diff --git a/README.md b/README.md index 7a756244..02fd9c36 100644 --- a/README.md +++ b/README.md @@ -11,11 +11,10 @@ Instead, *escnn* supports steerable CNNs equivariant to both 2D and 3D isometrie For instance, classical convolutional neural networks (*CNN*s) are by design equivariant to translations of their input. This means that a translation of an image leads to a corresponding translation of the network's feature maps. This package provides implementations of neural network modules which are equivariant under all *isometries* E(2) of the image plane -![my equation](https://chart.apis.google.com/chart?cht=tx&chs=19&chl=\mathbb{R}^2) -and all *isometries* E(3) of the 3D space -![my equation](https://chart.apis.google.com/chart?cht=tx&chs=19&chl=\mathbb{R}^3) +$\mathbb{R}^2$ +and all *isometries* E(3) of the 3D space $\mathbb{R}^3$ , that is, under *translations*, *rotations* and *reflections* (and can, potentially, be extended to all isometries E(n) of -![my equation](https://chart.apis.google.com/chart?cht=tx&chs=19&chl=\mathbb{R}^n) +$\mathbb{R}^n$ ). In contrast to conventional CNNs, E(n)-equivariant models are guaranteed to generalize over such transformations, and are therefore more data efficient. @@ -36,10 +35,10 @@ in our [paper](https://openreview.net/forum?id=WE4qe9xlnQw), we generalize the W [A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels](https://arxiv.org/abs/2010.10952) from G-homogeneous spaces to more general spaces X carrying a G-action. In short, our method leverages a G-steerable basis for unconstrained scalar filters over the whole Euclidean space -![my equation](https://chart.apis.google.com/chart?cht=tx&chs=19&chl=\mathbb{R}^n) +$\mathbb{R}^n$ to generate steerable kernel spaces with arbitrary input and output field *types*. For example, the left side of the next image shows two elements of a SO(2)-steerable basis for functions on -![my equation](https://chart.apis.google.com/chart?cht=tx&chs=19&chl=X=\mathbb{R}^2) which are used to generate two +$X=\mathbb{R}^2$ which are used to generate two basis elements for SO(2)-equivariant steerable kernels on the right. In particular, the steerable kernels considered map a frequency l=1 vector field (2 channels) to a frequency J=2 vector field (2 channels). @@ -154,7 +153,7 @@ y = relu(conv(x)) # 15 ``` Line 5 specifies the symmetry group action on the image plane -![my equation](https://chart.apis.google.com/chart?cht=tx&chs=19&chl=\mathbb{R}^2) +$\mathbb{R}^2$ under which the network should be equivariant. We choose the [*cyclic group*](https://en.wikipedia.org/wiki/Cyclic_group)