SlowGuess/NeurIPS_2023_OCR huggingface dataset:
At this point, after decomposition of node features  $Y^{(i)}$ , we define the messages sent from neighbors  $j$  to central atom  $i$  as  $M^{(ij)} = f_I^{(ij)}I^{(j)} + f_A^{(ij)}A^{(j)} + f_S^{(ij)}S^{(j)}$ , which get aggregated into  $M^{(i)} = \sum_{j\in \mathcal{N}(i)}M^{(ij)}$ . We use the irreducible decomposition of matrix products  $Y^{(i)}M^{(i)} + M^{(i)}Y^{(i)}$  between node embeddings and aggregated messages to generate new atomic scalar, vector, and tensor features. New features are generated in this way to guarantee the preservation of the original parity of scalar, vector, and tensor features. These new representations  $I^{(i)}, A^{(i)}, S^{(i)}$  are individually normalized dividing by  $||I^{(i)} + A^{(i)} + S^{(i)}|| + 1$  and are further used to compute independent linear combinations to get  $Y^{(i)} \gets W^I I^{(i)} + W^A A^{(i)} + W^S S^{(i)}$ . A residual update  $\Delta X^{(i)}$  for original embeddings  $X^{(i)}$  is computed with the parity-preserving matrix polynomial  $\Delta X^{(i)} = Y^{(i)} + (Y^{(i)})^2$ , to eventually obtain updated representations  $X^{(i)} \gets X^{(i)} + \Delta X^{(i)}$ .
Scalar output. The Frobenius norm  $\mathrm{Tr}(X^{\mathrm{T}}X)$  of full tensor representations and components in TensorNet is O(3)-invariant. For molecular potential predictions, total energy  $U$  is computed from atomic contributions  $U^{(i)}$  which are simply obtained by using the concatenated final norms of every atom's scalar, vector, and tensor features  $||I^{(i)}||, ||A^{(i)}||, ||S^{(i)}||$ ,
$$
U ^ {(i)} = \operatorname {M L P} \left(\text {L a y e r N o r m} \left(\left[ | | I ^ {(i)} | |, | | A ^ {(i)} | |, | | S ^ {(i)} | |\right]\right)\right), \tag {12}
$$
obtaining forces via backpropagation.
Vector output. Since interaction and update operations preserve the parity of tensor components, the skew-symmetric part of any full tensor representation  $X$  in TensorNet is guaranteed to be a vector, not a pseudovector. Therefore, from the antisymmetrization  $A^X$ , one can extract vectors  $\mathbf{v} = (v_x, v_y, v_z)$  by means of the identification given in (3).
Tensor output. Taking into account that rank-2 tensors have even parity and the skew-symmetric part  $A$  in TensorNet is a vector, not a pseudovector, one might need to produce pseudovector features before rank-2 tensor predictions can be built by combining irreducible representations. This can be easily done by obtaining two new vector features with linear layers,  $A^{(1)} = W^{(1)}A$  and  $A^{(2)} = W^{(2)}A$ , and computing  $\frac{1}{2} (A^{(1)}A^{(2)} - (A^{(1)}A^{(2)})^{\mathrm{T}})$ , which is skew-symmetric, rotates like a vector, and is invariant under parity, the simultaneous transposition of  $A^{(1)}$  and  $A^{(2)}$ .
We refer the reader to the Appendix (section A.3) for further training, data set and experimental details.
QM9: Chemical diversity. To assess TensorNet's accuracy in the prediction of energy-related molecular properties with a training set of varying chemical composition we used QM9 [33]. We trained TensorNet to predict:  $U_{0}$ , the internal energy of the molecule at  $0\mathrm{K}$ ;  $U$ , the internal energy at  $298.15\mathrm{K}$ ;  $H$ , the enthalpy, also at  $298.15\mathrm{K}$ ; and  $G$ , the free energy at  $298.15\mathrm{K}$ . Results can be found in Table 1, which show that TensorNet outperforms Allegro [18], and MACE [34] on  $U_{0}$ ,  $U$  and  $H$ . Remarkably, this is achieved with  $23\%$  of Allegro's parameter count. Furthermore, TensorNet uses only scalar, vector and rank-2 tensor features, as opposed to Allegro, which uses also their parity counterparts, and without the need of explicitly taking into account many-body terms as done in MACE.
