Equivalent Distance Geometry Error for Molecular Conformation Comparison
Authors: Shuwen Yang
Tianyu Wen
Ziyao Li
Guojie Song
Venue: ICLR 2022 Submitted (2022)
Keywords: molecule
molecular conformation
loss function

Abstract: \textit{Straight-forward} conformation generation models, which generate 3-D structures directly from input molecular graphs, play an important role in various molecular tasks with machine learning, such as 3D-QSAR and virtual screening in drug design. However, existing loss functions in these models either cost overmuch time or fail to guarantee the equivalence during optimization, which means treating different items unfairly, resulting in poor local geometry in generated conformation. So, we propose \textbf{E}quivalent \textbf{D}istance \textbf{G}eometry \textbf{E}rror (EDGE) to calculate the differential discrepancy between conformations where the essential factors of three kinds in conformation geometry (i.e. bond lengths, bond angles and dihedral angles) are equivalently optimized with certain weights. And in the improved version of our method, the optimization features minimizing linear transformations of atom-pair distances within 3-hop. Extensive experiments show that, compared with existing loss functions, EDGE performs effectively and efficiently in two tasks under the same backbones.

as the loss of neural networks Under review as a conference paper at ICLR 2022
EQUIVALENT DISTANCE GEOMETRY ERROR FOR MOLECULAR CONFORMATION COMPARISON
Anonymous authors
Paper under double-blind review
ABSTRACT
Straightffrwwrt conformation generation models, which generate 3-D structures directly from input molecular graphs, play an important role various molecular tasks with machine learning, such as 3D-QSAR and virtual screening in drug de- sign. However, existing loss functions in these models either cost overmuch time fail to guarantee the equivalence during optimization, which means treating different items unfairly, resulting in poor local geometry in generated conforma- tion. So, propose Equivalent Distance Geometry Error (EDGE) to calculate the differential discrepancy between conformations where the essential factors of three kinds in conformation geometry (i.e. bond lengths, bond angles and dihedral angles) are equivalently optimized with certain weights. And in the improved ver- sion of our method, the optimization features minimizing linear transformations of atom-pair distances within 3-hop. Extensive experiments show that, compared with existing loss functions, EDGE performs effectively and efficiently in two tasks under the same backbones.
INTRODUCTION
Small organic molecules, conventionally a chemical or biomedical topic, are now being widely stud- ied by artificial intelligence researchers. Based on prior natural science knowledge, machine learning models aid in multiple tasks on molecules including 3-D Quantitative Atructur-Acttvitty Relation- ship (QSAR) analysis, virtual screening for drug discovery, and protein-ligand binding analysis. Before any real industrial applications. molecular conformation generation draws particular con- cern. The task is also challenging (1) conformation contains the 3-D geometry which can be translated or rotated, making it complicated to compare the generated one with ground-truth (2) the number of molecules found in nature presents combination explosive growth, so the procedure of generating conformations is supposed to be fast enough for applicability.
In the past few years, different machine learning models have been proposed to handle the molec- ular conformation generation task. Generally, the molecules are featurized into token sequences (e.g. SMILES (Weininger 1988)) or 2-D molecular graphs Gilmer et al. 2017) to feed the neural networks, while the way to obtain conformations is optional: (1) Straight- methods directly generate the 3-D coordinates of atoms from the inputs; (2) Step-wise methods first predict distances between atom pairs and then rebuild the conformations from the distances either via discrete al- gorithms (e.g. EDG (Havel) 1998)) bi-level (Xu et al. (02T). As the expense of avoiding the troublesome comparison between conformations, slep-wise methods usually adopt more complex architectures and endure relatively expensive time and space costs. Therefore, based on its in- nate advantage of high efficiency, forward methods require powerful assistance to perform equally great as step-wise ones. And a well-designed loss function can be helpful.
In recent studies, the simplest way to compare the conformations is to represent the conforma- tions with atom position matrices R(g) E Rax3 and calculate their Soot-Mean-Squrre Deviations (RMSD). Since conformation is invariant to spatial permutations (e.g. translation and rotation), universal approach to first find an optimal alignment. As an instance, Kabsch algorithm (Kabsch) 1976) provides a differentiable method to align two given conformations. We referred to the so- calculated RMSD as
However, using
'The atoms and bonds inside the molecule form the vertexes and edges respectively.

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is neither efficient nor effective: it needs to calculate the rotation matrix for every single confor- mation, which stifles batch processing and increases time cost, meanwhile, the rotation matrix is vulnerable misplaced atoms and poorly-predicter regions (Mariani et al., 2013), so the RMSD between aligned and reference conformation cannot correctly reffect molecular similarity when in- put conformations are less likely.
To further improve the applicability of loss function on molecular conformation comparing, Dis- tance Geometry (DG) is proposed to encode conformation into a vector of distances among the atoms, where the comparison is applied. The main idea of DG is to indirectly optimize bond lengths D. bond angles & and dihedral angle v/ by optimizing the distances among 1-hop. 2-hop and more distant neighbors respectively in the molecular graph. However, when balancing time and effects, traditional distance geometry methods are faced with a dilemma: (1) too many distant neighbors in the graph are considered, the loss calculation will be time-consuming. pairs (e.g. 3-hop at most) are considered, the factors in the geometry will be treated unfairly. Based on the theoretical and experimental demonstration in Appendix B.1 the gradient allocated on bond angles & and dihedral angles will become much smaller than that on bond lengths D when op- timizing itom-pair distances, resulting in contorted angles and poor local geometry. Therefore, the equivalence in optimizing different factors should be stressed in the comparison.
