-centered Caussian-basee methods (Haque and Pande 2010 de Lima and Nascimento 2013
Yan et al. 2013). surface- methods [Hoofbauer et al.) 2004 Mavridis et al.7 Cai et al.
2012 Karaboga et al. 2013 Venkatraman et al., 2009 Sael et al. 2008). etc. rescriptor-based
methods are notably represented by the Ultrafast Shape Recognition (USR) algorithm (Ballester and
Richards [007]. which uses statistical moments of the distance distribution to characterize molecular
shapes. Gaussian overlay-based methods, with ROCS (Rush et al.) 2005 Hawkins et al., 2007 being the most commonly used one, evaluate the maximum volume overlap between two molecules. Surface-based methods typically employ shape signatures (Zaubar et al.) [2013] or shape histograms to delineate molecular surfaces for shape similarity assessment. Despite the progress, principled and thcoretically ground similarity method for 3D molecular graphs is currently lacking.
4 Experiments
4.1 Experimental Setup
Implementation Details: We use two mainstream 3D GNNs SphereNet [Liu et al. 2022 and DimeNer++ Gasteiger et al. 2020a] as the backbone models of our BGGNN. We directly use the optimal network configurations from the original papers for both backbone models. We train the network for 200 epochs, unless otherwise specified. We use the Adam Optimizer with an initial learning rate 5 x 10 and scale it by factor of 0.5 every 15 epochs.
Data and Active Learning Setup: We first perform experiments on the QM9 benchmark dataset. Since SphereNet is more stable and incorporates more 3D information, we conduct experiments on mu, alpha, homo, and lumo for SphereNet, and mu and lumo for DimeNet*+ These properties have continuous values, making the prediction problem regression task. We randomly divide the training set of 110, 000 molecules into three splits of size 25, 000 each. From each split, we randomly select 5, 000 molecules as the initial labeled set and the remaining 20, 000 molecules as the unlabeled set. In each AL iteration, we query 1, 500 molecules from the unlabeled set, which are labeled and appended to the labeled set. The model's performance is evaluated on a held-out validation set containing 10, 000 molecules. We save the best best-performing model on the validation set and report its performance on the test set containing 10, 831 molecules. The process is repeated for 7 AL iterations, which is taken as the stopping criterion. The final results are averaged over the three splits to rule out
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the effects of randomness. À in Eq. is taken as 1. The Mean Absolute Error (MAE) is used as the evaluation metric. In addition, to study the generalizability of our framework to more geometric data, we also conduct experiments to predict atomic forces for Aspirin in MD17 using our framework. Comparison Baselines:
We use four classic AL methods as basclines: Random Sampling, Core-
ser (Sener and Savarese 2018), Learning Loss (Yoo and Kweon 2019). and Evidential Uncer- tainty [Beluch et al. 2018 Amini et al. 2020]. Random Sampling is the default comparison baseline in AL research. Coreset and Learning Loss are two extensively used deep active learning algorithms for regression applications. Evidential Uncertainty is also commonly used technique to quantify uncertainty for molecular property prediction and was hence included a comparison baseline. Note some existing studies (Kulichenko et al., 2023 Gusev et al. 2023 Craig and Marcia-Melchor [021] have applied AL to molecule research and chemistry. However, these works focus on 2D molecules without considering 3D geometry. which is the focus of our work. Additionally, the techniques used in existing studies can arguably fall into the aforementioned AL categories. Hence, we think comparing with these classic AL methods is sufficient to demonstrate the superiority of our pipeline.
MU
ALPHA
4.2 Active Learning Performance
2200
2600
The active learning performance with SphereNet is depicted in Fig. 3 In
MAE (x 1=300000 (0000)
2000
Corent
Learing Loes
30,000)
2400
Laaning Less
Cornst
1800
2200
1600
2000
each graph, the r-axis denotes the
iteration number and the y-axis de-
1400
1800
notes the MAE on the test set. Our analysis revealed that Evidential Un- certainty depicted the worst perfor- mance and furnished significantly high error values for all four proper- ties, which obscured the difference in performance among the other meth-
1200
1600
1000
1400
4
2
Iteration Number
Iteration Number
HOMO
LUMO
1400
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Randuw
MAE (x 10,000)
Covost
10,000)
Cornet
1200
900
ods in the plots. For better interpreta- tion and understanding, we exclude the Evidential Uncertainty method from the plots here and present the results with this baseline in Sec. A.6 of the Appendix. The other base- line methods depict more or less similar performance, with Coreset marginally outperforming the other iteration onwards, to focus on the comparative performance of baselines. Our method comprehen- the methods after they start selecting samples using AL. Best sively outperforms all the baselines. viewed in color.
800
1000
700
800
600
Iteration Number
Iteration Number
Figure 3: Active learning performance results with SphereNet. The graphs show the mean (averaged over 3 runs) and the error- bars for all the methods. We plot the MAE values from the first
At any given AL. iteration, it consistently attains a lower MAE compared to all the baselines.
We also conducted statistical tests of Table 1: The table shows the p-values obtained using paired t-test between the results our method against each of the base- lines for all the properties studied. Here, I. Loss refers to Learning Loss.
significance using paired t-test to as-
sess whether the improvement in per- formance achieved by our method is statistically significant. For this pur- pose, we compared the average MAE achieved by our method against each of the baselines individually. The re- sults are reported in Table D each en- try in the table denotes the v-luuue of the paired t-test between our method and the corresponding baseline (denoted in the columns) for the property studied (denoted in the rows). From the table, we note that the improvement in performance achieved by our method is statistically significant (p- < 0.05) compared to all the baselines, consistently for all the four properties studied. These results unanimously corroborate the promise and potential of the proposed active sampling method to tremendously reduce the annotation cost in inducing a robust 3D graph neural network for molecular property prediction.
Basclinos
Properties
Random
L. Loss
Coreset
Evidential
mH
7.54x10
5.09x10-3 8
1-
2.19x10
alpha
1.06x10-b
8,11x10--
1.27x10-b
2.72x10
homo 2.26x10-5
8.36x10-7
4.23x10-6
1.71x10-5
lumo
4.48x10-5
1.25x10-o
3.12x10-1
2.39x 10-6
In addition, to study the robustness of our framework to the underlying network architecture and generalizability to the underlying geometric graph data, we have the following results: 1. To study the
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robustness of our framework to the underlying network architecture, results on the mu and lumo properties of the QM9 dataset using DimeNett [Gasteiger et al., 2020a as the backbone model are presented in Section A.7 of the Appendix due to space constraints. The results depict a similar pattern as Figure 3 with the proposed method consistently outperforming all the baselines for both the properties. A paired t-test, presented in Table reveeleed that the performance improvement achieved by our framework is statistically significant. 2. To study the generalizability of our framework to the underlying geometric graph data, results on predicting atomic forces for Aspirin molecules in the MD17 benchmark dataset [Chmiela et al. 2017b] using our framework are depicted in Section A.8 the Appendix and further corroborate the potential of our framework.
Query Budget1100 1,000
Query Budget 1,500
Query B Budget 2,000
2400
2200
2200
0X (0,00,00)
2200
Rardon
Coroed
MAE [X 10,000
2000
Kandonn
Leaning
2000
Rendom
Coresst
2000
Prepeeed
1800
1800
1800
1600
1600
1400
1600
14400
1200
1400
1200
1000
1200
1000
6
Iteration Number
Iteration Number
Iteration Number
Figure 4: Study of query budget on the active learning performance. The graphs show the mean (averaged over 3 runs) and the errorbars for all the methods. The results with budget 1500 are the same as the those presented in Figure 3 and are included here for comparison. Best viewed in color.
4.3 Study of Query Budget
Ablation Study:
Ablation Study: LUMO
2200
The goal of this experiment is study the effect of query bud- get (batch size) on the AL per-
10,000)
2000
Carmuet
Bleensiy Only
MAE 10,000)
1000
Carmot
1800
900
1600
formance. The results on the
800
mu property with SphereNet for query budgets 1, 000, 1, 500 and 2, 000 are depicted in Fig.
700
1000
600
Since Evidential Uncertainty
Iteration Number
Iteration Number
depicted much worse perfor- Figure 5: Ablation study results on the mu and lumo properties with mance than all the methods, it SphereNet. Best viewed in color.
was excluded from this comparison. Our framework once again outperforms all the baselines consis- tently for all the query budgets. As before, we conducted a paired I-test and the results are presented in Appendix A.9 From the p-values, we conclude that the error values furnished by our method are statistically significantly better (p- < 0.05) than all the baselines, consistently for all the query budgets. These results are particularly significant from a practical standpoint as the available query budget in a woaldworld application is dependent on time, resources, and other constraints.
4.4 Ablation Studies
We conduct ablation studies to examine the power of our diversity computing method, as it is our primary contribution in this research. We perform experiments the mu and lumo properties with SphereNet from two aspects. Firstly, we compare our framework with only the diversity term in Eq. lagainst Coreset, the state-of-the-art diversity-based AL technique. The results are reported in Fig. from which we note that the diversity component of our framework consistently furnishes much lower MAE values than Coresel over all the AL iterations, for both properties. Secondly, we also conducted experiments where we compared the performance of our overall framework (using both uncertainty and diversity) against the bascline where only the uncertainty term in Eq. was used for active sampling. The results revealed that removing the diversity term ad adversely affected the performance of our framework. A paired t-test revealed that the improvement in performance achieved by our diversity component is statistically significant (p < 0.05) for both these properties (p = 0.0001 for mu and p 0.04 for lumo). These results show the effectiveness of the proposed diversity metric for AL framework to train a 3D GNN for molecular property prediction. Additionally, we examine the individual impact of diversity and uncertainty components in Appendix [A.10] We also compare our proposed diversity component with the SOAP-based diversity, and test our method against BatchBALD Kirsch et al. 2019]. greedy clustering- based Bayesian uncertainty approach in Appendix A.10
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4.5 Computation Time Analysis
In this experiment, we analyze the computation time of all the methods studied this paper. The average time taken to query batch of unlabeled samples and update the SphereNet model (one active learning iteration) are shown in Table For fair comparison, all the methods were run on the same NVIDIA RTX A4500 20GB GPU.
Table 2: Average (+ std) time (minutes) taken by each method for sample selection and training the SphereNet model (one iteration of AL). Here, L. Loss refers to Learning Loss.
The computation time of our framework is much less than Coreset, which needs to solve mixed integer programming (MIP) problem. The other three methods have similar computation time, as they don't ' involve iterative algorithms. Our method takes only slightly more time than them, owing the implementation of faster QP solver mentioned in Sec. 2.3 as well as our vectorized implementation to enable the use of GPUs to perform diversity matrix computation. The performance studies in Sec. 4.2 show that our framework is much more accurate than these baselines, and the ablation studies in Sec. 4.4]indicate both the diversity and uncertainty components are necessary form QP problem. Given the large margin of performance improvement, we think the efficiency of our method is acceptable.
Random L. Loss
Coreset
Evidential
Ours
53 + 4.5 56 + 2.1 127 +
3.5 56 + 2.3 64.9 + 7.5
5 Conclusion, Limitations, Future Work, and Broader Impacts
We present a principled active learning framework with the goal of reducing the annotation cost for learning from 3D molecules represented as 3D graphs. The sample selection is posed as QP problem, which selects samples with high mutual diversity and high uncertainty. Novel diversity and uncertainty components are proposed for 3D graphs, with strong empirical results presented.
We present a model-agnostic diversity component for 3D graphs, and our method provably at least as powerful as any existing 3D GNN for learning geometric information. Even though our method can set the upper bound of the accuracy of diversity sampling for 3D molecules, it remains unexplored if such advantage can be incorporated into 3D GNN models for diversity sampling. For example, molecular similarity might be incorporated into 3D GNNs to achieve comparable AL performance. Moreover, experimental studies focus small molecules in this work.
As part of future work, we plan to apply our methods to problems where much more accurate but expensive annotation is required, such as computing molecular systems" ground states using the Schrödinger equation. DFT calculations are widely used but still involve approximations, as Schrödinger equation is prohibitively expensive and its use is limited in very small molecules. Our AL pipeline is anticipated to unleash greater potential in such extreme-scale applications. Additionally, given AL needs several interactions with each requiring the model is well-trained, we test our methods on the commonly used but medium-scale QM9 and MD17 datasets in this work. Even though we think the empirical studies sufficient to support our theory, still plan to test the scalability of our methods on large-scale molecule datasets, such as OC20 [Chanussot et al. 2021], in the future.
This work facilitates new avenue in graph analysis by effective and efficient representation of 3D geometric information, thereby dramatically advancing graph learning and mining. Our methods can reduce the annotation cost for molecular data and also have the potential in a broad set of scientific data types, such as materials and proteins, facilitating various disciplines including basic biology, material science, and quantum chemistry. This work is anticipated to have strong impacts on drug discovery and material design by enabling low-cost representation learning. Any positive and negative societal impact associated with those applications and domains can be applied to our methods.
6 Acknowledgment
This work of Y. Liu has used the computational equipment supported by the US Army Research Office under the award 1911 Y. Liu and W. Gao would like to thank Xiaolin Li at Stony Brook University for providing this computational equipment. The work of S. Chakraborty is partially supported by the National Science Foundation under the award (IS-2143422 (NSF CAREER Award).
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Randy Zauhar, Elconora Gianti, and William J Welsh. Fragment- based shape signatures: a new tool for virtual screening and drug discovery. Journal of Computer-Aided Molecular Design, 27: 1009-1036, 2013.
Beichen Zhang, Liang Li, Shijie Yang, Shuhui Wang, Zheng-Jun Zha, and Qingming Huang. State- relabeling adversarial active learning. IEEE Conference on Computer Vision and Puttern Recognition (CVPR), 2020.
Xuan Zhang, Limei Wang. Jacob Helwig, Youzhi Luo, Cong Fu, Yaochen Xie, Meng Liu, Yuchao Lin, Zhao Xu, Keqiang Yan, Keir Adams, Maurice Weiler, Xiner Li, Tianfan Fu, Yucheng Wang, Haiyang Yu, YuQing Xie, Xiang Fu, Alex Strasser, Shenglong Xu, Yi Liu, Yuanqi Du, Alexandra Saxton, Hongyi Ling, Hannah Lawrence, Hannes Stärk, Shurui Gui, Carl Edwards, Nicholas Gao, Adriana Ladera, Tailin Wu, Elyssa Hofgard, Aria Mansouri Tehrani, Rui Wang, Ameya Daigavane, Montgomery Bohde, Jerry Kurtin, Qian Huang, Tuong Phung, Minkai Xu, Chaitanya K. Joshi, Simon V. Mathis, Kamyar Azizzadenesheli, Ada Fang. Alán Aspuru-Guzik, Erik Bekkers, Michael Bronstein, Marinka Zitnik, Anima Anandkumar, Stefano Ermon, Pietro Liò, Rose Yu, Stephan Günnemannn Jure Leskovec, Heng Ji, Jimeng Sun, Regina Barzilay, Tommi Jaakkola, Connor W. Coley, Xiaoning Qian, Xiaofeng Qian, Tess Smidt, and Shuiwang Ji. Artificial intelligence for science in quantum, atomistic, and continuum systems. arXiv preprint MoXi:23307.08423.. 2023.
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Ting Zhou, Karine Lafleur, and Amedeo Caflisch. Complementing ultrafast shape recognition with an optical isomerism descriptor. Journal of Molecular Graphics and Modelling. 29(3):443 2010.
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A Appendix
A.1 Proofs of the Theorems and Propositions
Theorem 1. If A and B are triangular isometric, then A and are reference distance isometric; If A and B are cross-angle isometric, then A and B are triangular isometric.
Proof. Suppose that A and B are triangular isometric, then there exists collection of global group elements 7/2 E SE(3) such that
(r2, S , [(:)) = (1971, Malar,, Rai), Va, E A.
It follows immediately that for each point d; E A. (r2, f(a:)) (NTT1, 7a, also holds. Thus, if A and B are triangular isometric, then and are reference distance isometric.
Suppose that A and B are rross-nggular isometric, then there exists collection of global group elements Yig E SE(3) such that
7oaj, = (,77,,777 nyass, Yas, E A,i j.
By fixing Oj to be afar and the corresponding Ky to be it follows immediately that for each point a; A, there exists a collection of global group elements % SE(3) such that (r2, I (0tar) f(a))) (7/F1, na Ra;) Thus, if A and B are cross-angular isometric, then A and B are triangular isometric.
Proposition Roars least as expressive as the GWI. test. other words, Rours suffices to distinguish any non-isomorphic molecular structures that are distinguishable by any 3D GNN.
Proof. We prove the case when the reference points F1 and are the centroids of the point clouds A and B. respectively. The proof for other choices of reference points follows analogously. First, we will show that C, which is the function that gives us the geometric representation Gouus given point cloud, -orb-t injective.
Without loss of generality, assume that the centroids of these two point clouds are at the origin. Otherwise, they can be fixed by translation in T(3) = F(3)/O(3). For simplicity, we denote the point f(a:) as b; and bold symbol represents the corresponding vectors. Suppose that G Rours is the same for both points clouds A and B, that say, we have the following conditions:
|| |asl| = bbsl, Vi E Nen
(5)
(aiar:
a4)
(Brar, b²)
Vi E Nun
(6)
[ffarl]
||a/||
||ball'
(Arfar; aii
a.))
(biars bi bj)
Vi,j N5n, # j.
(7)
[
||a:
agl ||
|
bbb bill"
It follows from
55 and
66 that for any lo Ngn.
(arfary ax) (biars bx)-
Then, for all i E N i }j.
(Giar ag aa)) (brar, b; by)-
Thus, it is clear from for all i.j E Nen,
7 that all the pair-wise distances the same, i.e., ||a; a, || = ||b; b Thus
|| la; - ail1 = -||azll2 (a,, ag) + ||a,||
bbb 11 ||bb|]] (b, by) |bbl
follows that (a,, a.) (b, b,) from (5)
It is safe to assume that (a1, a2.............................. an) spans E3, otherwise the proof is trivial when all points are co-planer. Without loss of generality, let (a1, a2 a3) be a basis for E33 It is easy to see that (b,, b2, by) is also a basis for E
18