Table 1: QM9 results. Mean absolute error on energy-related molecular properties from the QM9 dataset, in meV, averaged over different splits. Parameter counts for some models are found between parentheses.
Property	DimeNet++ [25]	ET [16]	PaiNN [15]	Allegro (17.9M)[18]	MACE [34]	TensorNet (4.0M)
U0	6.3	6.2	5.9	4.7	4.1	3.9(1)
U	6.3	6.3	5.7	4.4	4.1	3.9(1)
H	6.5	6.5	6.0	4.4	4.7	4.0(1)
G	7.6	7.6	7.4	5.7	5.5	5.7(1)
rMD17: Conformational diversity. We also benchmarked TensorNet on rMD17 [35], the revised version of MD17 [36; 37], a data set of small organic molecules in which energies and forces were obtained by running molecular dynamics simulations with DFT. We report the results in Table 2. In the case of energies, TensorNet with two interaction layers (2L) is the model that achieves state-of-the-art accuracy for the largest number of molecules (6 out of 10), outperforming all other spherical models for benzene, with a parameter count of  $770\mathrm{k}$ . Energy errors are also within the range of other spherical models, except for ethanol and aspirin, and reach state-of-the-art accuracy for the case of toluene, with just one interaction layer (1L) and a parameter count of  $535\mathrm{k}$ . Force errors for 2L are also mostly found within the ranges defined by other spherical models, except for ethanol, aspirin, and salicylic acid, in which case these are slightly higher. However, for one interaction layer, force errors are increased and in most cases found outside of the range of accuracy of the other spherical models. We note that the smallest spherical models have approximately 2.8M parameters, and therefore TensorNet results are achieved with reductions of  $80\%$  and  $70\%$  in the number of parameters for 1L and 2L, respectively. Also, TensorNet is entirely based at most on rank-2 tensors.
SPICE, ANI1x, COMP6: Compositional and conformational diversity. To obtain general-purpose neural network interatomic potentials, models need to learn simultaneously compositional and conformational degrees of freedom. In this case, data sets must contain a wide range of molecular systems as well as several conformations per system. To evaluate TensorNet's out-of-the-box performance without hyperparameter fine-tuning, we trained the light model with two interaction layers used on rMD17 on the SPICE [38] and ANI1x [39; 40] data sets using the proposed Equivariant Transformer's SPICE hyperparameters [41] (for ANI1x, in contrast to the SPICE model, we used 32 radial basis functions instead of 64, and a cutoff of  $4.5\AA$  instead of  $10\AA$ ), and further evaluated ANI1x-trained models on the COMP6 benchmarks [39].
For SPICE, with a maximum force filter of  $50.94\mathrm{eV} / \mathring{\mathrm{A}}\approx 1$  Ha/Bohr, TensorNet's mean absolute error in energies and forces are  $25.0\mathrm{meV}$  and  $40.7\mathrm{meV} / \mathring{\mathrm{A}}$ , respectively, while the Equivariant Transformer achieves  $31.2\mathrm{meV}$  and  $49.3\mathrm{meV} / \mathring{\mathrm{A}}$ . In this case, both models used a cutoff of  $10\mathring{\mathrm{A}}$ . Results for ANI1x and model evaluations on COMP6 are found in Table 3. We note that for ANI1x training, which contains molecules with up to 63 atoms, TensorNet used a cutoff of  $4.5\mathring{\mathrm{A}}$ . The largest rMD17 molecule is aspirin with 21 atoms. The light TensorNet model shows better generalization capabilities across all COMP6 benchmarks.