(2) If only close atom-
To push the boundary of efficient and effeetive loss function for molecular conformation generation, we propose Equivalent Distance Geometry Error (EDGE). Firstly, EDGE exploits the essential factors inside the conformation geometry, including series of bond lengths D, bond angles & and dihedral angles V. Then the factors are jointly optimized with certain weights to guarantee that they are fairly treated. To avoid the high complexity of directly calculating the angles, we introduce Taylor Expansion to our method to approximate the angles with linear transformations of atom-pair distances within 3-hop. Extensive experiments prove that the improved version of EDGE performs as fast as optimizing 1/2/3-hop atom-pain distances and achieves better performance than existing popular loss functions on conformation comparison when applied to truight-forwad backbones.
To conclude, our main contributions are threefold:
We propose a novel loss function, EDGE, for straight-forward models on molecular con- formation generation, which optimizes the essential factors in the conformation geometry equivalently, realizing high consistency of generation.
To avoid the high complexity and gradient explosion caused by direct calculation of angles, we introduce Taylor Expansion and Multiplier Truncation which achieves efficient model optimization with improved EDGE and guarantees its applicability.
We conduct comprehensive experiments on several recently proposed benchmarks, in- cluding QM et al.] 2018). GEOM-QM9 and GDOM-Drrgs(Axelrod & Gómez- Bombarelli 2020). The outstanding performance on different metrics shows that EDGE is powerful on both definite conformation prediction and multiple conformation generation tasks, compared with existing loss functions. Besides, EDGE's efficiency is guaranteed via theoretical analysis and empirical test.
2 PROPOSED METHOD
2.1 PRELIMINARIES
Notations In the statements and equations below, we use italic letters for scalars and indices, bold lower-case letters for (column) vectors, bold upper-case letters for matrices, calligraphic letters for sets, and normal letters for annotations. Given certain molecule, M (V, E, n, n,m, XV, X*) is the corresponding molecular graph. Here, V is the set of all n atoms, È C V x V is the set of all m chemical bonds, X (xY, ..., xx)T E Rgxd, is the matrix of atomic features, and X" = (#f, )T,)' E = that of bond features. G is the geometry of its conformation and R(G) is the matrix of atom positions. d(uu, v; M) is the graph distance between atom v in M and d(u, v; 9) is their Fuclidean distance in G. conformation geometry & has three types of factors
D(G), (G), (G), which refer to the set of bond length d, bond angle & and dihedral angle vi
Dihedral angle referred torsion angle in some works.
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SE
SE
d
h
Ad
Reference
Taylor
(ogtional)
Σ
Av
Generated
"SE Squared Error
Figure 1: Given a generated conformation and its reference, different factors in the geometry d, Φ. rb are taken to calculate the Squared Errors, which are then added up with certain weights Xa, As Ag- As improvement, Taylor Expansion can be introduced as an optional approximation of SE, guaran- teeing the high efficiency of EDGE. See 2.3 for details.
respectively. For operators, @ is the operator concatenations and input vector.
denotes the l2 norm of the
Problem Definition We consider following straighttforward molecular conformation generation process: the models input molecular graph M and output molecular conformation R(g). Our task is to find an efficient and effective loss function to calculate the differential discrepancy be- tween generated and reference conformation G, G. which can be utilized to optimize the models backwardly.
2.2 EQUIVALENCE MOLECULAR GEOMETRY
When applied to straight-forward backbones on molecular conformation generation, existing DG- based loss functions such Conn-k (Yang et al. 2021) can't fairly distribute the gradient differ- ent factors DG)). (G) and (g) in the conformation during optimization, resulting in contorted geometry. Simply calculating the average error between the generated D(G), (G)), (G) and the reference ones has the same deficiency, because the three types of factors are in different dimensions and equivalence can hardly be guaranteed while training the model. To fairly treat all the factors, necessary to jointly optimize them with certain weights which are determined by their corresponding degrees freedom and value distributions in the dataset.
Degree of Freedom Consider 3-D molecular geometry G of molecule with atoms, it has [g 3n degree of freedom due to the existence of rotation and translation (Simm Hernandez- Lobato 2020). It means that geometry can be just recovered from certain combination of bond lengths, bond angles and dihedral angles with a total number 3n - 6. Note that three types of factors occupy ÍD, fo, J degrees of freedom out of fg respectively, and have
fg = fD + so + fo = 3n
(1)
To completely perceive the local structure of molecular geometries, the number of bond lengths, bond angles and dihedral angles sampled in the geometry (i.e. f4, fw, fw) may exceed their exact degrees of freedom fD, fe, fv during optimization. So we need to balance them through a simple proportion (c.g. fD/ fo for bond lengths D).
Value Distribution When jointly optimizing bond lengths D, bond angles & and torsion angles V, which are in different dimensions, can hardly achieve equivalence to manually give hyper- parameters to balance the gradients. Therefore, we rescale D, &, T with the reciprocals of their deviations OD, σz, σv respectively to uniformly approximate a Gaussian distribution with o =
The calculation of degrees of freedom for each kind of factors be found in Appendix C.1
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2.3 OUR METHOD: EQUIVALENT DISTANCE GEOMETRY ERROR
We propose Equivalent Distance Geometry Error (EDGE) as a loss function for straight-orwwrd methods of molecular conformation generation. As illustrated in Fig EDGE calculates the dis- crepancy between generated conformation & and reference conformation I following
L) = A
Σ (d d) + X
Σ
(Ø
(0--))++1 +
Σ SC SE((E)
(6)
de dCD(G) deD(C)
SC +(0))
$E#(1)
(2)
fD 1
fo 1
fw 1
where
AD
X
fD
OD
f
ae
fy 0V
To be more efficient, we can encode molecular conformation into vector z:
[Ana]d E D(G)
24 o E (G)) = A v E (()
(3)
z nD
+
ZΦ
+ Z
Then Eq 2 can be represented as the discrepancy between two encodings of conformation, which achieves fast comparison among multiple conformations:
L(
Calculation of For conformation G, first calculate the distance ma- trix D:
Day R()); R(g)
(5)
where R(G); E R is the 3-D position of i-th atom in g. Then the value of sampled bond lengths D can be found in D.
c
Calculation of + For certain bond angle & determined by three atoms in G. b are the lengths of its two sides and is the length of the side opposite it, which is shown in Fig ((a) As a, b, e can be found in D, the value of & can be calculated through the cosine theorem
(a)
a² + b² c²
arccos
(6)
2ab
Calculation of y For certain dihedral angle s determined by four scquentially-comeetee atoms in G, a, b, e are the lengths of the sides in the atom chain and d, e, are 2/3-hop distances among the atoms, which shown in Fig 2(b) Following Euler's 's formula in the tetrahedron, we have
(b)
4a2b2e2 - 62r d2r3 -c2t? + r2t1t2
Figure 2: ó and vi
arcsin
4a212 a27?
in the conformation geometry.