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Let X and Y denote the matrices whose columns are (a1, 0z) and (b1. b2. by). respectively. Let G denote the associated Gram matrix, i.e. G = y"y, then is symmetric and positive semi-definite. Moreover, as both X and Y are full-rank, there exist orthogonal matrices Qx, Qy and upper triangular matrices Rx, Ry such that
X Qx {
= Ry
then
G x= - Ry Rx Ry
- R
The form above follows the pattern of Cholesky decomposition. As G is symmetric and positive semi-definite, the Cholesky decomposition is unique. Thus, Rx Ry. Thus, X QxQv' where QxQy' is orthogonal matrix. Thus, there exists 9 E 0(3) such that gX Y. Ifn s 3. this completes the proof.
When n 2 4, for any k: 2 4, ak Eim caa, where { are uniquely determined by c = (ak, ai). Then, gax = 9 (2:1 cia;) = Ei es (ga:) = Σ: 33 c;b; bj.
Now, without loss of generality, we can conclude that if Ç (A) < (B), then there exists g E (3) such that g = B. As we have an injective map, our method is naturally at least as expressive as the GWL test for E(3) isomorphism. As result, our method surpasses all existing 3D GNNs in terms of distinguishing non-isomorphic point clouds.
A.2 Geometric Weisteller-Leman (GWL) Test
For the Geometric Weisfeiler-Leman (GWL) test, consider graph G with its set of vertices repre- sented as V(G) and its set of edges as E()). A vertex in graph G is denoted by i, and Ai signifies the set of vertices adjacent to i. The color of vertex i at iteration t is given by c and the geometric object for vertex é at iteration is represented by g," The procedure for the GWI. test is as follows:
1. Initialization: Each vertex i is assigned an initial color ci and a geometric object g, typically based on its local property or geometric attributes.
2. Iterative Aggregation: For each iteration & 2 1, the geometric object of each vertex i is updated aggregate geometric information from its t-hop neighborhood, represented as which includes the colors and geometric objects from the previous iteration of vertex i and its neighbors.
3. Color Update: The color of each vertex i at iteration è is computed by aggregating the geometric information around vertex using G-orbit injective and (-invarriant function, denoted by I-HASH, i.c., c( 1 (ASH(()
4. Termination:
The procedure terminates when colors do not change from the previous iteration or predetermined maximum number of iterations reached.
5. Graph Comparison:
Finally, two geometric graphs G and H. are geometrically non- isomorphic if there exists some iteration for which the sets of colors of their vertices are not equal, i.e., {(<() E v(G) + {{c
(0) |ie }(10)).
A.3 Statistical Moments
The equations that used for calculating four moments follows.
The mean, often referred as the average, represents the sum of all data points divided by the number of data points and is given by
>n
Fz
Mean
(8)
n
19

