1 Graph Convolutional Layer (GCL) mij=ϕe(hil,hjl,aij)hil+1=ϕh(hil,∑j∈N(i)mij) permutation equivariant on the set of nodes V 2 Equivariant Graph Convolutional Layer (EGCL) mij=ϕe(hil,hjl,‖xil−xjl‖2,aij)xil+1=xil+C∑j≠i(xil−xjl)ϕx(mij)hil+1=ϕh(hil...
\qquad Equivariant Graph Convolutional Layer (EGCL)将节点嵌入\bm h^l = (\bm h^l_0 , \ldots,\bm h^l_{M-1}),坐标嵌入\bm x^l = (\bm x^l_0 , \ldots,\bm x^l_{M-1})和边信息E = (e_{ij})作为输入,并输出对\bm h^{l+1}和\bm x^{l+1}的转换。简而言之:\bm h^{...
{ij})\)denotes the corresponding convolutional filter. It should be noted that all learnable weights in the filter lie in the rotationally invariant radial functionR(rij). This radial function is implemented as a multi-layer perceptron which outputs together the radial weights for all filter-...
Specifically, we propose a spherical graph construction criterion showing that a graph needs to be regular by evenly covering the spherical surfaces in order to design a rotation equivariant graph convolutional layer. For the practical case where the perfectly regular graph does not exist, we design...
In particular, we propose two constructions, one based upon a hierarchical clustering of the domain, and another based on the spectrum of the graph Laplacian. We show through experiments that for low-dimensional graphs it is possible to learn convolutional layers with a number of parameters ...
The Allegro architecture, shown in Fig.1, is an arbitrarily deep equivariant neural network withNlayer ≥ 1 layers. The architecture learns representations associated with ordered pairs of neighboring atoms using two latent spaces: an invariant latent space, which consists of scalar (ℓ =...
including graph representation learning [1,2], especially deep generative models [3,4,5], have become increasingly popular in the field of molecular design due to their exceptional ability to learn intricate data distributions and generate novel data samples. By training on a dataset that consists...
In this paper, we propose a scale-equivariant convolutional network layer for three-dimensional data that guarantees scale-equivariance in 3D CNNs. Scale-equivariance lifts the burden of having to learn each possible scale separately, allowing the neural network to focus on higher-level learning ...
study of their performance, proving that they do not suffer from barren plateaus, quickly reach overparametrization, and generalize well from small amounts of data. To verify our results, we perform numerical simulations for a graph state classification task. Our work provides theoretical guarantees ...
then combined together into complex vectors and passed to the Wigner–Eckart layer. The Wigner–Eckart layer uses the rules in Eq. (3) and Eq. (6) to convert these vectors to tensors of the form in Eq. (4), except that the tensors here have rank 4 instead of 2. After that, the...