Learning Two Layer Rectified Neural Networks in Polynomial Time

Understand

Consider the following fundamental learning problem: given input examples $x \in \mathbb{R}^d$ and their vector-valued labels, as defined by an underlying generative neural network, recover the weight matrices of this network.

• We consider two-layer networks, mapping $\mathbb{R}^d$ to $\mathbb{R}^m$, with $k$ non-linear activation units $f(\cdot)$, where $f(x) = \max \{x , 0\}$ is the ReLU.
• Such a network is specified by two weight matrices, $\mathbf{U}^* \in \mathbb{R}^{m \times k}, \mathbf{V}^* \in \mathbb{R}^{k \times d}$, such that the label of an example $x \in \mathbb{R}^{d}$ is given by $\mathbf{U}^* f(\mathbf{V}^* x)$, where $f(\cdot)$ is applied coordinate-wise.
• Given $n$ samples as a matrix $\mathbf{X} \in \mathbb{R}^{d \times n}$ and the (possibly noisy) labels $\mathbf{U}^* f(\mathbf{V}^* \mathbf{X}) + \mathbf{E}$ of the network on these samples, where $\mathbf{E}$ is a noise matrix, our goal is to recover the weight matrices $\mathbf{U}^*$ and $\mathbf{V}^*$.