---

Variance measures the spread or dispersion of a dataset and is defined as
Ei= (r; Mean)²
Variance
(9)
n
Skewness gauges the asymmetry of a dataset's distribution. Here we sightly change its definition to be positive for convenience
E J; Mean //
Skewness
(10)
{=== (2t Mean) 1)3 3/2"
Kurtosis assesses the "tailedness" of dataset's distribution as
E Mean) //1
Kurtosis
(11)
12 (z. Mean) 1)}
A.4 Schematic Diagram of our Framework
schematic diagram of our active sampling framework is depicted in Fig. We are given a labeled training set T., an unlabeled set U and query budget & for each active learning iteration. The SphereNet model is first trained on the labeled set L. In the second step, the trained model is applied on the unlabeled to populate the uncertainty vector r; the diversity matrix D is also computed in this step where D(i, J) the diversity between unlabeled molecules 2; and zy. Next, the QP problem solved to select unlabeled molecules for annotation. These molecules are removed from the unlabeled set U and appended the labeled L. The active sampling process is continued iteratively until some stopping criterion is satisfied (taken Note that, computing the diversity matrix process. Once we have the initial D, as more and more samples are queried through AI,, we keep deleting the corresponding rows and columns from / derive the updated matrix.
compute a prediction uncertainty of each unlabeled molecule, which used
iterations in our work).
Step needs be executed just once for the whole
Given: Labeled set L, unlabeled set U, active query budget k
Train the BGGNN model
Step
Step 2
Step
Step
Apply the trained model on U and compute uncertainty vector
Compute the diversity matrix on U
AL Iterations
Solve the QP problem to select batch with k samples from U and get them annotated
Step
Step 5a
Step 5b
Update the labeled set:
Update the unlabeled set:
L= U
U U-
Figure 6: Schematic diagram of the proposed active learning framework.
A.5 Related Work for Active Learning
Active Learning (AL) well-researched problem in the machine learning community [Settles] 2009]. Uncertainty sampling is an important strategy for AL, where unlabeled samples with the highest prediction uncertainties are queried for annotation. Several techniques have been explored to
compute the uncertainty, such as Shannon's entropy [Guo and Schuurmans, 2007 Li and Guo 201 3 the distance of sample from the separating hyperplane for SVM classifiers [TTng and Koller 2001 the disagreement among committee of classifiers regarding the label of sample [Frunnd et al.
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MBU
2.229
ALPHA
HONO
LUNO
MAS Dt 10,000
MAE Dt 10,000
MAE It 10,000
2000
1500
1400
1900
1200
1000
1000
800
1000
600
Heration Number
Iteration Number
Heration Number
Reration Number
Figure 7: Active learning performance results. The graphs show the mean (averaged over 3 runs) and the errorbars for all the methods. Best viewed in color.
1997
Filad-Bachrach et al.
2005 , among others (Hoi et al. 2006 HOI et al. 2008 Guo and Greiner
2007 Freytag et al.] 20144. With the advent of deep learning. Deep AL has attracted significant
research attention [Hnoo 2020 Ren et al.] [02T]. Entropy-based methods are developed as well
WWng and Shang 2014 Ranganathan et al.) 2017]. Yoo and Kweon [2019] cascaded task-agnostic
loss learning module that queries samples with the highest predicted loss values. Huang et al. [202]]
proposed strategy based on temporal output discrepancy. Techniques based on adversarial training
have also been explored [Sinha et al.] 2019 Mayer and Timofte, [2200] [Zhang et al. 2020 Zhu and
Bento 2017]. Bayesian neural networks (BNNs) [Lampinen and Vehtari [001] Titterington 2004
Goan and Fookes 2020) and deep model ensemble LLakshminarayanaal et al. 2017 Huang et al.
2017 generally achieve superior performance but may induce excessive computational cost.
core-set sampling technique proposed by ? queries a batch of samples such that a model trained on the queried subset is competitive for the remaining data samples. Diversity sampling has also been exploited in the context of Bayesian neural networks (Kirsch et al., 2019). [Buchert et al. [0023] uses diversity sampling, together with self-supervised representation learning to select an informative seed set for AL. Combinations of arrrraaniiiivesiiilreeeeeearreee
Diversitynepeesenaatvveess based AL sampling has also been exploited.
criteria have also been used as
query functions in AL research (Chakraborty et al. 2015 Wu et al. 2022
Ash et al. 2020].
A.6 Results with the Evidential Uncertainty Baseline
The active learning performance results on the four properties studied (mu, alpha, homo, and lumo) are depicted in Fig. 7 As mentioned in Sec. 4.2, we note that Evidential Uncertainty depicts significantly high error values than the other methods, for all the four properties.
A.7 Performance using the DimeNet** ++ Backbone
The objective of this experiment to study the performance of our framework with DimeNet + Gasteiger et al. 2020a] as the backbone model of our active learning approach. We use the mu and lumo properties from the QM9 dataset in this experiment. We use the same experimental setup as detailed in Section 4.1 of the paper. The results are depicted in Figure 8 The proposed framework consistently outperforms all the baselines at each AL iteration across both the datasets.
The results of the statistical tests of significance are reported in Table Each entry in the table denotes the p-value of the paired t-test between our method and the corresponding baseline (denoted in the columns) for the property studied (denoted in the rows). We note that the performance improvement achieved by our method is statistically significant (p < 0.05) compared all the baselines for both the properties. These results corroborate the robustness of our framework to the underlying GNN backbone.
Table 3: The table shows the v-vuuues obtained using paired t-test between the results our method against each of the baselines for the mu and lumo properties, using Dimenet* backbone.
Baselines
Properties
Random
Learning Loss
Coreset
mu
1.56 x 10-6
1.87
10-4
1.25x 10-4
lumo
8.55 x 10
4.24 10-4
1.16 x 10-2
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MU: DIMENET++
LUMO: DIMENET++
1000
1800
Random
Random
MAE (X 10,000)
Learning Loss
MAE (X 10,000)
900
Learning Loss
Coreset
Coreset
1600
Proposed
Proposed
800
1400
700
1200
600
1000
500
2
2
6
Iteration Number
Iteration Number
Figure 8: Study of our framework using the DimeNet backbone. The graphs show the mean (averaged over 3 runs) and the errorbars for all the methods. Best viewed in color.
ASPIRIN
3000
Random
Learning Loss
Coreset
2500
Proposed
MAE
2000
1500
Iteration Number
Figure 9: Study of our framework on Aspirin molecules using the SphereNet backbone. Best viewed in color.
A.8 Generalization: Performance on the MD17 Dataset
The research of 3D molecular learning is new, and there are only few reliable benchmark datasets for 3D molecules (containing atom types as well as XYZ coordinates for all atoms for each molecule). To test the generalization ability of our proposed method, we study the performance of our framework on the MD17 dataset [Chmiela et al. 2017a]. QM9 consists of molecules in equilibrium, while MD17 contains several thermalized (i.e., non-equilibrium, slightly moving) molecular systems. Additionally, QM9 contains various quantum properties for molecules, like the important homo and lomo orbitals. MD17 for dynamic system simulation, thus it contains labels for both the energy and atomic forces. In summary, test methods molecule systems in both equilibrium and coneeuuilibrium covering various quantum properties and molecular dynamics tasks.
We used 300 samples as the initial training set, 700 samples as the unlabeled set and 1, 000 test samples; we used 100 as the batch size and conducted iterations of active learning. For this dataset, we train the network for 500 epochs. The results on Aspirin molecules using the SphereNet as the backbone of our GNN are depicted in Fig. 9 Our framework once again depicts promising performance and attains the lowest MAE values across all the AL iterations compared to all the baselines. These results further demonstrate the promise and potential of our method for scientific applications.
A.9 Statistical Tests of Significance for the Query Budget Experiment
Table reports the results of the statistical tests of significance the study of query budget (presented in Sec. 4.3). Each entry in the table denotes the p-lluue of the paired t-test between method
22