2.4 IMPROVEMENT
2.4.1 DEFICIENCY OF ORIGINAL EDGE
High Complexity Although obtaining the exact value of d, d in D(G), D(G) is straightforward, calculating and optimizing the factors (G)) and (G) will bring high computational complexity due to the existence of complicated operations like inverse trigonometric functions.
Gradient Explosion When Φ - π or + 0, π 7/2, the derivative of inverse trigonometric func- tions in reaches co. Even if we introduce Taylor Expansion to the calculation of each factor's error, which will be shown in 2.4.2 the same situation occurs when calculating the partial deriva- tives. So, gradient explosion is also considerable problem during the optimization with EDGE.
*The calculation the weights for different datasets can be found in Appendix C
'In Eq. T1 82 + 22 e², r2 = e², a² 12 L f2 2 f. And details of the calculation are available Appendix A.
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2.4.2 TAYLOR EXPANSION
To solve the problem of high computational complexity and make caleulation in Eq efficient, we utilize Taylor Expansion to approximate the errors of bond angles and dihedral angles y with linear transformation of atom-pair distances in D from Eq
For each pactor-pair & and in (g) and + (g). considering the bond lengths à, b, é in 9 which generate ø in the way mentioned above, they have the corresponding bond lengths a, b, c in g. We can compose the vectors = (a, b, ê) and u. = (a, b,c) with them respectively. Like Eq. 6
can be taken as function w.r.t u. So, we can apply multivariate Taylor expansion to $ - 6:
0Φ tu
(u- - -u) ==u)
(8)
where u +0 and dó
0o
0o dΦ Ub dc
du
Da
Similarly, for each factor-pair and v in (g) and L (G), we can get the vectors à (a, b, é, f) and e = (a, b, c, d, e, f) Like Eq. 7 v can be taken as a function w.r.t v. So we can apply multivariate Taylor expansion to va th:
Ovp
(==b)
8)
Ov
(9)
Dv Obb 2y Oy dΦ abb
where v v and
)T
Dv
Da
Ob" Oc Od de 0s
As the generated conformation G is changeable while reference conformation is fixed during training, the partial derivatives Og 0e processing
Tu lum dh lo= which is invariant to S. can be obtained in data o Therefore, the discrepancy of bond angles and dihedral angles in Eq can be replaced with Eq which are simply linear transformations of atom-pair distances. Besides, the atom-pairs considered here is most 3-hop connected, so the number of variables in linear transformation (i.e. hair-pair distances) in the same order magnitude with atom count n (see Appendix B.22) So, the problem of high complexity of EDGE is resolved by Taylor Expansion.
2.4.3 MULTIPLIER TRUNCATION
Taylor Expansion solves the problem of high computational complexity, but gradient explosion still exists. Here, we apply Multiplier Truncation strategy to truncate the partial derivatives finite values. Instead of limiting the final values, the problematic multipliers hidden in the formulas are found and then truncated to prevent gradient explosion, while the useful parts the values are mostly preserved.
For á w.r.t. u, we have:
OΦ
Ob
Cos
Coss Φ du sin du
(10)
du
cos Φ
When ④ n. sin 6 cause of gradient explosion in Eq 10 Here, we truncate sin to [-10 - from (-oo, -1 -1. Similarly, for vé w.r.t. v, we have:
0o. As stable values can always be acquired by multiplier
@ 1
the
du
sin
Ohb 8
i 0sin² 2 vb Ov
1 8sin v/
(11)
dv
sin
bb
2in sa cos tp Ov
When vi + 0 or π. /2, s v cas +
+ 0o, so we truncate 2sin d cas v to [1, 10] from [1, oo). Moreover,
the value of a denominator 452c² - (b2 + c² e²)² will tend to 0 when e b (i.e. the bond angle between b, e is т), resulting in that the elements in the vector reaches oo and triggers explosion. To avoid this, we truncate 2 (b² + 2² ee² to [10, oo) from [0, co).
Asin ø inside the multiplier Sy
Detailed formulas of the partial derivatives in Eq 8 are available in Appendix A.2

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The improved version of EDGE with Taylor Expansion and Multiplier Truncation resolves the pre- vious deficiency and achieves time-friendee frieendly model training with equally powerful performance in the experiment.
3
EXPERIMENT
3.1
EXPERIMENT SETUP
3.1.1 TASKS
We evaluate our loss function on two molecular conformation tasks: definite conformation predic- tion and multiple conformation generation.
Definite conformation prediction aims to predict the optimal conformation for each molecule. Here, a popular dataset QM9 (Wu et al., 2018), which contains the conformations of about 133k molecules. These reference conformations are generated through Density Functional Theory (DFT), making that they are in relatively stable status. The backhone model where apply the loss functions is PhysNet (Yang et al.) 2021). a learnable force field designed to optimize the molecular conformations. To show the performance of the loss functions on various aspects, they are evaluated on following metrics:
D RMSE, & RMSE and v RMSE: the root-mean-squared error the bond lengths, bond angles and dihedral angles sampled from molecular graphs, respectively.
A-RMSD (Mansimov et al., 2019): the root-mean-sque squared derivation between reference conformation G and generated conformation G after applying alignment function A i.e.