Scalar, vector and tensor molecular properties for ethanol in a vacuum. We next tested TensorNet performance for the simultaneous prediction of scalar, vector, and tensor molecular properties: potential energy, atomic forces, molecular dipole moments  $\mu$ , molecular polarizability tensors  $\alpha$ , and nuclear-shielding tensors  $\sigma$ , for the ethanol molecule in vacuum [15; 42]. We trained TensorNet to generate atomic tensor representations that can be used by different output modules to predict the
Table 2: rMD17 results. Energy (E) and forces (F) mean absolute errors in meV and meV/Å, averaged over different splits.
Molecule	TensorNet 1L (535k)	TensorNet 2L (770k)	NequIP [17]	Allegro [18]	BOTNet [19]	MACE [20]
Aspirin	E	2.7	2.4	2.3	2.3	2.3	2.2
F	10.2(2)	8.9(1)	8.2	7.3	8.5	6.6
Azobenzene	E	0.9	0.7	0.7	1.2	0.7	1.2
F	3.8	3.1	2.9	2.6	3.3	3.0
Benzene	E	0.03	0.02	0.04	0.3	0.03	0.4
F	0.3	0.3	0.3	0.2	0.3	0.3
Ethanol	E	0.5	0.5	0.4	0.4	0.4	0.4
F	3.9(1)	3.5	2.8	2.1	3.2	2.1
Malonaldehyde	E	0.8	0.8	0.8	0.6	0.8	0.8
F	5.8(1)	5.4	5.1	3.6	5.8	4.1
Naphthalene	E	0.3	0.2	0.9	0.2	0.2	0.5
F	1.9	1.6	1.3	0.9	1.8	1.6
Paracetamol	E	1.5	1.3	1.4	1.5	1.3	1.3
F	6.9	5.9(1)	5.9	4.9	5.8	4.8
Salicylic acid	E	0.9	0.8	0.7	0.9	0.8	0.9
F	5.4(1)	4.6(1)	4.0	2.9	4.3	3.1
Toluene	E	0.3	0.3	0.3	0.4	0.4	0.5
F	2.0	1.7	1.6	1.8	1.9	1.5
Uracil	E	0.5	0.4	0.4	0.6	0.4	0.5
F	3.6(1)	3.1	3.1	1.8	3.2	2.1
Table 3: ANI1x and COMP6 results. Energy (E) and forces (F) mean absolute errors in meV and meV/Å for Equivariant Transformer and TensorNet models trained on ANI1x and evaluated on the ANI1x test set and the COMP6 benchmarks, averaged over different training splits.  $43\mathrm{meV} = 1$  kcal/mol.
Model	ANI1x	ANI-MD	GDB7-9	GDB10-13	DrugBank	Tripeptides	S66x8
ET	E	21.2	249.7	17.8	51.0	95.5	57.9	30.7
F	42.0	50.8	29.2	57.4	47.7	37.9	19.0
TensorNet	E	17.3	69.9	14.3	36.0	42.4	40.0	27.1
F	34.3	35.5	23.1	41.9	32.6	26.9	14.3
desired properties. The specific architecture of these output modules can be found in the Appendix (section A.3).
Table 4: Ethanol in vacuum results. Mean absolute error for the prediction of energies (E), forces (F), dipole moments  $(\mu)$ , polarizabilities  $(\alpha)$ , and chemical shifts for all elements  $(\sigma_{all})$ , averaged over different splits, with corresponding units between parentheses.
Model	E (kcal/mol)	F (kcal/mol/Å)	μ (D)	α (Bohr3)	σall (ppm)
PaiNN [15]	0.027	0.150	0.003	0.009	-
FieldSchNet [42]	0.017	0.128	0.004	0.008	0.169
TensorNet	0.008(1)	0.058(3)	0.003(0)	0.007(0)	0.139(4)
Results from Table 4 show that TensorNet can learn expressive atomic tensor embeddings from which multiple molecular properties can be simultaneously predicted. In particular, TensorNet's energy and force errors are approximately a factor of two and three smaller when compared to FieldSchNet [42] and PaiNN [15], respectively, while increasing the prediction accuracy for the other target molecular properties, with the exception of the dipole moment.