---

Table The table shows the p-values obtained using paired t-test between the results our method against each of the baselines for the mu property for query budgets 1, 000, 1, 500 and 2, 000.
Baselines
Budget
Random
Learning Loss
Coreset
Evidential
1000
7.58 x 10
1.05 x 10-5
5.32x10
2.46 x 10 10
1500
7.54 x 10-6 6
5.09 x 10-5
1.51 x 10
2.19 x 10
2000
7.90 x 10-5 1.74 x 10-3
1.94 x 10-4
1.77 x 10-8
and the corresponding baseline (denoted in the columns) for the query budget (denoted the rows) for the mu property, From the table, we note that the improvement in performance achieved by our method is statistically significant (p 0.05) compared all the baselines, consistently for all the query budgets.
A.10 Addtional Ablation Studies
The Individual Impact of Diversity and Uncertainty Components. In Fig. 10 present the result on the individual impact of the diversity and uncertainty components. Our proposed method outperforms the individual in our method's dual focus on both geometric importance and chemical contexts. Moreover, it can be observed that the diversity component alone shows strong performance; it is only slightly less effective than our method because it captures the geometries of molecules, which are fundamental in distinguishing different molecules with different properties. On top of this, we also conduct statistical tests to conclude that the improvement of our method is significant compared to only diversity or only uncertainty in Table 5
of diversity or uncertainty alone. The key this outperformance lies
Table The table shows the p-values obtained using paired t-test between the result of our method against uncertainty only and diversity only components in ablation study for mu and lumo prediction.
Components
Properties
Uncertainty Only
Diversity Only
mu lumo
1.24 10-5
1.82x 10
7.46 10
2.26x10
Allaioon Study:
Ablation Study: LUMO
1800
900
Diversity Only
Diversity Only
MAE (X 10,000)
1600
Uncertainty Only
Proposed
MAE [X 10,000
Proposed
Uncertainty Only
800
1400
700
1200
1000
600
2
Iteration Number
Iteration Number
Figure 10: Ablation study results studying the individual impact of uncertainty and diversity on the mu and lumo properties with SphereNet. Best viewed in color.
A Comparison between Our Diversity Component and SOAP. We include a comparison between our method and one that uses well-known geometric descriptor in chemistry, the SOAP descriptor [Bartók et al., 2013). SOAP produces descriptors that characterize local atomic environments using spherical harmonics and radial basis functions. incorporates both geometric information and elemental (species) details. Table 6 presents the results on the QM9 dataset for two important properties, mu and lumo. These results demonstrate that our diversity component clearly outperforms SOAP. Consequently, overall method, which combines both diversity and uncertainty, also
23

---

surpasses the performance of the SOAP descriptors. The p-values in Table further illustrate that our proposed method significantly improves the selection strategy compared to SOAP. This improvement can be attributed to the more localized nature of the SOAP descriptors and the inability to maintain permutation invariance.
Table 6: The table shows comparison between our proposed descriptor and the SOAP descriptor for the properties mu and lumo, using SphereNet the backbone.
Iteration
mu
lumo
SOAP D
Our D
SOAP B
Our B
SOAP D
Our D
SOAP B
Our B
2154
1769
2057
1741
1016
890
1013
876
1901
1576
1877
1550
921
805
902
799
1732
1440
1721
1412
839
740
838
732
1701
1352
1539
1315
797
692
791
677
1587
1225
1456
1205
759
680
744
660
1414
1157
1345
1121
699
629
711
615
1322
1092
1280
1072
667
604
681
594
Abbreviations: means using diversity Only; B means using both uncertainty diversity.
The results from the method with superior performance highlighted in bold.
Table 7: The table shows the p-values obtained from paired t-test comparing the results of our method against those of SOAP for the properties mu and lumo, using SphereNet as the backbone.
lumo
p-value 4.20 10-6 2.51 x 10-6
A Comparison between our method and BatchBALD. We investigate the impact of our quadratic programming formulation compared to the greedy, clustering- baaseedeng--eedd Bayesian uncertainty baseline, BatchBALD Kirsch al.) 22199, which selects diverse batch of samples by maximizing mutual information. Our method provides more structured approach to uncertainty, particularly tailored for 3D molecular data. The results, presented in Fig. [T] demonstrate the effectiveness of our approach. Additionally, we conducted statistical tests, as shown in Table 8 which confirm that the improvement of our method over the baselines is statistically significant. This outperformance can be attributed to the components specifically designed for 3D molecular graphs.
Table 8: The p-values obtained using paired t-test between the results our method against each of the baselines for all the properties studied. Here, L. Loss refers Learning Loss.
Baselines
Properties
Random
L. Loss
Coreset
Evidential
BatchBALD
mu
7.54x10-6
5.09x10
1.51x10
2.19x10-7
7.20x10
alpha homo
1,06x10
8.14x10
4.27x10
2.72x10
4.86x10
2.26x10-1
9.36x10-1
4.23x10-6
1.71x10-8
1.54x10-4
lumo
4.48x10-5
1.25x10
3.12x10-4
2.39x10-6
3.96x10
A.11 Licenses for Existing Assets
We list all the licenses for existing assets in Table
24

