  Matrix completion problems arise in many applications including
recommendation systems, computer vision, and genomics. Increasingly larger
neural networks have been successful in many of these applications, but at
considerable computational costs. Remarkably, taking the width of a neural
network to infinity allows for improved computational performance. In this
work, we develop an infinite width neural network framework for matrix
completion that is simple, fast, and flexible. Simplicity and speed come from
the connection between the infinite width limit of neural networks and kernels
known as neural tangent kernels (NTK). In particular, we derive the NTK for
fully connected and convolutional neural networks for matrix completion. The
flexibility stems from a feature prior, which allows encoding relationships
between coordinates of the target matrix, akin to semi-supervised learning. The
effectiveness of our framework is demonstrated through competitive results for
virtual drug screening and image inpainting/reconstruction. We also provide an
implementation in Python to make our framework accessible on standard hardware
to a broad audience.