A-RMSD 9) = RMSD(A(R((), R())), R()))
(12)
where A is the alignment function.
IDDT-Score (Mariani et al. 2013): Compute over all pairs of atoms in the reference struc- ture at distance closer than I5A, and the distances in generated conformations are consid- ered preserved if they are within tolerance threshold compared reference ones. IDDT- Score is the average fraction of preserved distances over thresholds 0.5A, 1A, 2 and 4A, i.e.
IDDT-ScoreeG,.o
average 2E(0.5.1.24) V.V..dd.ssss
Σ
1(d((, 9) d(u, 9) < r)
(13)
Multiple conformation generation aims to generate realistic and diverse conformations for molecule, Here, we use the recent proposed GEOM-QM9 and GEOM-Drugs (Axelrod & Gómez- Bombarelli 2020), both of which contain multiple conformations for each molecule. Following the settings from Xu et al. (202T), we sampled 50,000 molecule-confoormaioon pairs as training set and another 17813 3 pairs covering 150 molecules as test set from GEOM-QM9 As for GEOM-Drugs, we sampled 50,000 molecule-conformatton pairs as training set and another 9161 pairs covering 100 molecules as test set. The backbone models selected to handle conformation generation are PhysNet and CVGAE (Mansimov et al. 2019), an auto-encoder designed for generating molecular conformations. Following metrics used to evaluate the performance of loss functions:
COV-8 (Xu al.) 2021): the fraction of conformations in reference set S, covered by at least one conformation in generated set Sy within a RMSD threshold 6 after alignment, i.e.
COV(S,.S,)
( e S,JA-RMSD(C 9) < 6, Ig e &
(14)
MAT et 2021): y the average RMSD over conformations in reference set S,,, each of which is compared with the most similar conformation in generated set S, after alignment, i.c.
MAT(S g. S.)
Σ min A-RMSD(G.9) GeS,
(15)
SES
TThe alignment function A1 AnMo from RDKit package v1.6.11

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Multi-IDDT-Scon (Mariani al. 2013): the average IDDT-Score between each confor- mation in reference set Sy and the conformation in generated set S with highest IDDT- Score compared to the former, i.c.
Mulli-IDDII-Scor((B,,, S,)
max IDDT-Scorre(,, gE3,
(16)
GES,
3.1.2 BASELINES
We compare EDGE with its variants and multiple loss functions for molecular conformation com- parison as listed below:
Naive RMSD: root-mean-squared error of the atom positions, i.e. RMSD( R(())
Kabsch RMSD (Li et al. 2021 root-mean-squree error of the atom positions after align-
ing the conformations using Kabsch algorithm, i.c. RMSSD(Kassch(() (R(g), R(C)), R()))
IDDT-y (Mariani et al.
2013) : not-meen-ssqared error of atom-pair distances whose val- (d(u, v; 9), d(u, 0; G))
ues are less than in reference conformation, i.e. RMSE
v,vEVAd(u..=$5
Conn-k (Yang 2021:: pairs, i.e. RMSE
root-mean-squaree error of distances of k-hop connected atom- (d(u, v; (). d(u, un 9))
u.v.VAd(u.v..M)ck
EDGE-w/o f: an invariant of EDGE, where degree of freedom is not considered when calculating hyper-parameters ÅD, λo,
EDGE-w/o a: an invariant of EDGE, where value distribution not considered when calculating hyper-parameters ÅD, Ag.
EDGE: our loss function.
To further reflect how the loss function improves the performance, we also consider following base- lines which are simply conformation generation strategies rather than learning models:
RDKit: directly using the conformation generated by RDKit without further training.
Random Guess: sampling the atom positions from the standard Gaussian distribution and constructing conformation.
3.1.3 DETAILS IMPLEMENTATION
For PRyKN initialized), we use RDKit (package version 1.6.1) to generate an initial confor- mation and then refine it with PhysNet Block. The number and duration of time-step are 5 and 0.2 respectively. The hyper-paramett setup inside PhysNet follows Yang et al. (021).
For CVGAE, follow the settings of UFF-free version from Mansimov al. 2019), show that EDGE can also perform better on generative backbones where derivative tine-tuning with force field is not considered.
The value of hyper-parameters Äp, Àg. Ay is calculated through sampling the degrees of freedom and standard deviation of D, o, from the train set of datasets QM9, GEOM-QNE and GEOM- Drugs The explicit process and the results calculation can be found in Appendix C.2
When optimizing the backbones, set learning rate to 2e-6 for PhysNet and 5e-5 5 for CVGAE, and learning rate decay is 0. 95. We train all the models for 100 epochs on the train set.
3.2 RESULT
The result on three datasets and two backbones is listed Tab I in Appendix C
2 and More tables are presented
As for efficiency, EDGE can optimize the molecular conformations as efficiently as Naive RMSD and Conn-3, and it's much faster than distance geometry methods considering distant atom-pairs

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Table 1: Definite Conformation Prediction QM9 with backbone PhysNei(RDKii initialized). Time refers to the seconds used to process around 106k molecules in training epoch. 1 means "higher is better".
Global Geometry
Local Geometry
Baseline
Time
A-RMSD
IDDT-Score (%)T
D RMSE
+ RMSE
v RMSE
RDKit
0.750
83.3
0.084
0.088
0.354
Random Guess
1.420
38.3
0.709
0.825
0.526
Naive RMSD
439
1.151
42.0
0.452
0.702
0,476
Kabsch RMSD
3208
0.747
75.5
0.142
0.141
0.345
Conn-3
478
0.736
85.4
0.038
0.093
0.326
Conn-oo
1248
0.730
85.3
0.057
0.096
0.324
IDDT-5
1105
0.733
85.4
0.043
0.096
0.325
EDGE-w/o f
554
0.742
79.7
0.288
0.187
0.325
EDGE-w/o 0
553
0.759
78.5
0.352
0.253
0.339
EDGE
554
0.710
85.4
0.069
0.097
0.318
Table 2: Multiple Conformation Generation GEOM-Drugs with backbone PhysNet(RDKit ini- tialized). Time refers to the seconds used to process 50k molecules in training epoch. means "higher is better". An NaN result occurs in Naive RMSD.