---

MU
ALPHA
2600
Random
Random
MAE (X: 10,000)
2000
Learning Loss
Coreset
MAE [X 10,000)
2400
Coreset
Learning Loss
BatchBALD
2200
BatchBALD
Proposed
Proposed
1500
1800
1000
1400
6
2
lteration Number
Iteration Number
HOMO
LUMO
1400
MAF (X 10,000)
Learning Loss
Random
Coreset
MAE 10,000)
1000
Learning
Coreset
Random
1200
900
BatchBALD
Proposed
Proposed
800
1000
700
Iteration Number
Iteration Number
Figure 11: Active learning performance results with SphereNet QM9 Dataset. Best viewed in color.
Table Assets, Licenses, and Descriptions
Asset
License
Description
SphereNet Liu et al
2022
GNU General Public License v3.0
GNN Model
DimeNet Gastegger
Hippocratic License v2.1
GNN
QM9 Ramakrishiah et al
201
CC BY-NCC-A 4.0 International License
Benchmark Dataset
Kamakrishiot
BY-NC-SA International MIT License N/A N/A. Apache-2.0 License
Benchmark Dataset
Coreset Soner lild Salvire 2018
Active Leaming Scheme
Leaming Loss Y00 and Kwou
2019
Active Learning Scheme
Evidential Unoeraiity [Beue
2018 Amini al. 2020
Active Learning Scheme
25

