# Convex Relaxations of Convolutional Neural Nets

## TL;DR

This paper introduces convex relaxations for one-hidden-layer convolutional neural networks with fixed output weights, enabling efficient global optimization and revealing phase transition phenomena under certain data assumptions.

## Contribution

It develops convex second order cone relaxations for CNNs with rectified linear units and proves their ability to recover global minima under a planted model.

## Key findings

- Relaxations are convex second order cone programs solvable efficiently.
- The relaxation recovers the global minimum under Gaussian data assumptions.
- A phase transition phenomenon in global minimum recovery is identified.

## Abstract

We propose convex relaxations for convolutional neural nets with one hidden layer where the output weights are fixed. For convex activation functions such as rectified linear units, the relaxations are convex second order cone programs which can be solved very efficiently. We prove that the relaxation recovers the global minimum under a planted model assumption, given sufficiently many training samples from a Gaussian distribution. We also identify a phase transition phenomenon in recovering the global minimum for the relaxation.