Baseline
Time
MAT
COV-0.5(%)
COV-1.25(%)1
Multi-IDDTT-Sore( %
RDKit
1.105
11.4
65.8
86.1
Random Guess
3.350
1.0
1.0
33.5
Naive RMSD
488
Kabsch RMSD
2412
1.172
9.9
62.2
76.6
Conn-3
528
1.098
10.3
66.1
86.2
EDGE
579
1.076
11.9
68.6
86.3
(Conn-oc and IDDT-5) and traditional RMSD with time-consuming alignment algorithm (Kabsch RMSD). This is proved empirically in all three groups of experiments in the tables.
For Definite Conformation Prediction QM9 (Tab EDGE has the best performance A- RMSD and IDDT-Score, two universal metrics to compare the conformations from a global view. To further demonstrate that EDGE outperforms the baselines through equivalently optimizing the bond lengths D. bond angles & and dihedral angles V, we evaluate their RMSE on the three fac- tors. Compared traditional DG-based methods (Conn-k and IDDT->), EDGE better retains the dihedral angles, which is significant in preserving complicated local structures like atom chains (see Sec 3.3 Moreover, EDGE's variants (w/o f and w/o σ) break the balance on degrees of freedom or
Table 3: Multiple Conformation Generation GEOM-QM9 with backbone CVGAE. Time refers to the seconds used to process 50k molecules training epoch. 1 means "higher is better".
Baseline
Time
MAT
COV-0.5(%))
COV-1.25(%)1
Multi-IDDTT-Soree(93 (
Random Guess
0.945
21.2
62.9
34.3
Naive RMSD
115
0.716
27.8
88.7
68.9
Kabsch RMSD
1254
0.670
28.7
95.8
72.6
Conn-3
169
0.641
39.4
91.8
79.9
EDGE
180
0.582
41.0
99.7
81.6
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Graph
RDKit
Kabseh
Conn-3
EDGE
real
Figure 3: Results of visualization with PhysNet on QM9.
value distribution and lose the equivalence on D, $ V, so they obtain much worse results. sum- mary, EDGE treats D, 2, V equivalently, so it can avoid contorted local geometries and optimize the conformation more effectively.
The result of Multiple Conformation Generation shows that EDGE can also assist optimization in the generative conformation task and still has better performance compared to traditional Kabsch RMSD and DG-based Conn-3. Moreover, EDGE is adaptive multiple types of backbones, matter it's generative (CVGAE) or derivative (PhysNet).
In Tab 12 where the conformations are fine-tuned from RDKit-initialzed ones, Kabsch RMSD and traditional DG-based methods may end up with results worse than RDKir in optimization. The main cause is that they can't measure the accurate discrepancy among the conformations, and the back- bone falls into a solution which is optimal for the given loss function but suboptimal for universal metrics e.g. A-RMSD. Contrastively, EDGE finds the essential factors in the conformation geometry and treats them equivalently when calculating the discrepancy, achieving better perfor- mance.
3.3 VISUALIZATION
We visualize the conformations generated by PhysNet under multiple loss functions dataset QM9 and compare them with RDKit-initialized conformations and reference ones, as shows Fig Kabsch RMSD and Conn-3 retain appropriate bond lengths and bond angles while the atom chains inside are severely contorted because dihedral angles are significant in preserving local geometries, especially the atom chains in molecules (Winter al. (021). Contrastively, EDGE equivalently optimizes the essential factors conformation geometry and focuses more on the angles. Therefore, PhysNet leveraged by EDGE produces conformations with better-structured atom chains as well as fine atom rings and achieves better performance.
CONCLUSION AND FUTURE WORK
We analyze the shortcomings of existing methods on measuring conformation discrepancy e.g. com- paring the atom positions or atom-pair distances. Then we propose EDGE, which can find the essen- tial factors in conformation geometry (i.e. bond lengths, bond angles and dihedral angles) and treat them equivalently by balancing the degrees of freedom as well as standard deviations. Theoretical analysis and extensive experiments demonstrate EDGE's efficiency and effectiveness when being applied to diverse backbone models. For future work, with the assistance of EDGE, the end-to-end conformation generation models are prevented from being misled by time-consuming or ineffec- tive optimization strategies, and thus more well-designed models can be proposed and improved to handle various tasks molecular conformation.

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Wolfgang Kabsch. A solution of the best rotation relate two sets of vectors, Acta Crystallograph- ica A, 32:922-922 1976.
Ziyao Li, Shuwen Yang, Guojie Song, and Lingsheng Cai. Conformation-guian molecular repre- sentation with hamiltonian neural networks. In International Conference Learning Represen- tations, 2021.
Elman Mansimov, Omar Mahmood, Seokho Kang. and Kyunghyun Cho. Molecular geometry pre- diction using deep generative graph neural network. Scientific reports, 9(1):1-13, 2019.
Valerio Mariani, Marco Biasini, Alessandro Barbato, and Torsten Schwede. Iddt: local superposition-free score for comparing protein structures and models using distance difference tests. Bioinformatics, 29(21):2722-2728, 2013.
Gregor Simm and Jose Miguel Hernandez-Lobate A generative model for molecular distance ge- ometry. In Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pp. 8949-8958, 2020.
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Robin Winter, Frank Noé, and Djork-Amé Clevert. Auto-encoding molecular conformations, In Machine Learning for Molecules Workshop at NeurIPS, 2021.
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Shuwen Yang, Ziyao Li, Guojie Song, and Lingsheng Cai. Deep molecular representation learning via fusing physical and chemical information. 2021.