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Reviews:
[{"review_id": "xlmCMtl96r", "reviewer": "Reviewer_yNCZ", "summary": "This paper describes a way of sampling 3D graphs for active learning on\nmolecules, leveraging isometries.", "questions": "* How does sampling the graphs relate to sampling the interactions that are\nresponsible for properties?\n\n* Is chirality considered?\n\n* Why do the graphs in Figure 3 not start at the same loss?  Before any active\nlearning has occured, shouldn't that the situation?\n\n* Are the other AL techniques also designed to be used in this iterative manner?\nIf not, I think a better baseline test would be to use the same number of\nsamples.", "limitations": "The paper does not adequately address limitations. I would imagine one would be that it does not consider the actual elements of each atom. In other words, two molecules with similar geometries are considered similar, even though they could be very different chemically.\n\nWhat about periodic structures?\n\nAlso, what if there is not a pre-existing source of valid, but unlabeled molecules? A common situation is that there are only a small number of molecules known at all.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "I think this work is interesting, and potentially useful.\nClearly some kind of active learning would have useful use cases.\nAlso, the use of isometries could also be relevant, depending on the exact\nnature of the sampling.", "weaknesses": "I think the main weakness of this work is that it needs stronger connections\nto the actual chemistry.\n\n* The actual elements are important. Two molecules can have similar\ngeometries, yet very different electronic interactions that give rise to very\ndifferent properties.\n\n* The paper would benefit from explaining the relationship between the\ndifferent isometries and actual molecules and chemical properties.\n\n* The space of molecules is very large.  The tests hold back part of known\ndatasets as unlabeled molecules.  But the most useful new labels may not be\nin the QM9 dataset, but may be in some other dataset, or even a novel\nmolecule.  Thus, a more interesting algorithm would be one that identified a\nmolecule X as as important new molecule to add, allowing some search for it.\nOf course, this would raise a number of other important issues, such as\ngenerating the new molecule X, etc.\n\n* The paper does not test against 2-D methods.  Not sure why.  Perhaps they\nperform well.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">W1: about actual atoms\n\nOur framework contains two components for selecting important molecules: diversity and uncertainty. **We do consider the actual atoms in the uncertainty part, as the atom types are embedded into node features.** \n\nOur diversity component focuses on molecular geometries, **which essentially reflect the interactions among atoms in a molecule.**\nThus, 3D shapes significantly influence molecular properties [1, 2]. As a concrete example, when predicting the important HOMO-LUMO gap of QM9 using MAE (lower is better), using data without 3D geometries produces *1.23-3.28* [3], while using data with 3D info significantly improves it to *0.03-0.06* [4]. The GWL test also focuses on geometries, and 3D GNNs aim to integrate geometric information into the learning process.\n\n**In summary, by combining both uncertainty and diversity, we take both the chemical contexts and geometric contexts into consideration.** We will emphasize this in the next version.\n\n> W2: the relationship between different isometries, actual molecules, and chemical properties\n\n**Full geometric information is important for computing chemical properties.** As we discussed in lines 172-179, different isometries are mainly related to the completeness of the geometric information and the universal approximation of 3D GNNs. For example, simply using the distances (Reference Distance Isometry) cannot recognize two molecules with the same edge lengths but different angles, which is expected to produce suboptimal property predictions. Using angle information (Triangular Isometry), for example, improves the MAE of the HOMO-LUMO gap from *0.063 to 0.033* [4].\n\n>W3: extend to different datasets and even generate new molecules\n\nVery insightful view!\n\nFirst, we can definitely extend QM9 to new datasets like MD17, as our active selection is applicable to all 3D molecules. We do it separately because we follow the fashion of the community and it's easy for evaluation.\n\nSecond, this study falls into molecular property prediction, and molecular generation is a different topic. We agree that extending our algorithm to identify novel molecules can be highly beneficial. **As you mentioned, this approach could raise other issues, such as the validation of the generated molecules, which is far beyond the scope of this work.** Therefore, we kindly request not to view this as a weakness of our work.\n\n>W4: does not test against 2-D method\n\nThe scope of this work is 3D molecular learning based on 3D GNNs. As we have pointed out previously, the 3D conformations of molecules determine their properties; thus, **3D GNNs outperform 2D GNNs by a large margin**. Formulating molecules as 3D graphs introduces new challenges for active selection criteria, which motivates our current work.\n\n\n> Q1: sampling the graphs and sampling the interactions\n\nInteractions of atoms are reflected by the 3D positions of all atoms in a graph, which is exactly the reason why we formulate molecules as 3D graphs. We design strategies to select the most informative molecules for efficient learning given a limited budget, this is, we select molecules with the most salient interatomic features.\n\n> Q2: chirality\n\nNo. Our isometries are defined in the E(3) group instead of the SE(3) group, and the only difference is reflection, i.e., chirality in chemistry. First, AL selects more informative samples, and we want chiral molecules to have similar informativeness (diversity scores). Chirality can significantly affect molecular properties. If we treat a molecule as an important sample to learn from but don't treat its chiral counterparts as important, the chiral molecules are less likely to be seen in training. This can result in a GNN that is very biased due to the lack of diverse chiral data during training. \n\nMoreover, it is for efficiency. Some recent GNNs (like GemNet [5]) can recognize chiral molecules, but **the complexity is $O(n^3)$ while ours is $O(n^2)$.**\n\n> Q3: Figure 3 not start at the same loss\n\nSince all the methods start with the same initial labeled training set, their starting MAE values on the test set will be the same. **We have therefore plotted the MAE values from the first iteration onwards, to focus on the comparative performance of the methods after they start selecting samples using AL.**\n\n> Q4: Iterative manner of AL\n\n**Active Learning (AL) techniques are designed to operate in an iterative manner (see mainstream AL papers mentioned in the Related Work section).** A budget $k$ is imposed on the number of unlabeled samples that can be queried for annotation in each iteration.**We followed this conventional setup**.\n\n\n\n> L1: actual elements of each atom\n\nPlease refer to W1; **our method consists of two parts that take into account both geometric and atomic information**. Diversity in this paper focuses on geometries. Actually, we did conduct experiments for diversity that, **a vector for atom types** was concatenated to the current geometric vector, but the performance remained the same. This is mainly because, considering chemical priors like interatomic forces, it's nearly impossible for two different stable molecules to have the same 3D shape.\n\n> L2: periodic structures\n\n**Periodic structures are only for crystal materials** (see famous methods CGCNN, MEGNET, ALIGNN etc). Our focus is on small molecules, so it's beyond our scope of interest. We will discuss this.\n\n> L3: unknown molecules\n\nThis can be a very interesting perspective and a new research direction that is outside the scope of this work. We will make sure to include this in our limitations in the paper.\n\nReference\n\n[1] G. T., et al. Principles governing amino acid composition. Journal of Molecular Biology\n\n[2] B. J., et al. Torsional Diffusion for Molecular Conformer Generation. NeurIPS'22\n\n[3] Y. L., et al. Spherical message passing for 3D molecular graphs. ICLR'22\n\n[4] J. G., et al. Neural message passing for quantum chemistry. ICML'17\n\n[5] J. G., et al. GemNet. NeurIPS'21", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6XNNSJcb8U", "reviewer": "Reviewer_UZqH", "summary": "The paper proposes an active learning scheme for molecular property prediction using uncertainty estimates from Dropout Monte Carlo and diversity metrics. In each active learning iteration, molecules are selected by maximizing uncertainty and diversity in the batch by solving a quadratic programming problem. The authors propose a 48-dimensional vector that encodes statistical moments of reference distances, angles and cross-angles for 4 reference points. Experiments on QM9 and MD17 datasets are performed to benchmark the effectiveness of the proposed method against several baselines.", "questions": "Does the 48-dimensional vector contain any information about the atom-species involved in distances, angles and cross angles or is this metric purely geometry-based?", "limitations": "The authors discuss some limitations of the proposed approach, including the scaling to large datasets such as OC20. However, the scaling to large atom sizes could be discussed more as requiring further research given that it is non-obvious that the statistical moments measured at 4 reference points are sufficient to distinguish protein-size geometries or large crystal structures. Additionally, it should be stated more clearly that the proven power of the proposed descriptors exceeding the GWL test only holds before compressing the descriptors into the 48-dimensional vector.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.\tThe proposed 48-dimensional vector that encodes statistical moments of reference distances, angles and cross-angles for 4 reference points is a promising and novel metric to quantify geometric diversity of 3D molecular structures.\n2.\tThe proposed quadratic programming formulation of the molecule selection step allows to trade-off uncertainty and diversity of selected conformations.\n3.\tThe performed experiments demonstrate the usefulness of the proposed diversity metric and clearly outperforms the random selection baseline and other, non-Bayesian-uncertainty-based active learning approaches.", "weaknesses": "1.\tMany of the claims in the paper are too bold, neglecting several important related works in the field:\n     - There are several active learning applications for 3D GNNs already in the literature, especially in the neural network potential literature:  https://www.nature.com/articles/s41467-021-21376-0 ,   https://www.nature.com/articles/s41524-023-01104-6 \n     - In the review of alternative 3D structural descriptors, the most common methods such as SOAP ( https://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.184115  ), or ACE ( https://journals.aps.org/prb/abstract/10.1103/PhysRevB.99.014104 ) are neglected. \n     - Unlike claimed by the authors, the proposed Dropout Monte Carlo scheme for active learning in molecular property prediction with 3D GNNs is not novel, but has been proposed previously: https://www.nature.com/articles/s41524-024-01277-8 \n\n2.\tThe considered baselines might not be the most relevant to the method and do not allow to attribute the origin of the empirical performance increase given that neither Coreset nor Learning loss uses a Bayesian uncertainty. BatchBALD would be a much more relevant baseline given that the same uncertainty estimates can be used. This would also allow to investigate the effect of the quadratic programming formulation against the greedy clustering-based approach of BarchBALD. It would also allow to investigate the benefit of the proposed diversity metric vector against, for example, classical descriptors such as SOAP. This would allow to clarify the underlying source of the outperformance: Is it the Bayesian UQ estimate, the quadratic programming formulation of the proposed novel structure descriptor vector?\n\n3.\tThe quadratic scaling of the computational cost of the method is concerning. The pool sizes considered in the experimental section are very small (15,000) in the field of 3D molecular property prediction, where datasets can easily span several million conformations (PubChemQC, ANI-2x, GEOM). Given that there is already a computational overhead visible over competing AL methods, this difference will only increase for realistically-size datasets.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> W1: Many of the claims in the paper are too bold, neglecting several important related works in the field.\n\nThank you for your comments. We will modify the language in the paper to clarify our intentions.\n\nThe first two active learning (AL) papers do not specifically consider 3D geometry information and **differ from our paper/method**, which focuses on 3D molecules. The 3D geometry of molecules is crucial in determining molecular properties, but entangles unique challenges for designing AL schemes, which motivates our work.\nWe will make sure to include them in the literature review.\n\nRegarding the descriptors, our method produces a 48-dim vector **describing the geometry of a molecule**. Our method is equivariant to roto-translations and **invariant to permutations**, as the statistical quantities do not change due to permutation. Meanwhile, SOAP produces vectors that **describe local atomic environments** using spherical harmonics and radial basis functions in an atom-wise manner. In general, they are equivariant to roto-translations but **not invariant to permutations**. ACE involves a systematic expansion that can describe various orders of interactions (e.g., two-body, three-body); however, it is **less of a conventional descriptor compared to our method and SOAP**. We will make sure to include these two papers in our discussions in the paper.\n\nRegarding the Monte Carlo scheme, this paper was published on May 3, 2024, which is only around 20 days before the submission deadline. We were not aware of this paper at the time. We would like to say that this is **concurrent work**, and we will discuss this paper in our next version.\n\n> W2: Bayesian baseline and SOAP descriptors\n\nWe have included the results for BatchBALD in the global rebuttal (please refer to Figure 1 of the attached PDF). It can be clearly seen that **our method outperforms BatchBALD**. Moreover, we have performed statistical tests to confirm that **our improvement compared to BatchBALD is significant** with a $p$-value << $0.001$ (please refer to Table 1 of the attached PDF).\n\nIn regards to the SOAP descriptors, as we discussed in W1, SOAP produces vectors that describe local atomic environments, whereas we aim to measure diversity based on global features. There are ways to concatenate and compress the SOAP descriptors into lower-dimensional global descriptors; however, given the short time frame of the rebuttal period, we cannot provide any results on this. We will definitely consider working on this in the future.\n\n> W3: computational cost\n\nFirst, an active sapling strategy will rely on the backbone 3D GNN models for downstream molecular learning. However, existing mainstream 3D GNNs have the complexity of $O(n^2)$ (DimeNet [1], SphereNet [2]) or even $O(n^3)$ (GemNet [3]). Hence, the proposed sampling approach with $O(n^2)$ does not increase the order of the overall complexity, and might not be the major computational overhead of the whole learning pipeline.\n\nAs mentioned in Sec. 2.3 of the paper, we implemented a solution to execute the QP problem on the GPU (instead of the CPU) using the parallel implementation of the alternating direction method of multipliers, as detailed in [4]. Furthermore, we have vectorized our implementation and utilized the GPU to perform calculations to make the quadratic diversity matrix computation faster (find more details in item 3 the global rebuttal). Overall, our efficiency is slightly worse than that of the basic Random sampling.\n\nFor larger datasets, we can also address scalability by first sub-sampling from the unlabeled pool using only the uncertainty criterion (which is linear and thus scalable) to select the most uncertain samples. We then apply the proposed active sampling criterion only to the selected subset. This strategy has been used in previous AL research with promising results [5]. We plan to explore this strategy on large-scale molecular datasets in the future. \n\n> Q1: 48-dim vector\n\nIt is purely geometry-based. However, we want to emphasize that our framework contains two components for selecting important molecules: diversity and uncertainty. The diversity component focuses on the geometric perspective of view.