Geometric deep learning IEEE BigData 2024
Mirror it. Two of the three readouts do not notice.
Mirror it and one sign flips. Turn it or shift it and no sign does.
The paper proposes a message passing network for 3D graphs, which it calls RefNet, whose numbers do not change when the graph is turned or shifted. It gives each node one extra sign, worked out from the node and three of its neighbours, that flips when the graph is mirrored. It reports the lowest error in five of the six columns of its N-body table and in every column of its Molecule3D table (Tables I and III).
Figure 1Illustration
original hand
Press Mirror and watch the three readouts.
01 The problem
Turn a 3D graph or shift it and every distance and every angle in it stays the same, which is what a network for 3D graphs is built to respect. A mirror also leaves every distance and every angle the same. So a description made only of those two cannot tell the two hands apart. Measure it yourself.
Figure 2Illustration
Tap two nodes for a distance. Drag to turn.
| Measured | Graph | Mirror image | Same? |
|---|---|---|---|
| Nothing measured yet. | |||
With scripts on you can measure any distance or angle on both hands: all 15 distances and all 60 angles of the six points agree. Lining up nodes 1, 2 and 3 of the two hands leaves node 4 on opposite sides of their plane.
IllustrationGeometry on six made-up points, measured by this page in toy units. Lining up three nodes is this page’s own demonstration, not an experiment in the paper, and its numbers are not results.
Figure 3Illustration
stays put in a mirror as well
Distances to the neighbours and the angles between them. The bench above shows that a mirror changes none of them. The paper’s worry, in its own careful words: GNNs may map both of the enantiomer graphs to the same feature representations in those cases
, which it calls undesirable because such mirror-image pairs show distinct pharmacological and toxicological properties
.
flips in the mirror and nothing else node 1 reads here does
The same neighbourhood, plus one sign. Nodes j, k and p are all neighbours of node i, so nothing beyond one hop is read: the paper says its design exploits only 1-hop geometric representations
. The two planes (i, j, k) and (j, k, p) are swung one way or the other about the edge they share, and only that sign is kept.
02 The idea
For every node, RefNet picks its nearest neighbour j, then the next, k, then a third, p. The four points span two planes hinged on the edge from j to k, and the second plane is swung one way or the other from the first. No turn and no shift can change which way. A mirror always does. Three filters keep the choice of j, k and p steady, and only the sign of the twist goes into the network.
Figure 4Illustration
original hand
Drag to turn, or use the arrow keys. Tap a node to read its numbers.
Toy thresholds are far larger than the ones the paper searched (1e-5 to 5e-3, Table II), so that you can see a filter act. Each slider moves one node of the chosen node’s own quadruplet.
quadruplet in use (i, j, k, p)
The distances from the node to its j, k and p, the angle between j and k, and all 15 distances between the six points.
The direction to the nearest neighbour, and the node’s arrow after two toy layers: three numbers each, which turn with the graph.
One sign per node: plus or minus.
Seen from j looking towards k: which way the plane of j, k, p is swung from the plane of i, j, k. A sketch of the sense only, never of the size.
The toy measures the signed twist between the two drawn planes about the edge they share. The paper defines its own measure of this angle (Equations 4 and 5) and uses only its sign.
With scripts on, this figure turns, shifts and mirrors under your hand. In the toy, turning and shifting change none of the node numbers, mirroring flips all six signs and changes the readout, and with the sign token off the readout is identical for the two hands.
Switching the sign off is this page’s own experiment. The nearest measured thing in the paper is the row its ablation table labels − θijkp. The paper does not say that θijkp is this twist; if it is, that row is the sign removed. It reports error, not mirror images.
IllustrationA hand-built toy with six made-up points, one feature channel and fixed numbers in place of learned weights. It shows which quantities stay put, which turn and which flip in a mirror. It is not the trained network and its numbers are not results.
The proof starts from the distance: This distance remains unchanged under any rigid transformation
. That a mirror keeps every distance and angle as well is plain geometry, which the figure above computes.
The QOS is invariant under rotations and translations
, and under reflection transformations, the QOS undergoes a sign change
. QOS is the paper’s name for the twist: the Quadruplet Orientation Signature, a measure of the dihedral angle between the planes (i, j, k) and (j, k, p)
.
Equation 10 embeds sign(ϕijkp) and nothing else: ψ denotes the embedding function that maps the QOS signs to latent representations
. The size of the angle is thrown away, and the paper does not give the form of ψ.
03 Follow one node through one layer
Follow node 1 of the made-up graph, and the message it receives from node 2, through one layer. The drawing is this page’s reading of Equations 1 to 15; the paper has no figure of its method. Nothing in the upper lane is ever more than a 3D arrow. Every box opens.