A DETAILS OF EDGE
A. THE FORMULA OF vé
For certain dihedral angle & determined by four sequentially cooneeeced atoms in G which is shown Fig2(b) we want to calculate the value of using a, c, d, e, f. which are sides' lengths of the tetrahedron generated by the four atoms. Supposing that is the angle between b and c, the volume of the tetrahedron can be calculated following
sisin a a sin eb abe sin sin vb
V Sh =
(17)
3
3
2
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Under review as a conference paper at ICLR 2022
Using Euler's formula in the tetrahedron, V can also be calculated following
d²
d
2
2
V
6 te
c
6
o²
(18)
6
+c"
a
2
Note that
62
+ c2 62
sin
/1
cos or
(19)
2bc
Combining Eq
17 18 and 19
can get the formula of t/:
4a2 1/21 d2r3 22
r21112
3 aresin
(20)
4a - -
where
F1 = 12 + 2 e²
122 = 12 6²
(21)
4 = a² + b d
1 0 + e f
A.2 PARTIAL DERIVATIVES IN TAYLOR EXPANSION
A.2.1
For each in (9)) considering the bond lengths b, in & which generate following 2(a) can compose the vectors u (a, b, c) with them. Note that ø can be taken as a function w.r.t u.
a + 62 c²
c = o(u) arccos
(22)
2ab
To implement Taylor Expansion, need calculate
0Φ
0Φ
0Φ
(23)
du
Da
0b de
Following Eq 22 we can get the elements in Eq which are the partial derivatives w.r.t. each
23
variable:
06 a² + +22 ²
da 2a á
d a² 6² d
(24)
0b 2al2 sin 6
d6
dc
ab sin
A.2.2 vb
For each d in (g), considering the bond lengths a, c. d, e, in 9 which generate Φ following 2(b) we can compose the vectors v = (a, b, c, d, e, f) with them. Note that & can be also taken as a function (v) following Eq 20 To implement Taylor Expansion, we need to calculate
d
Oxb
Oh 0 Ob O dy T
(25)
dv
Da
ab dc Od de 0f
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Under review as a conference paper at ICLR 2022
Following (v),, get the elements in Eq 25 which the partial derivatives w.r.. each
variable:
dv (( ri)sinn s S18² - s2 - -836² + c2²
da
M
Ogp ((2za sin? 21 s2a²
83e2 + +2b2
0b
M
Qv.
(2r2a sin 1/ 2r2a² + 1162)
0c
M
(26)
d( - 21 + r2(22)
0d
M
0v
el
(2r ssin vp - 81 81 a²
8282 8zb2 22
de
M
11=2t + +r221)
0f
M
where
T1 = 13 + 6 6
P2 = 8 - 2 2
T3 b + te
81 e 8 2
$2 2 f
(27)
83 2² + + +2
t = 2 + 8 -8
t2 = 2 + + f2
M = a (r² 4b2 sin de cos
B PROVES
B.1 PROVE OF INEQUITABLE GRADIENTS OVER D, Φ, V IN DISTANCE GEOMETRY
We first analyze the gradients allocated on bond lengths D and bond angles & when optimizing 2- hop atom- atom-pair distances. Suppose two bonds a, with bond angle Φ, note that the third side of the triangle is c, 2-hop distance (see Fig ((a). We have:
a + 2ab cos Φ
(28)
The partial derivatives of c on variables a, b and & E & are:
Ōc b cos Φ
dc b COS &
dc
absin Φ
(29)
da
Ōb
To further quantify the derivatives in Eq 29 we sample the distributions of D, & from QM9 dataset
value of jad 1.452 and standard deviation oD R 0.103, and bond angles has as 108 and ag R 22 As op is relatively small compared to μD, we suppose that b HD. Then the values the derivatives are only variant to the temporary é, and we get:
(see more details in Appendix C.2). We found that bond lengths has
dc
de ah
π cos
dc
π
uD sin
(30)
Da
2
06
2
Eq 30 is visualized in Fig 4 When optimizing the 2-hop distances, large portion of gradient is allocated to fine-tumin minintunig the bond lengths D rather than the bond angles 4 especially for bigger bond angles. Therefore, traditional loss functions based distance geometry, where the bond angles expected to be optimized by preserving 2-hop distances, didn't treat D, equivalently and would cause contorted local geometries.
12

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Under review as a conference paper at ICLR 2022
do/Ca,0c/āb
partial derivative
dcl0p
1.452
1.000
0.500
60
108
180
Figure 4: Visualization of partial derivatives in 3q Ticks with μr 1..52 and μ4 108 highlighted.
When optimizing 3-hop atom-pair distances, the inequity of leveraging D, 4. V is proved empiri- cally from the result in Tab as the DG-based loss functions (Conn-k: and IDDT-y) focus more on bond lengths and less dihedral angles.
B.2 PROVE OF LIMITED ATOM-PAIR NUMBER
For an atom in common molecule, the number of neighbors is at most 6 (e.g. sulfur atom). There- fore, an atom has most 30 2-hop neighbors and 150 3-hop neighbors, which means that at most 186 atoms are 3-hop connected to it. For molecule with atoms, the number of 3-hop connected atom-pairs will not exceed 186n 93n.
Note that the number of 3-hop connected atom-pairs is y times of atom number n. For molecules in train set GEOM-Drugs where the average atom number is around 44.2 (Xu et al. 2021), the estimated value of is 4.6. This empirical result shows that optimizing the 1/2/3-hop atom-pair distances is constant times more expensive compared to directly optimizing atom positions, and be times faster than considering all atom-pairs, the constant 7 4.6 is much smaller than atom number n 44.2.
C IMPLEMENTATION DETAILS
C.I CALCULATION OF DEGREE OF FREEDOM
Consider 3-D conformation geometry G and the corresponding molecular graph M (with atoms), the degrees of freedom is fs 3n due to translation and rotation. We can figure out the exact number of degrees on three types of factors by:
Generate spanning tree T for the molecular graph M, and fo edges in T.
is the number of
For every in T, if the number of its neighbors N(v) are more than 1. it provides 2[W(v) - 3 degrees of freedom for fa In brief, we have so = ΣVENTTAN() ([)(s) -3
Obviously, fv fg fo fo
For atom v whose distances to its neighbors (v) given, needs 2)N( 3 exact values of bond angles (v as the axis) determine their relative position in 3-D space, if (W()) > (Havel 1998).