Equivalence, interaction and cutoff ablations. TensorNet can be straightforwardly modified such that features are SO(3)-equivariant and scalar predictions are SE(3)-invariant by modifying the matrix products in the interaction mechanism. An interaction product between node features and aggregated messages  $2Y^{(i)}M^{(i)}$ , instead of  $Y^{(i)}M^{(i)} + M^{(i)}Y^{(i)}$ , gives vector and tensor representations which are combinations of even and odd parity contributions. We refer the reader to the Supplementary Material for detailed derivations. Furthermore, the norms  $\mathrm{Tr}(X^{\mathrm{T}}X)$  used to produce scalars will only be invariant under rotations, not reflections. This flexibility in the model allows us to study the changes in prediction accuracy when considering O(3) or SO(3) equivariant models. We also evaluated the impact on accuracy for two rMD17 molecules, toluene and aspirin, when modifying the receptive field of the model by changing the cutoff radius and the number of interaction layers, including the case of using the embedding and output modules alone, without interaction layers (0L), with results in Table 5.
Table 5: Equivariance, interaction and cutoff ablations results. Energy (E) and force (F) mean absolute errors in meV and meV/Å for rMD17 toluene and aspirin, averaged over different splits, varying the number of interaction layers, the cutoff radius, and the equivariance group.
TensorNet0L O(3)	TensorNet 1L O(3)	TensorNet 2L O(3)	TensorNet 1L SO(3)	TensorNet 2L SO(3)
Molecule	4.5Å	9Å	4.5Å	9Å	4.5Å	9Å	4.5Å	4.5Å
Toluene	E	3.3	2.0	0.33	0.36	0.26	0.32	0.50	0.42
F	15.7	11.5	2.0	2.2	1.7	2.0	2.9	2.4
Aspirin	E	9.8	7.8	2.7	2.9	2.4	2.8	3.7	3.4
F	32.7	28.3	10.1	11.0	8.9	10.5	13.1	11.8
The inclusion or exclusion of equivariance and energy invariance under reflections has a significant impact on accuracy. The consideration of the full orthogonal group O(3), and therefore the physical symmetries of the true energy function, leads to higher accuracy for both energy and forces. Furthermore, the use of interaction products produces a drastic decrease in errors (note that TensorNet 1L  $4.5\mathring{\mathrm{A}}$  and TensorNet 0L  $9\mathring{\mathrm{A}}$  have the same receptive field). In line with rMD17 results, a second interaction layer in the case of  $r_c = 4.5\mathring{\mathrm{A}}$  gives an additional but more limited improvement in both energy and force errors. For forces, the use of a second interaction layer with  $r_c = 9\mathring{\mathrm{A}}$  encompassing the whole molecule provides a smaller improvement when compared to  $r_c = 4.5\mathring{\mathrm{A}}$ . We note that for 0L, when the model can be regarded as just a learnable aggregation of local atomic neighborhoods, TensorNet with both cutoff radii achieves for aspirin (the rMD17 molecule on which the model performs the worst) lower mean absolute errors than ANI (16.6 meV and 40.6 meV/Å) [7; 43] and SchNet (13.5 meV and 33.2 meV/Å) [22; 44].
Computational cost. We found that TensorNet exhibits high computational efficiency, even higher than an equivariant model using Cartesian vectors such as the Equivariant Transformer [16] in some cases. We provide inference times for single molecules with varying numbers of atoms in Table 6, and in Table 7 we show training steps per second when training on the ANI1x data set, containing molecules with up to 63 atoms.
For molecules containing up to  $\sim 200$  atoms, TensorNet 1L and 2L are faster or similar when compared to the ET, even when its number of message passing layers is reduced (ET optimal performance for MD17 was achieved with 6 layers, found to be one of the fastest neural network potentials in the literature [16; 45]), meaning that energy and forces on these molecules can be predicted with rMD17 state-of-the-art TensorNet models with a lower or similar computational cost than the reduced ET. For larger molecules with thousands of atoms, TensorNet 2L becomes significantly slower. However, TensorNet 1L, which still exhibits remarkable performance on rMD17 (see Table 1), performs on par the reduced ET in terms of speed even for Factor IX, containing 5807 atoms. For training on ANI1x, TensorNet 1L and 2L are faster or comparable to the ET up to a batch size of 64, being the speed for the 2L model being significantly slower for a batch size of 128. Nevertheless, the model with 1 interaction layer is still comparable to the reduced ET.