\nThe uncertainty part considers atom types, which are embedded into node features. By combining both uncertainty and diversity, **we take both the chemical contexts and geometric contexts into consideration**.\n\n\n> L1: scaling to large atom sizes with statistical moments and expressive power after compressing the descriptors\n\nThank you for your comments, we will include such a discussion in the limitation section.\n\nUSR is a well-known work for recognizing similar molecular shapes [6], where they use distance information only and employ the first three moments (mean, variance, skewness) to approximate the distribution of distances. The authors state in the paper (page 5, right column, 2nd paragraph) that \"Such an approach is based on a theorem [7] from statistics, which proves that a distribution is completely determined by its moments.\"\n\nWith this theorem in mind, we can reconstruct the distribution with high fidelity by increasing the number of translated moments as well as the order of computed moments. We found that **distance information alone cannot describe a complete isometry space**, so we study a complete isometry space by **considering angular isometries**, eventually resulting in a theoretically guaranteed solution for precise molecular diversity computing. With the proposed complete isometry space and sufficient statistical moments, our method can be at least as expressive as the GWL test. However, in practice, we compress into a lower-dimensional descriptor for efficiency.  \n\nThis also addresses the scaling issue. For protein-sized geometries or large crystal structures, one can choose to include more statistical moments to better distinguish different structures. As our work focuses on small molecules, we use four statistical moments.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "k9yLvGAfgG", "reviewer": "Reviewer_JSdU", "summary": "This paper introduces a principled active learning (AL) paradigm tailored for molecular learning. The proposed AL approach aims to alleviate the hurdle of human annotation by automatically querying labels for the most informative samples. The authors treat molecules as 3D molecular graphs and they introduce a set of new 3D graph isometries for 3D graph isomorphism analysis, which are shown to be as expressive as the Geometric Weisfeiler-Lehman (GWL) test. To ensure the selection of samples with maximal uncertainties, the authors design a Bayesian geometric graph neural network specifically for 3D molecular graphs. Active sampling is formulated as a quadratic programming (QP) problem integrating these components. Experimental results demonstrate the effectiveness of the proposed AL paradigm, highlighting the advantages of the diversity and uncertainty methods introduced.", "questions": "1. **Clarification Needed on Line 175**: The sentence starting in line 175 needs to be more carefully justified. Specifically, how can you assert that SchNet is upper bounded by distance isometry but not by triangular and cross-angular isometries? Please provide a more detailed explanation or supporting evidence for this claim.\n\n2. **Generalization in Section 4.2**: It is not clear how the experiment in Section 4.2 demonstrates generalization. Could you provide more information and insight on the choice of graphs used in this section? Additionally, explaining the rationale behind the experimental setup and how it supports your conclusions would be helpful.\n\n3. **Complete Results for Section 4.4**: Please report all the results from Section 4.4.", "limitations": "Limitations addressed", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. **Innovative Isometries for Graph Discrimination**: The introduction of a new set of 3D graph isometries is a significant contribution, demonstrating important efficacy in distinguishing between different graphs. This advance enhances the expressiveness and applicability of the proposed method.\n\n2. **Theoretical Support and Clarity**: The methodology is explained with theoretical support\n\n3. **Comprehensive Experimental Evaluation**: The experiments clearly highlight the method's improvements over existing baselines. The inclusion of statistical analysis further strengthens the validity of the results, providing evidence of the method's effectiveness.", "weaknesses": "1. **Overly Strong Claims**: Some of the claims made in the paper are quite strong and require more careful justification. For detailed points of concern, please refer to the Questions section. Providing additional evidence or more nuanced discussions could strengthen these claims.\n\n2. **Need for More Detailed Experimental Insights**: While the experimental section is comprehensive, it could benefit from additional insights and details. Specific experiments would be clearer and more informative with further explanation. Providing more context and interpretation of the results would enhance the reader's understanding and the overall impact of the findings.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> W1: Overly Strong Claims\n\nThank you for your comments. We will use more precise language and include additional discussions in the paper to clarify our claims. We will address your concerns in the Questions section.\n\n> W2: Need for More Detailed Experimental Insights\n\nIn the current version of our paper, we have provided the necessary information for the overall experimental setup, as well as detailed analyses for each individual experiment. Even so, we agree with you that some rationales and high-level insights are missing, especially for readers outside this field to easily understand the paper. As we cannot revise the papers for now, we summarize the key rationales and insights in this rebuttal; **we will integrate them in our next version.**\n\n- 4.1 Experiment setup: We choose different types of molecular graphs for various tasks to demonstrate the generalizability of our methods.  We test our methods on molecular systems in both equilibrium and non-equilibrium states, covering various quantum properties and molecular dynamics tasks. This will be further detailed for your Question 2 below.\n\n- 4.2 The performance comparison of our method with the baselines: This shows that our active sampling approach is significantly better than mainstream active learning methods. This further means that, given a budget to select more informative samples for wet-lab annotation (usually expensive for molecules), our method can select the most informative subset, thereby significantly improving the learning efficacy and efficiency.\n\n- 4.3 An examination of the query budget study: This shows that for various annotation budgets, our method is consistently better than baselines, which indicates that the superior capability of our sampling approach is robust in practice.\n\n- 4.4 An ablation study focusing separately on the diversity and uncertainty components: This indicates that both the proposed sampling approaches are novel and effective. Combining them considers both the geometry and chemical contexts in 3D molecules, and thus achieving the best active learning performance.\n\n- 4.5 An evaluation of the computation time: This shows our method is not only effective but also efficient. Our sampling approach has similar efficiency to the Random sampling, and is much more efficient than the important diversity-based active learning baseline Coreset.\nThis further indicates our approach is readily applicable to macromolecules like materials and proteins, which will be conducted as our future work.\n\n\n\n> Q1: Clarification Needed on Line 175\n\nTo clarify, we mean that SchNet has at most the same expressive power in distinguishing different geometric structures as our method. Since both methods rely solely on distance information, they can distinguish structures uniquely defined by their distances alone. Our method is deterministic, meaning that, with the same distance information, **a perfectly trained SchNet can be at most as powerful as our method**. In practice, it is almost impossible for a SchNet to be perfectly trained. \n\n> Q2: Generalization in Section 4.2\n\nThe research of 3D molecular learning is new, and there are only a few reliable benchmark datasets for 3D molecules (containing atom types as well as XYZ coordinates for all atoms for each molecule). We choose our datasets based on the following two criteria:\n\nQM9 consists of molecules in equilibrium, while MD17 contains several thermalized (i.e. non-equilibrium, slightly moving) molecular systems. Additionally, QM9 contains various quantum properties for molecules, like the important HOMO and LOMO orbitals. MD17 is for dynamic system simulation, thus it contains labels for both the energy and atomic forces. In summary, we test our methods on molecule systems in both equilibrium and non-equilibrium, covering various quantum properties and molecular dynamics tasks.\n\nQM9 and MD17 are necessary benchmark datasets in the field of 3D molecular learning. Almost all representative works in the field use them for evaluation [1-4]. We follow this conventional setup in our work. Besides QM9 and MD17, other reliable benchmark datasets for 3D scientific data would go beyond molecules, like the Materials Project dataset for crystal materials and the Fold Dataset for proteins. These data contain special structures (like periodic structures for crystals), and thus they are out of the scope of this work. We will explore the potential of our active learning methods on these data as future work.\n\nWe will include these discussions in the next version of our paper.\n\n> Q3: Complete Results for Section 4.4\n\nWe have included additional results from the ablation study, which can be found in the global response (please refer to Figure 2 and Table 2 of the attached PDF file ). We've also included more discussions in item 2 of the global rebuttal. Basically, the results show that both the uncertainty-only and diversity-only approaches outperform all baselines. Additionally, we show both diversity and uncertainty components significantly contribute to the overall performance. We did not include other baselines in this plot to avoid overwhelming it, but if by \"report all the results from Section 4.4\", you mean including all the baselines in this plot as well, we will certainly do so in the next version of the paper.\n\nReference\n\n[1] K.S., et al.  SchNet: A continuous-filter convolutional neural network for modeling quantum interactions. NIPS'17\n\n[2] J.K., et al. Directional message passing for molecular graphs. ICLR'20\n\n[3] Y. L., et al. Spherical message passing for 3D molecular graphs. ICLR'22\n\n[4] J. G., et al. GemNet: Universal Directional Graph Neural Networks for Molecules. NeurIPS'21", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kfkS54IHXC", "reviewer": "Reviewer_Abqb", "summary": "This paper proposes a principled AL paradigm to alleviate the annotation hurdle of 3D molecular graphs. It introduces a novel diversity component for 3D molecular graphs, which is provably at least as expressive as the GWL test. Furthermore, the authors develop an effective and efficient pipeline to compute uncertainties for 3D molecular graphs rooted in Bayesian inference.", "questions": "+ The authors formulate a criterion based on uncertainty and diversity. How do these two factors respectively impact performance?", "limitations": "Please see the weaknesses part.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "+ The authors develop an effective method for computing diversity among different 3D molecular graphs and introduce a Bayesian Geometric Graph Neural Network (BGGNN). The BGGNN takes a 3D graph as input and produces the desired properties along with uncertainty values.\n+ The motivation of the paper is clear, and it includes well-defined theoretical proofs.", "weaknesses": "+ The paper lacks some necessary explanations, such as those for \"USR\" and \"GSL,\" making it difficult to understand.\n+ Experiments were conducted only on the QM9 and MD17 datasets, so the effectiveness and efficiency on larger datasets remain unknown.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> W1: lack of necessary explanations\n\nThank you for your comments.\n\nFor \"USR\", the primary concept involves using statistical moments to approximate the geometry of the molecules, capturing essential features of their shapes. Detailed explanations of this method are provided in Section 2.1.3 (lines 189 to 214). If you feel that additional background information is needed, please let us know the specific points where more details would be beneficial so that we can improve our explanation.\n\nWe do not have the abbreviation \"GSL\" in the paper. If you mean \"GWL,\" we present the high-level ideas of the GWL test from lines 159 to 164. The key message we want to convey is, as stated in the paper, \"the GWL test imposes an upper bound on the expressive power of 3D GNNs.\" We also provide the details of this test in Appendix A.2. \n\nWe believe that the high-level explanations highlighted in the main paper are sufficient to understand our claims. We also provide detailed explanations in Appendix. Therefore, we kindly request not to view this as a weakness of our work.\nHowever, we will definitely take your suggestions and further clarify the context.\n\n> W2: experiments on larger datasets\n\nWhile our experiments were conducted exclusively on the QM9 and MD17 datasets, this choice was made because our primary focus is on small molecules and to validate our method on **well-established** and **most commonly used** benchmarks.\nQM9 and MD17 are necessary benchmark datasets in the field of 3D molecular learning. Almost all representative works in the field use them for evaluation [1-4].\n\nOther reliable benchmark datasets for 3D scientific data would go beyond molecules, like the Materials Project dataset for crystal materials and the Fold Dataset for proteins. These data contain special structures (e.g., periodic structures for crystals), and thus they are out of the scope of this work. We will explore the potential of our active learning methods on these macromolecules as future work.\n\nIf you have concerns about complexity, we can vectorize the diversity computation implementation for GPUs for faster computations. We have updated the results comparing selection times (please see Table 3 of the attached PDF) and included additional discussions in the global response (item 3), to reflect our new vectorized implementation. It can be seen that our sampling approach is highly effective and efficient. Note that the Coreset approach is an important diversity-based baseline, and our sampling approach is much more efficient than it (127 vs 64.9 minutes). The Random approach does not involve any sophisticated sampling strategy, and our method is just slightly more expensive than it (53 vs 64.9 minutes). Thus, we believe our sampling approach is readily applicable to larger datasets.\n\n\n> Q1: individual impact of uncertainty and diversity\n\nWe have shown in the ablation study (Sec. 4.4) that our proposed diversity metric alone is highly effective. We can statistically conclude that with only the proposed diversity metric, we outperform all other baselines. Additionally, the uncertainty metric clearly further improves our method as all the results suggest. **In summary, our ablation study (Sec. 4.4) demonstrates that our proposed diversity metric is highly effective on its own. Statistically, we outperform all other baselines using only this metric. Furthermore, incorporating the uncertainty metric enhances our method even more.**\n\nAdditionally, **we have presented new ablation results in the global response** (Figure 2 and Table 2), please see the details there. We compared our full method (combining uncertainty and diversity) against methods using only uncertainty or only diversity. The results indicate that our diversity component is highly effective in selecting informative samples for model training. Moreover, the performance improvement achieved by our method compared to using only uncertainty or only diversity-based selection is statistically significant ($p < 0.001$), as shown in Table 2 of the PDF file in the global rebuttal. Incorporating uncertainty further refines our method by integrating additional chemical contexts, such as atom types. **This clearly demonstrates that our approach is superior, as it not only distinguishes different geometries with precision but also accounts for uncertainties in chemical contexts.**\n\nReference\n\n[1] K.S., et al. SchNet: A continuous-filter convolutional neural network for modeling quantum interactions. NIPS'17\n\n[2] J.K., et al. Directional message passing for molecular graphs. ICLR'20\n\n[3] Y. L., et al. Spherical message passing for 3D molecular graphs. ICLR'22\n\n[4] J. G., et al. GemNet: Universal Directional Graph Neural Networks for Molecules. NeurIPS'21", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank the reviewers for their invaluable comments and suggestions. In this global response, we would like to clarify a few points and present new results based on the feedback received.

