as point clouds, where an ordering of the data points can be enforced.

Neural 3D Morphable Models
--------------------------

Let $\bm{F}=[\bm{f_0}| \bm{f_1}|..., \bm{f_N}]$, $\bm{f_i} \in \mathbb{R}^{d*m}$ the matrix of all the signals defined on a set of meshes in dense correspondence that are sampled from a distribution $\mathcal{D}$, where $d$ the dimensionality of the signal on the mesh (vertex position, texture etc.) and $m$ the number of vertices. A linear 3D Morphable Model [@vetterMM] represents arbitrary instances $\mathbf{y}\in \mathcal{D}$ as a linear combination of the k largest eigenvectors of the covariance matrix of $\bm{F}$ by making a gaussianity assumption: $$\mathbf{y} \approx \Bar{\bm{f}} + \sum_{i}^k \alpha_i \sqrt{d_i} \mathbf{v_i}$$

where $\Bar{\bm{f}}$ the mean shape, $\mathbf{v_i}$ is the $i$th principal component, $d_i$ the respective eigenvalue and $\alpha_i$ the linear weight coefficient. Given its linear formulation, the representational power of the 3DMM is constrained by the span of the eigenvectors, while its parameters scale linearly w.r.t the number of the eigencomponents used, leading to large parametrisations for meshes of high resolution.

In contrast, in this paper, we use spiral convolutions as a building block to build a fully differentiable non-linear Morphable Model. In essence, a Neural 3D Morphable Model is a deep convolutional mesh autoencoder, 