TensorNet's efficiency is given by properties that are in contrast to state-of-the-art equivariant spherical models. In particular, the use of Cartesian representations allows one to manipulate full tensors or their decomposition into scalars, vectors, and tensors at one's convenience, and Clebsch-
Table 6: Inference time. Inference time for energy and forces for single molecules (batch size of 1), in ms, on an NVIDIA GeForce RTX 4090.
Molecule	N	TensorNet 0L	TensorNet 1L	TensorNet 2L	ET 4L	ET 5L
Alanine dipeptide	22	10.0	22.1	26.5	27.2	29.0
Chignolin	166	10.4	22.5	26.9	26.8	28.9
DHFR	2489	27.3	66.9	106.7	52.4	67.7
Factor IX	5807	53.1	149.8	248.6	110.6	136.4
Table 7: Training speed. Number of batch training steps per second for AN1x dataset on an NVIDIA GeForce RTX 4090.
Batch size	TensorNet 0L	TensorNet 1L	TensorNet 2L	ET 4L	ET 5L
32	20.5	13.2	10.1	9.5	8.4
64	19.1	12.9	9.1	9.4	8.3
128	17.3	8.9	5.9	9.1	8.0
Gordan tensor products are substituted for simple  $3 \times 3$  matrix products. As detailed in the model's architecture, state-of-the-art performance can be achieved by computing these matrix products after message aggregation (that is, at the node level) and using full tensor representations, without having to individually compute products between different irreducible components. When considering an average number of neighbors per atom  $M$  controlled by the cutoff radius and the density, given that matrix products are performed after aggregation over neighbors, these do not scale with  $M$ . This is in contrast to spherical models, where tensor products are computed on edges, and therefore display a worse scaling with the number of neighbors  $M$ , that is, a worse scaling when increasing the cutoff radius at fixed density. Also, the use of higher-order many-body messages or many message-passing steps is not needed.
We have presented TensorNet, a novel O(3)-equivariant message-passing architecture leveraging Cartesian tensors and their irreducible representations. We showed that even though the model is limited to the use of rank-2 tensors, in contrast to other spherical models, it achieves state-of-the-art performance on QM9 and rMD17 with a reduced number of parameters, few message-passing steps, and it exhibits a low computational cost. Furthermore, the model is able to accurately predict vector and rank-2 tensor molecular properties on top of potential energies and forces. Nevertheless, the prediction of higher-rank quantities is directly limited by our framework. However, given the benefits of the formalism for the construction of a machine learning potential, TensorNet can be used as an alternative for the exploration of the design space of efficient equivariant models. TensorNet can be found in https://github.com/torchmd/torchmd-net.
We thank Raul P. Pelaez for fruitful discussions. GS is financially supported by Generalitat de Catalunya's Agency for Management of University and Research Grants (AGAUR) PhD grant FI2-00587. This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 823712 (RG, AV, RF); the project PID2020-116564GB-I00 has been funded by MCIN / AEI / 10.13039/501100011033. Research reported in this publication was supported by the National Institute of General Medical Sciences (NIGMS) of the National Institutes of Health under award number GM140090. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
A.1 Architecture diagrams
(a) Embedding
Figure A.1: Diagrams of the embedding and interaction and node update modules in TensorNet, described in Section 3.2. Models are built from an initial embedding module, the concatenation of several interaction layers, and the output module. The dotted line in Interaction and node update indicates the point where, after node level transformations, neighboring atom's features are used to create messages.
