A 1x1 convolution or a network in network is an architectural technique used in some convolutional neural networks.

The technique was first described in the paper Network In Network.

A 1x1 convolution is a convolutional layer where the filter is of dimension $1\times 1$.

The filter takes in a tensor of dimension $n_h\times n_w\times n_c$, over the $n_c$ values in the third dimension and outputting a $n_h\times n_w$ matrix. Subsequently, an activation function (like ReLU) is applied to the output matrix.

If we have $p$ $1\times 1$ filters, then the output of the layer is a tensor of dimension $n_h\times n_w\times p$. This is useful if the number of channels $n_c$ in the previous layer of the network has grown too large and needs to be altered to $p$ channels.

The $1\times 1$ convolution technique was featured in paper introducing the Inception network architecture, titled Going Deeper With Convolutions.