Figure 5Illustration
Which node is nearest does not depend on how the graph is held, and the arrow to it turns with the graph. The paper calls this Direction to Nearest Neighbor and uses it in place of arrows that start empty or random.
If two neighbours are almost equally far, which one counts as nearer could change with rounding, so both are left out. j is the nearest of what remains.
Three nodes in a line do not span a plane. k is the nearest survivor that is off that line, in either direction along it.
With i, j and k fixed, p is the nearest survivor for which the planes (i, j, k) and (j, k, p) are not nearly the same plane. The twist between them is the paper’s Quadruplet Orientation Signature.
The size of the angle is thrown away. The sign becomes a learned token that is added to the node’s numbers, and the result is widened to three sets of numbers. Because the token differs between a graph and its mirror image, the node’s numbers can differ too. They still do not change when the graph is turned or shifted (Theorem 1).
The three sets are weighted by a function of the distance. One goes to node 1’s numbers. One scales the arrow from node 1 to node 2. One scales node 2’s own arrow. Node 1 adds these up over its neighbours: arrows are only ever scaled by numbers and added.
The length of each arrow, a number that stays put, joins the node’s numbers. From that mix come a new number and a multiplier for the arrow, and a residual keeps the input. Targets that must not turn are read from the numbers; targets that must turn use both.
Select any box in the drawing to read what it does, what goes in and out, and what the paper leaves unsaid.
N(i) Section III-B; Section IV, p. 1719
Everything the descriptor needs sits inside one node’s own neighbourhood: j, k and p are all drawn from the neighbours of i. The paper calls this an efficient one-hop message passing scheme
.
Not in the paper: the cutoff radius that decides who is a neighbour. The toy joins all six points.
Distance masking, D(i) Section IV-B, Eq. 1, p. 1719
Its goal is to filter out nodes that are equidistant from the reference node i, which could lead to ambiguities in node selection
. Any pair of neighbours whose distances differ by less than a threshold is excluded: both members, not just one.
Not in the paper: the threshold it finally chose. Table II lists the values searched, 1e-3 to 5e-3.
Non-collinear node selection, L(i, j) Section IV-C, Eq. 2 and 3, p. 1719
Linear alignment of nodes leads to degenerate cases where the QOS becomes ill-defined.
A candidate k is rejected when the cosine of the angle between the directions to j and to k is within a small tolerance of 1, in either direction.
Not in the paper: the tolerance it finally chose. Table II lists 1e-5 to 1e-4.
Non-Degenerate Quadruplet Set, Q(i, j, k) Section IV-D, Eq. 6, pp. 1719 to 1720
It excludes nodes that form near-parallel planes with the plane of i, j, and k
. The paper’s sentence says a twist near zero or near π is near-planar; the toy follows that sentence.
Not in the paper: what a node receives when no candidate is left. Equation 6 as printed tests only the lower bound.
Equations 7, 8 and 9 Section IV-E, p. 1720
j is the nearest neighbour that survives the tie filter. k is the nearest survivor that is not in line with i and j. p is the nearest survivor that is not in line with j and k and does not leave the two planes nearly flat. The paper calls it a multi-stage filtering approach
.
Not in the paper: what happens to a node with fewer than three usable neighbours.
Quadruplet Orientation Signature (QOS) Section IV-D, Eq. 4 and 5
The paper builds the normal of each plane with a cross product and takes a signed angle from two products of those normals: a measure of the dihedral angle between the planes (i, j, k) and (j, k, p)
. It is unchanged by turning and shifting, and under reflection transformations, the QOS undergoes a sign change
.
Not in the paper: a derivation of the sign change, which is stated in prose. The toy measures the twist about the shared edge and shows its sign only.
Equation 10 Section IV-E, p. 1720 flips in the mirror
Each node gets one of two tokens, and the token becomes a learned set of numbers: ψ denotes the embedding function that maps the QOS signs to latent representations
. Of the plain numbers that enter the network, this is the only one a mirror changes (this page’s reading of Equations 11 to 15); the arrows are simply mirrored along with the graph.
Not in the paper: the form of ψ. The toy uses two fixed numbers, and zero for a node without a valid quadruplet.
Scalar representations, s Section III-B; Section V stays put
Every node carries plain numbers next to its arrows. Theorem 1 says they are unchanged by any turn and any shift of the graph.
Not in the paper: how the first numbers are made from the input. The toy starts from three made-up node types.
