# Efficient Long-Range Convolutions for Point Clouds

arXiv: https://arxiv.org/abs/2010.05295 · HTML: https://mlsanity.com/paper/2010.05295/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2020.

## Built on by (most cited first, 3 of 3)

- [Physics-inspired Equivariant Descriptors of Non-bonded Interactions](https://mlsanity.com/paper/2308.13208/) (2023)
- [Solving the Wide-band Inverse Scattering Problem via Equivariant Neural Networks](https://mlsanity.com/paper/2212.06068/) (2022)
- [GeoSpark: Sparking up Point Cloud Segmentation with Geometry Clue](https://mlsanity.com/paper/2303.08274/) (2023)

## Abstract (arXiv)

The efficient treatment of long-range interactions for point clouds is a challenging problem in many scientific machine learning applications. To extract global information, one usually needs a large window size, a large number of layers, and/or a large number of channels. This can often significantly increase the computational cost. In this work, we present a novel neural network layer that directly incorporates long-range information for a point cloud. This layer, dubbed the long-range convolutional (LRC)-layer, leverages the convolutional theorem coupled with the non-uniform Fourier transform. In a nutshell, the LRC-layer mollifies the point cloud to an adequately sized regular grid, computes its Fourier transform, multiplies the result by a set of trainable Fourier multipliers, computes the inverse Fourier transform, and finally interpolates the result back to the point cloud. The resulting global all-to-all convolution operation can be performed in nearly-linear time asymptotically with respect to the number of input points. The LRC-layer is a particularly powerful tool when combined with local convolution as together they offer efficient and seamless treatment of both short and long range interactions. We showcase this framework by introducing a neural network architecture that combines LRC-layers with short-range convolutional layers to accurately learn the energy and force associated with a N-body potential. We also exploit the induced two-level decomposition and propose an efficient strategy to train the combined architecture with a reduced number of samples.