13

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Under review as a conference paper at ICLR 2022
Because there might be more than possible spanning trees in molecular graph, the output fD. fæ. fv is diverse. However, no matter how fD. fa. fv compose the total degrees of freedom fg. it always embodies set of 3n 6 factors (including bond lengths, bond angles and dihedral angles) which can determine the conformation geometry G. Therefore, the correctness of output ÍD, f f insensitive process of generating T.
C.2 CALCULATION HYPER- PARAMETERS
To capture the complete local geometry, we consider all possible bond length, bond angles and dihe- dral angles when sampling them from conformation G, and their numbers fD. fa, fv may exceed the
actual degrees of freedom fD- fa- fu. So we balance them with weights fn. In. La The weights fn fn fo
are calculated through sampling from a certain dataset, whose exact values are listed in Tab
Table The proportion degrees freedom over number sampled factors on D, 2, v. To save time, only sampled 1/10 molecules in datasets GEOM-QM9 and GEOM-Drugs.
Q.9
GEOM-QM9
GEOM-Drugs
of molecules
133885
5000
5000
Sum. of fD
830932
38959
119701
Sum. of fo 1030264
48274
153801
Sum. of f 311779
14644
70601
Sum. fp 1002205
46946
133582
Sum. of fo 1570148
75635
209399
Sum. of fw 1927138
94642
271308
Su/ʃp fo/fo fulfv 0.162
0.829
0.830
0.896
0.656
0.638
0.734
0.155
0.260
We also sampled the standard deviation of D, 4b \/ for all three datasets, it shows in Tab
Table 5: The standard deviation of sampled D, Φ V/ in each dataset.
QM9
GEOM-QM9
GEOM-Drugs
dD
0.103
0.101
0.109
04
0.387
0.380
0.122
0
0.433
0.437
0.394
Then we calculate the hyper-parameters via Eq
and the result is listed in Tab
Table 6: The hyper-parameters weighting D, 4 V for each dataset.
QM9
GEOM-QM9
GEOM-Drugs
Ap
8.063
7.689
8.394
Xa
1.692
6.257 1.676
X
0.378
0.347
0.681
C.3
MORE RESULTS ON MULTIPLE CONFORMATION GENERATION
See tables 7 and g
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Under review as a conference paper at ICLR 2022
Table 7: Multiple Conformation Generation on GEOM-QM9 with backbone PhysNet. Time refers the seconds used process 50k molecules training epoch. f means "higher is better".
Loss
Time
MAT
COV-0.5(%)
COV-1.25(%)
Multi-DDDT-Score %)f
RDKit
0.399
77.2
93.3
64.5
Random Guess
0.945
21.2
62.9
34.3
Naive RMSD
203
0.993
28.7
45.8
47.1
Kabsch RMSD
1880
0.596
29.1
93.3
56.9
Conn-3
232
0.458
69.1
93.3
62.1
EDGE
260
0.394
81.6
93.3
64.8
Table Multiple Conformation Generation on GEOM-Drugs with backbone CVGAE. Time refers
to the seconds used process 50k molecules in training epoch. 1 means "higher is better".
Loss
Time
MAT
COV-0.5(%)
COV-1.25(%)
Multi-IDDD-Scor((20
Random Guess
3,350
1.0
1.0
33.5
Naive RMSD
440
2.565
2.4
8.7
61.4
Kabsch RMSD
2201
2.520
1.7
8.2
61.6
Conn-3
469
2.540
1.6
8.0
61.6
EDGE
498
2.492
1.7
8.8
62.0
15


Reviews:
[{"review_id": "uldIH6tSZM7", "reviewer": "Reviewer_fdL6", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "**Originality:** The idea of the work is novel, the area of *differentiable* advanced conformation generation loss function are quite promising and poorly explored.\n\n**Clarity:** The author describes the motivation, goals and contribution of the work. The text, however, needs improvement (quality of graphs is not good, language needs spell-checking).\n\n**Significance:** The novelty of the idea is high, but the significance is not proven. The experiment does not show a significant gap between the proposed loss and baselines.\n\n**Drawbacks / questions**:\nThe claim of this work is that the new loss function is as accurate as best of its alternatives, and is faster to evaluate at the same time. The main accomplishment of the work is the comparison of different loss functions under various metrics. However, there are multiple issues that have to be addressed to improve the manuscript:\n\n1. The proposed loss is dependent on multiple hyperparameters (ratios of degrees of freedom, dispersion, gradient clipping thresholds), which are derived from data or set manually by the authors. This makes questionable the claims on the accuracy of the loss, because the tests were done only on 2 datasets: QM9 and GEOM. More thorough testing should be done to justify the procedure.\n\n2. The method is at most marginally better than the Conn-k loss, as is shown in the experimental section. The proposed loss is much more complex than Conn-k on the other hand and requires dataset-derived quantities. More arguments should be provided to show the advantage.\n\n3. The effect of Taylor approximation on the accuracy of the loss is not studied. The authors provide too few details about this aspect of the method.\n\n4. Plots are sloppy (see Fig. 4 which contains unnecessary whitespace). This creates a bad impression. Overall, the current form of the manuscript can not proceed until the mentioned issues are addressed.\n\nOverall, the current form of the manuscript can not proceed until the mentioned issues are addressed.", "summary_of_the_paper": "The work  introduces a custom loss function for the molecular conformation similarity. The authors combine this loss function with the PhysNet and GraphDG models to either predict a single conformation or to generate an ensemble of conformations. Other loss functions, including Naive RMSD, Kabsch algorithm, Conn-k and lDDT-k are compared under the same settings.\n\nThe main contributions of the paper are as follow: authors proposed a novel loss function for conformation generation, that can be used in *straight-forward* models, also provide Taylor Expansion and Multiplier Truncation techniques to speed up this metric.", "summary_of_the_review": "The idea of the paper is novel, still experiments do not confirm practical significance of the method. The method is dependent on hyperparameters choice, still authors do not provide an advice how to find suitable ones.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "pS3alm-bWI", "reviewer": "Reviewer_AFf9", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "Strengths \nThe approach is new and relatively effective. It sidesteps costly operations, including trigonometric functions, and applies some insightful approximations to improve speed and avoid exploding gradients. It is the fastest / best approach to including key dihedral information in the loss function of conformer generation approaches.