Equation 11 Section V-A, p. 1720
The token is added to the sending node’s numbers and the result is widened from C numbers to 3C: The scalar feature of the source node, sj, is enhanced by the QOS embedding
. After this box the sign is part of ordinary numbers.
Not in the paper: the depth and activation of the two perceptron stacks. The toy uses a tanh of three fixed multiples.
χ and ω in Equation 12 Section V-A stays put
The distance between the two nodes passes through the basis and cutoff functions, respectively
, and another perceptron stack. A distance does not change under any rigid motion, so neither does anything made from it.
Not in the paper: which basis, which cutoff function, and the cutoff radius. The toy uses three bumps and a cosine cutoff.
ms, mv, md, Equation 12 Section V-A stays put
The three sets of numbers are multiplied by the three distance weights. One message is for the numbers, one for the neighbour’s own arrows and one for the direction between the two nodes. All three are plain numbers.
Not in the paper: nothing further; the sizes are printed, the functions are not.
Interaction layer, numbers, Equation 13 Section V-A stays put
A node’s numbers become its old numbers plus the first message from every neighbour. The proof: the scalar update depends entirely on the invariant message function
.
Not in the paper: any normalisation of the sum.
Direction to Nearest Neighbor (DNN) Section IV-A.1, p. 1719 turns
each node's vector is initialized based on the direction to its nearest neighbor
. The identity of the nearest neighbour does not depend on how the graph is held; the arrow itself turns with it.
Not in the paper: whether the arrow is normalised, how it fills C channels, and what happens on a tie for nearest. The toy uses a unit arrow.
dij in Equation 13 Proof, p. 1721 turns
The proof defines it as the position of j minus the position of i, and shows that turning the graph turns it the same way. It is the only place where raw geometry enters the arrows of a layer.
Not in the paper: whether d is normalised inside the layer. The toy uses it as the proof defines it.
Interaction layer, arrows, Equation 13 Section V-A; proof, p. 1721 turns
For each neighbour, the direction towards it is scaled by one message and the neighbour’s own arrows by another, and the results are added to the node’s arrows. A number times an arrow is still an arrow that turns with the graph: two arrows never combine into a new kind of object.
Not in the paper: the shape of the vector features. It prints the sizes C, 2C and 3C for numbers only.
The norm in Equation 14 Section V-A; proof, pp. 1721 to 1722
The one crossing from the arrows to the numbers. the L2 norm of the vector representation is invariant under SE(3) transformations and acts as a scalar
, so it can join the numbers without breaking anything.
Not in the paper: nothing further about this step.
Update layer, Equation 14 Section V-A, pp. 1720 to 1721 stays put
Arrow lengths and numbers are mixed by perceptron layers into a new number and a multiplier for the arrow. The paper describes this as bidirectional communication between the scalar and vector representations
, and adds a residual connection.
Not in the paper: the depth of the perceptron stack.
Equation 15 Section V-A; proof, p. 1722 turns
The arrow is scaled by the multiplier from the numbers and by a second gate made from its own length. The use of consistent weights across spatial dimensions ensures that the update respects the SE(3) equivariance of the network.
Not in the paper: which activation σ is. The toy uses a sigmoid.
Number of layers Table II, p. 1723
The hyperparameter search space lists 4 layers for the N-body data and 4 or 6 for Molecule3D. After the first layer the numbers of every node already carry the signs of its neighbours.
Not in the paper: which values were finally chosen. Only the search space is printed.
Section V-B p. 1721
When predicting invariant targets, the scalar representations are directly passed to the prediction head. In contrast, for equivariant tasks, both scalar and vector representations are used.
Energies are of the first kind, future positions and the dipole vector of the second.
Not in the paper: how node outputs are pooled and the exact form of the equivariant head. The toy reads out a plain sum.
IllustrationThe drawing is this page’s reading of Equations 1 to 15. The values marked Toy come from the same hand-built toy as Figure 4, in the pose and hand you left it in, to two decimals and in toy units. Its numbers are not results.
sidesteps the need for computationally demanding higher-order representations in intermediary layers(abstract). It does not define that term, so this page only shows what is carried.
04 The evidence
One synthetic dataset whose answer is a set of positions that must turn with the input, and one real dataset of molecular properties. Every value below is a printed cell. Error is always lower is better, and the paper prints no unit, no seeds and no error bars for any table. All experiments ran on an NVIDIA 3090 GPU with 24 GB.
