: Spirals vs soft-attention operators[]{data-label="spiral_vs_attn"}

### Comparison to Lim et al. [@spirals] {#spiral_comaparison}

In order to showcase how the operator behaves when the ordering is not consistent, we perform experiments under four scenarios: the original formulation of [@spirals], where each spiral is randomly oriented for every mesh and every epoch (*rand mesh & epoch)*; choosing the same orientation across all the meshes randomly at every epoch (*rand epoch*); choosing different orientations for every mesh, but keeping them fixed across epochs (*rand mesh*); and fixed ordering (Ours). We compare the LSTM-based approach of [@spirals] and our linear projection formulation (Eq ). The experimental setting and architecture is the same as in the previous section. The proposed approach achieves over 28% improved performance compared to [@spirals], which substantiates the benefits of passing corresponding points through the same transformations.

  operation     rand mesh & epoch   rand mesh   rand epoch    fixed ordering
  ------------ ------------------- ----------- ------------ ------------------
  LSTM          0.888 [@spirals]      0.880       0,996           0.792
  lin. proj.          0.829           0.825       0.951      **0.635 (Ours)**

  : Importance of the ordering consistency

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Neural 3D Morphable models {#3dmm_comparison}
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![Colour coding of the per vertex euclidean error of the reconstructions produced by PCA (2nd), COMA (3rd), and our Neural3DMM (bottom). Top 