### Clarification
To start our rebuttal, we would like to clarify a few points about our method.
> Our model considers both geometry and chemical contexts to achieve the best performance

Firstly, our method consists of two parts: uncertainty and diversity. The diversity component, based on our proposed geometric isometries, aims to focus on diverse molecules, thereby capturing a wide range of chemical properties. The diversity part is our major contribution. As we will discuss later, the results of studying the individual contributions of uncertainty and diversity to performance show that the diversity component is highly effective.

Additionally, our method includes an uncertainty component, which aims to quantify and incorporate the uncertainty in our model's predictions. The uncertainty part also takes chemical contexts (like atom types) into account as node features, enhancing the model's ability to recognize and better learn uncertain chemical interactions. **Therefore, our method considers both geometry and chemical contexts to achieve the best performance.**

### Additional Results
To this end, we would like to discuss some additional results we obtained, which can be found in the **attached PDF file**. References will be made to this attached file rather than the paper unless otherwise specified.

> 1. Comparison with a new baseline, BatchBALD (as suggested by Reviewer UZqH)

In Figure 1, we included the results for a new baseline, BatchBALD[1], adapted for our regression setting following [2]. Similar to the uncertainty component in our method, BatchBALD is based on Bayesian uncertainty, and it serves as an important baseline in this venue. As observed in the results, **our method consistently achieves a lower MAE (the lower the better) at any given AL iteration compared to BatchBALD**. For our analysis of other baselines, please refer to the paper. In addition, as a complement to the result above, we updated Table 1 with p-values obtained using paired t-tests between our method and all baselines, now including BatchBALD. **The performance improvement achieved by our method is statistically significant** ($p << 0.001$) for all $4$ properties tested against BatchBALD.

Given that BatchBALD is also a Bayesian-based uncertainty approach, this comparison further highlights the robustness and effectiveness of our method, especially the combination of uncertainty and diversity. To this end, it is important to emphasize that our method has two components: uncertainty and diversity. The diversity component, which is our main contribution, plays a more fundamental role in improving performance. This is because capturing 3D atomic geometries is crucial for accurately modeling and understanding molecular interactions. By incorporating diversity, our method ensures a more comprehensive selection of informative samples, leading to better overall results.

> 2. On the individual impact of diversity and uncertainty components (as suggested by Reviewer Abqb and Reviewer JSdU)

In Figure 2, we present a study on the individual impact of the diversity and uncertainty components. It is clear that our proposed method outperforms the individual use of diversity or uncertainty alone. The key to this outperformance lies in our method’s dual focus on both geometric importance and chemical contexts. Moreover, it can be observed that the diversity component alone shows strong performance, it is only slightly less effective than our method because it **captures the geometries of molecules, which are fundamental in distinguishing different molecules with different properties**. On top of this, we also conducted statistical tests to conclude that **the improvement of our method is significant** compared to only diversity or only uncertainty ($p << 0.001$) in Table 2. 

> 3. Vectorized implementation of our methods and updated timing results (in response to the concerns of Reviewer UZqH and Reviewer Abqb)

In Table 3 of the PDF attached, we provided updated results on the average time taken by our method. Compared with the original results in the paper (see *Table 2* in the paper), the computational time is reduced from 15 minutes slower than random to 11 minutes slower than random. This improvement in calculation time is due to **vectorizing our implementation and utilizing the GPU to perform calculations for diversity matrix computation**. Vectorization accelerates computations by processing multiple data points parallelly, a task at which GPUs excel. For instance, deep neural networks also benefit from parallelization, achieving faster performance on GPUs compared to CPUs. A simple example is calculating the inner product of two vectors. Without vectorization, we compute the products of corresponding elements one by one and then sum them sequentially. With vectorization, the GPU performs element-wise multiplications in parallel and then computes the sum in a single, efficient reduction step.

It can be seen that our sampling approach is highly effective and efficient. Note that the Coreset approach is an important diversity-based baseline, and our sampling approach is much more efficient than it (127 vs 64.9 minutes). The Random approach does not involve any sophisticated sampling strategy, and our method is just slightly more expensive than it (53 vs 64.9 minutes). Thus, we believe our sampling approach is readily applicable to larger datasets.


[1] Andreas Kirsch, et al. BatchBALD: Efficient and Diverse Batch Acquisition for Deep Bayesian Active Learning, NIPS

[2] D.H.,et al. A framework and benchmark for deep batch active learning for regression, JMLR

Meta-review: This paper presents an active learning method for 3D molecular graphs, where the diversity of molecules are measured according to 3D graph isomorphism analysis. Experimental results on two data sets prove the effectiveness of the proposed approach. 

In general, the reviewers like the novelty of the proposed approach. There are a few aspects that the authors need to address before publication: (1) the claims are too strong; (2) more baselines should be included, e.g., 2D-graph based approaches.
Decision: Accept (poster)