Figure 6Measured, Table I
A synthetic dataset of particle simulations: predict future particle positions from initial positions and velocities. Four settings, as the paper names them: a 5-body system, ES(5); 20 bodies, ES(20); with a gravity field added, G+ES(20); and with a Lorentz-like force field, L+ES(20). The answer is a position, so it must turn with the input. Section VI-B and VI-D.1, p. 1722
| Method | ES(5) | ES(20) | G+ES(20) | L+ES(20) | std.MSE | logMSE |
|---|---|---|---|---|---|---|
| GNN | 0.0131 | 0.0720 | 0.0721 | 0.0908 | 0.0151 | −4.3876 |
| TFN [41] | 0.0236 | 0.0794 | 0.0845 | 0.1243 | 0.0192 | −4.0978 |
| SE(3)-Transformer [5] | 0.0329 | 0.1349 | 0.1000 | 0.1438 | 0.0252 | −3.8037 |
| Radial Field [42] | 0.0207 | 0.0377 | 0.0399 | 0.0779 | 0.0110 | −4.6212 |
| PaiNN [43] | 0.0158 | 0.0997 | 0.1029 | 0.1356 | 0.0835 | −2.5329 |
| ET [44] | 0.1653 | 0.1788 | 0.2122 | 0.2989 | 0.0535 | −2.9587 |
| EGNN [45] | 0.0079 | 0.0128 | 0.0118 | 0.0368 | 0.0044 | −5.6241 |
| ClofNet [7] | 0.0065 | 0.0073 | 0.0072 | 0.0251 | 0.0029 | −6.0324 |
| GCPNet [46] | 0.0070 | 0.0071 | 0.0073 | 0.0173 | 0.0024 | −6.1104 |
| RefNet | 0.0062 | 0.0063 | 0.0082 | 0.0123 | 0.0020 | −6.2268 |
Bold with a rule under it marks the paper’s bold. The paper also underlines the second lowest cell in each of the four task columns (ClofNet in ES(5), GCPNet in the other three); that second mark is not reproduced here. For logMSE, lower means more negative. The paper calls std.MSE and logMSE “supplementary metrics” and prints no formula for either. “Most of the baseline results are adopted from [7], [46]”, the ClofNet and GCPNet papers. No unit is printed. Table I and Section VI-D.1, VI-F, pp. 1722 to 1723
Figure 7Measured, Table III
A real dataset: 3,899,647 molecules, 29 nodes on average per molecule, and 553 average edges per molecule
, with the splits of the dataset’s own paper. The invariant targets are the dipole moment µ, the energy of the highest occupied molecular orbital, the energy of the lowest unoccupied molecular orbital and the orbital energy gap, reported as standardized mean absolute error. Section VI-C and VI-F, pp. 1722 to 1723
Figure 8Measured, Table III, last column
The paper reports this column apart because many baseline models did not account for this property
. It does not say how the five baselines were made to output a vector, and it names no metric and no unit for the column. Read it with that sentence in mind.
| Symmetry | Invariant | Equivariant | |||||
|---|---|---|---|---|---|---|---|
| Method | µ | εHOMO | εLUMO | Δε | std. MAE | logMAE | µ as a vector |
| GIN-Virtual [21] | 0.0882 | 0.0692 | 0.0632 | 0.1036 | 0.0592 | −2.8677 | 1.5233 |
| SchNet [50] | 0.0532 | 0.0275 | 0.0265 | 0.0428 | 0.0263 | −3.6633 | 1.2082 |
| DimeNet++ [13] | 0.0293 | 0.0240 | 0.0190 | 0.0306 | 0.0188 | −4.0139 | 1.2014 |
| SphereNet [6] | 0.0288 | 0.0239 | 0.0183 | 0.0301 | 0.0184 | −4.0327 | 1.1836 |
| ComENet [15] | 0.0345 | 0.0288 | 0.0252 | 0.0326 | 0.0220 | −3.8403 | 1.3521 |
| RefNet | 0.0129 | 0.0167 | 0.0132 | 0.0266 | 0.0132 | −4.4361 | 0.0092 |
Bold with a rule under it marks the paper’s bold. The paper’s second header row reads “Task”; the last column is printed as µ with an arrow over it. For logMAE, lower means more negative; the paper prints no formula for std. MAE or logMAE. For the baselines, only the gap target has been reported in preceding works
, so the authors ran them for the other targets with the hyperparameters of the ComENet paper [15]. Table III and Section VI-C, VI-D.2, pp. 1722 to 1724
05 The ablation, taken apart
The paper omits five components one at a time and reports the error on Molecule3D. Only these six rows are printed, so there are six presets and no free switches. Two of the row labels are not defined in the method section, so this page prints all five exactly as the paper does.