(b) Interaction and node update
A.2 Proofs and derivations
Parity and interaction products
Consider some tensor  $X$ , with its corresponding decomposition  $I^{X} + A^{X} + S^{X}$  and initialized as in TensorNet (see Section 3.2), that is, with vector features obtained directly by the identification of vector components with matrix entries (3). When considering the parity transformation, the components of the vector used to build the initial features flip sign, and for any full tensor initialized in this way
$$
X = I ^ {X} + A ^ {X} + S ^ {X} \rightarrow X ^ {\prime} = I ^ {X} - A ^ {X} + S ^ {X} = X ^ {\mathrm {T}}. \tag {A.1}
$$
Consider now another tensor  $Y = I^{Y} + A^{Y} + S^{Y}$  initialized in the same way, and the product  $XY = (I^{X} + A^{X} + S^{X})(I^{Y} + A^{Y} + S^{Y})$ . One can compute the resulting decomposition  $I^{XY} = \frac{1}{3}\mathrm{Tr}(XY)\mathrm{Id}$ ,  $A^{XY} = \frac{1}{2} (XY - (XY)^{\mathrm{T}}) = \frac{1}{2} (XY - Y^{\mathrm{T}}X^{\mathrm{T}})$  and  $S^{XY} = \frac{1}{2} (XY + (XY)^{\mathrm{T}} - \frac{2}{3}\mathrm{Tr}(XY)\mathrm{Id}) = \frac{1}{2} (XY + Y^{\mathrm{T}}X^{\mathrm{T}} - \frac{2}{3}\mathrm{Tr}(XY)\mathrm{Id})$  from the individual decompositions of  $X$  and  $Y$ . Taking into account that transposition is equivalent to sign reversal of skew-symmetric components, one obtains after some matrix algebra
$$
\begin{array}{l} I^{XY} = \frac{1}{3}\Bigg(\underbrace{\operatorname{Tr}(I^{X}I^{Y} + A^{X}A^{Y} + S^{X}S^{Y})}_{\text{scalar}} + \underbrace{\operatorname{Tr}(A^{X}S^{Y} + S^{X}A^{Y})}_{= 0} + \\ + \underbrace {\operatorname {T r} \left(I ^ {X} S ^ {Y} + S ^ {X} I ^ {Y} + I ^ {X} A ^ {Y} + A ^ {X} I ^ {Y}\right)} _ {= 0}) \mathrm {I d}, \tag {A.2} \\ \end{array}
$$
$$
\begin{array}{l} A ^ {X Y} = \underbrace {I ^ {X} A ^ {Y} + A ^ {X} I ^ {Y} + \frac {1}{2} (A ^ {X} S ^ {Y} - (A ^ {X} S ^ {Y}) ^ {\mathrm {T}}) + \frac {1}{2} (A ^ {Y} S ^ {X} - (A ^ {Y} S ^ {X}) ^ {\mathrm {T}})} _ {\text {v e c t o r}} + \\ + \underbrace {\frac {1}{2} \left(A ^ {X} A ^ {Y} - \left(A ^ {X} A ^ {Y}\right) ^ {\mathrm {T}}\right) + \frac {1}{2} \left(S ^ {X} S ^ {Y} - \left(S ^ {X} S ^ {Y}\right) ^ {\mathrm {T}}\right)} _ {\text {p s e u d o v e c t o r}}, \tag {A.3} \\ \end{array}
$$
$$
\begin{array}{l} S ^ {X Y} = \underbrace {I ^ {X} I ^ {Y} - \frac {1}{3} \operatorname {T r} (I ^ {X} I ^ {Y}) \mathrm {I d}} _ {= 0} + \\ + \underbrace {\frac {1}{2} \left(A ^ {X} S ^ {Y} + \left(A ^ {X} S ^ {Y}\right) ^ {\mathrm {T}} - \frac {2}{3} \operatorname {T r} \left(A ^ {X} S ^ {Y}\right) \operatorname {I d}\right) + \frac {1}{2} \left(S ^ {X} A ^ {Y} + \left(S ^ {X} A ^ {Y}\right) ^ {\mathrm {T}} - \frac {2}{3} \operatorname {T r} \left(S ^ {X} A ^ {Y}\right) \operatorname {I d}\right)} _ {\text {p s e u d o t e n s o r}} \\ + \underbrace {I ^ {X} S ^ {Y} + S ^ {X} I ^ {Y} + \frac {1}{2} \left(A ^ {X} A ^ {Y} + \left(A ^ {X} A ^ {Y}\right) ^ {\mathrm {T}} - \frac {2}{3} \operatorname {T r} \left(A ^ {X} A ^ {Y}\right) \operatorname {I d}\right) + \frac {1}{2} \left(S ^ {X} S ^ {Y} + \left(S ^ {X} S ^ {Y}\right) ^ {\mathrm {T}} - \frac {2}{3} \operatorname {T r} \left(S ^ {X} S ^ {Y}\right) \operatorname {I d}\right)} _ {\text {t e n s o r}}, \tag {A.4} \\ \end{array}
$$
where we have made explicit that the product  $XY$  gives rise to both even parity (transposing  $A^X$  and  $A^Y$  does not flip the sign of the expression) and odd parity (transposing  $A^X$  and  $A^Y$  does flip the sign of the expression) skew-symmetric and symmetric traceless contributions. However, when considering  $XY + YX$ , one gets
$$
I ^ {X Y + Y X} = \frac {1}{3} \left(\underbrace {\operatorname {T r} \left(2 I ^ {X} I ^ {Y} + A ^ {X} A ^ {Y} + \left(A ^ {X} A ^ {Y}\right) ^ {\mathrm {T}} + S ^ {X} S ^ {Y} + \left(S ^ {X} S ^ {Y}\right) ^ {\mathrm {T}}\right)} _ {\text {s c a l a r}}\right) \mathrm {I d} \tag {A.5}
$$
$$
\begin{array}{l} A ^ {X Y + Y X} = \underbrace {2 I ^ {X} A ^ {Y} + 2 A ^ {X} I ^ {Y} + \left(A ^ {X} S ^ {Y} - \left(A ^ {X} S ^ {Y}\right) ^ {\mathrm {T}}\right) + \left(A ^ {Y} S ^ {X} - \left(A ^ {Y} S ^ {X}\right) ^ {\mathrm {T}}\right)} _ {\text {v e c t o r}} + \\ + \underbrace {\frac {1}{2} (A ^ {X} A ^ {Y} - A ^ {Y} A ^ {X} + A ^ {Y} A ^ {X} - A ^ {X} A ^ {Y})} _ {= 0} + \\ + \underbrace {\frac {1}{2} \left(S ^ {X} S ^ {Y} - S ^ {Y} S ^ {X} + S ^ {Y} S ^ {X} - S ^ {X} S ^ {Y}\right)} _ {= 0}, \tag {A.6} \\ \end{array}
$$
that is, the undesired pseudovector contributions cancel out. For the symmetric traceless part pseudotensor contributions also cancel out, getting eventually
$$
\begin{array}{l} S ^ {X Y + Y X} = 2 I ^ {X} S ^ {Y} + 2 S ^ {X} I ^ {Y} + \left(A ^ {X} A ^ {Y} + \left(A ^ {X} A ^ {Y}\right) ^ {\mathrm {T}} - \frac {2}{3} \operatorname {T r} \left(A ^ {X} A ^ {Y}\right) \operatorname {I d}\right) + \\ + \left(S ^ {X} S ^ {Y} + \left(S ^ {X} S ^ {Y}\right) ^ {\mathrm {T}} - \frac {2}{3} \operatorname {T r} \left(S ^ {X} S ^ {Y}\right) \mathrm {I d}\right), \tag {A.7} \\ \end{array}
$$
which has even parity. These results can be summarized as expression (5) in the main text
$$
I ^ {X ^ {\mathrm {T}} Y ^ {\mathrm {T}} + Y ^ {\mathrm {T}} X ^ {\mathrm {T}}} = I ^ {X Y + Y X}, \quad A ^ {X ^ {\mathrm {T}} Y ^ {\mathrm {T}} + Y ^ {\mathrm {T}} X ^ {\mathrm {T}}} = - A ^ {X Y + Y X}, \quad S ^ {X ^ {\mathrm {T}} Y ^ {\mathrm {T}} + Y ^ {\mathrm {T}} X ^ {\mathrm {T}}} = S ^ {X Y + Y X}. \tag {A.8}
$$
Invariance of Frobenius norm