\n\nWeaknesses\n\nWhile the Taylor series approach is nifty, the results are not unexpected and do not represent a large methodological innovation. Likewise, the experimental improvements are there, but the improvement in either times or accuracy w.r.t. conn-3 is just too subtle.\n\n\n\nQuestions\n\n\"it needs to calculate the rotation matrix for every single conformation, which stifles batch processing and increases time cost\"  The rotation matrix can be found through a quaternion tensor operation. Is this not batchable?\n\nStandard deviations or confidence intervals across multiple experiments could help confirm the magnitude of the improvements\n\n", "summary_of_the_paper": "This paper proposes a new loss function for conformer generation tasks. These tasks essentially generated 3D point clouds from molecular graphs, and require a geometric similarity metric between generated samples and training data to drive the learning. The traditional metric of distance is RMSD based on aligned poses, but the authors suggest this is too time-consuming and instead propose a heuristic-weighted combination of bond distance between first other neighbors, and a Taylor-expansion of angles between second-order neighbors and dihedrals between 3rd order neighbors. \n\nThe metric is empirically found to perform well, based on both statistical metrics used in conformer generation tasks, by visual inspection of pathological cases, and because the timings are excellent", "summary_of_the_review": "All in all, i find this paper informative and useful, but it does not provide enough truly novel insights for acceptance. I think it would make an outstanding workshop paper, or complement to some new conformer generative approach, but in and of itself it does not provide enough novelty and applied impact", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "oogePMM_2xh", "reviewer": "Reviewer_vUYf", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "Novel & Significance:\n1.\tIn Table 2, the paper lacks a comparison with previous methods like ConfGF, CGCF, VAE based methods and so on, which makes me hard to evaluate the effectiveness of the method.\n2.\tAlthough it is a new perspective to use D-$\\Phi$-$\\Psi$ for molecular conformation generation, I did not see a clear evidence about why it is helpful and how much improvement over previous is brought.\n\nClarification:\n1.\tIn Eqn.(8), what does the $u$ stand for?\n\nThe authors should release their code for reproduction.\n", "summary_of_the_paper": "The authors proposed a new method for molecular conformation generation. The authors choose to use bond lengths $d$, bond angles $\\phi$ and dihedral angles $\\psi$ for molecular conformation generation. The authors use Taylor expansion and multiplier truncation to improve models.  The authors conduct experiments on QM9 to verify the algorithm. \n", "summary_of_the_review": "The paper lacks a clear motivation why this method, and many important baselines are missing.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "3S3jx36rgv-", "reviewer": "Reviewer_AM3U", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "Strength:\n- The paper is well-written and easy to follow. The method is clearly stated.\n- The experimental part is sound. The effectiveness of the proposed method is well supported by the experiments on several benchmarks.\n\nWeakness:\n- The most interesting point for me is combining the new objective and PhysNet to design a new molecular graph generation model in the multiple conformation generation setting. However, there seem some technical issues regarding the generative model. For a generative model, it should be learned by matching the distribution instead of matching every two data points. It seems with the PhysNet backbone, you can generate different conformations with different initialization from RDKit, but how do you determine which reference confirmation should the generated one be compared to? If you just compare it with a uniformly sampled one from the dataset, it would not be theoretically sound for maximizing the likelihood of the dataset. Besides, the numbers in Tab.2 and Tab.3 show that CVGAE is much better than your proposed PhysNet framework. I think this is due to this technical issue.\n- Some highly related reference is missed. You may do not need to compare with it in experiments, but the technical connection should be clarified. For example, in the abstract, you mentioned the bi-level method for learning 3D objectives. but a recent paper on this direction is missed in the reference. (Xu et al. \"An End-to-End Framework for Molecular Conformation Generation via Bilevel Programming.\" ICML 2021.)\n- Though improved over the RMSD-based method, however, there are many recent methods with more competitive results, such as the references you have cited \"A generative model for molecular distance geometry\" and \"Learning neural generative dynamics for molecular conformation generation.\". It will be better to also compare them because otherwise the proposed method will not be practically useful; or if the comparison is not meaningful the author should clearly clarify the reason in paper.", "summary_of_the_paper": "The papers proposed a new objective function for comparing two 3D structures, in the molecular conformation context. Specifically, they proposed to compare the bond lengths, angles, and torsions, and design reweights to balance the three components. Furthermore, they improved the efficiency by approximating the comparison of angles and torsions with Taylor expansion. Experiments demonstrate that compared with directly comparing the aligned RMSD and other metrics, the proposed method is both more efficient and effective.", "summary_of_the_review": "The paper proposed an interesting idea for molecular conformation comparison. However, comparison with some important baselines is missed, and there is some essential technical issue, and thus I vote for rejection. But I'm willing to change if I made any mistake or the author can address my concern.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a new loss function for molecular conformation comparison to be used in generation tasks. All reviewers found the research topic is interesting, but the work lacks in multiple aspects. Major concerns include limited contributions and novelty, lack of comparison with prior methods, limited improvements, writing and clarity, etc. The authors did not provide any response during discussion. Given the consistency and extent of concerns, and lack of response, I recommend this paper be rejected at this time.
Decision: Reject