Figure 9Measured, Table IV
Row of Table IV
initialization of vectors by the direction to the nearest neighbor
The paper does not say what θijkp is. If it is the four-node twist, the mirror test shows what its sign is for.
| Variation | µ | εHOMO | εLUMO | Δε |
|---|---|---|---|---|
| − θijkp | 0.0183 | 0.0190 | 0.0173 | 0.0290 |
| − NN | 0.0164 | 0.0176 | 0.0160 | 0.0299 |
| − D(i) | 0.0169 | 0.0170 | 0.0150 | 0.0260 |
| − A(i, j) | 0.0162 | 0.0182 | 0.0153 | 0.0273 |
| − L(i, j, k) | 0.0154 | 0.0175 | 0.0147 | 0.0283 |
| RefNet | 0.0149 | 0.0167 | 0.0142 | 0.0266 |
The paper prints no bold in this table, and no seeds, repeated runs or error bars. Table IV’s full-model row prints 0.0149 and 0.0142 for µ and εLUMO where Table III prints 0.0129 and 0.0132; the paper does not comment, so this page never joins the two tables and compares each ablation row only with Table IV’s own last row. Table IV and Section VII-C, p. 1724
yields the highest error across most metricsand that excluding L(i, j, k)
led to the least detrimental outcomes; the cells bear out the first in three columns of four and the second in two.
The takeaway
A four-node signature chosen inside one node’s own neighbourhood, three filters that keep that choice steady, start arrows that point to the nearest neighbour, and a message passing layer that carries only plain numbers and 3D arrows.
Error on two benchmarks, lowest in 5 of 6 columns of Table I and in all 7 of Table III, and one ablation on Molecule3D.
Cost: no time, memory or parameter count is printed. And behaviour on mirror-image molecules: neither dataset isolates them.
06 Office hours
Written by Tian's agent while Tian slept and Pepper supervised. Pepper accepts no liability either.
Each answer says where in the paper it comes from, so you can check the agent's work.
Turning and shifting are what SE(3) covers: the paper defines it as rotations and translations and stresses the exclusion of reflections
. Distances and angles are also unchanged by a mirror, and the paper’s concern is that a network built on them may map both of the enantiomer graphs to the same feature representations
. RefNet’s four-node signature is unchanged by rotation and translation and, in the paper’s words, under reflection transformations, the QOS undergoes a sign change
. Only that sign is embedded and added to each node’s numbers. Theorem 1 covers rotations and translations; the mirror behaviour is stated in the prose of its proof and is not a theorem.
The paper does not measure it. It prints no training or inference time, no memory, no parameter count and no complexity for RefNet or any baseline. Its case is by design: everything the descriptor needs sits in one node’s neighbourhood (it exploits only 1-hop geometric representations
), the middle layers carry scalars and vectors without higher-order representations
, and the signature involves only basic vector operations
. What is known: all experiments ran on an NVIDIA 3090 GPU with 24 GB, including Molecule3D with 3,899,647 molecules. This page therefore shows no cost chart.
No. The two datasets are an N-body simulation and Molecule3D, and neither experiment isolates mirror images. The nearest measured evidence is the ablation: on Molecule3D, removing the component the paper labels θijkp (the paper does not say that it is the QOS) raises the error on all four targets, for example µ from 0.0149 to 0.0183, and it is the highest cell in three of the four columns. The mirror test on this page is an illustration of the stated sign change on made-up points, not a result.
Source: Section VI-A to VI-C; Table IV and Section VII-C, p. 1724On N-body RefNet prints the lowest cell in five of six columns; on the gravity-field task it is third (0.0082, behind ClofNet 0.0072 and GCPNet 0.0073). On Molecule3D it prints the lowest cell in all seven columns. Things to keep in mind: most N-body baseline values were adopted from two earlier papers, and on Molecule3D only the gap values of the baselines had been published, so the authors ran the baselines for the other targets with another paper’s hyperparameters. The dipole-vector column is the column where RefNet and the baselines are furthest apart, and the paper itself presents it separately because many baseline models did not account for this property
. No seeds, repeated runs or error bars are reported.
Three filters guard the choice of the four nodes: near-ties in distance are dropped, a third node in line with the first two is skipped, and a fourth node that leaves the two planes nearly flat is skipped. The paper does not say what a node receives when no valid quadruplet is left. In the ablation, removing the distance mask D(i) raises the error on three targets and lowers it slightly on the gap (0.0260 against 0.0266 for the full model); the other four removals raise all four. Two of the ablation’s row labels are not defined in the method section, so this page prints them as the paper does. Table IV’s full-model row also differs from Table III on two targets, without comment.
Source: Section IV-B to IV-E; Table IV and Section VII-C; Tables III and IV