 Pointwise convolution

A pointwise convolution uses a spatial kernel of size one to combine neural network channels independently at each spatial position. For a two-dimensional input this is a one-by-one convolutional layer. In the ordinary dense-channel, unit-stride case, the same affine map acts on the channel vector at every grid point.^[ref]

Let $x_{i,j}\in {\mathbb{R}}^{C_{\mathrm{in}}}$ be the channel vector at position $(i,j)$. With weight matrix $M$ and neural network bias vector $b$, the output isThis is matrix multiplication across channels, not an average of neighboring spatial positions. The same $M$ and $b$ are reused throughout the grid.

For an example without biases, takeAt a point with input $(1,2,3{)}^{\mathsf{T}}$ the output is $(5,1{)}^{\mathsf{T}}$. At another point with input $(3,0,4{)}^{\mathsf{T}}$ it is $(3,4{)}^{\mathsf{T}}$. A three-channel neural network feature map has become a two-channel map without mixing the two positions.

There are $C_{\mathrm{in}}{C}_{\mathrm{out}}$ weights, plus $C_{\mathrm{out}}$ optional biases. Thus three input and two output channels require six weights, or eight parameters including biases. A dense three-by-three filter with the same channel counts requires $9(3)(2)=54$ weights, or fifty-six parameters with biases. This comparison counts parameters for the stated connectivity; it does not claim that the two layers compute equivalent functions.

Channel reduction can make a later spatial convolutional layer cheaper. The original Inception uses one-by-one reductions before some larger filters. A pointwise operation can also increase the channel count; size one refers to its spatial footprint, not to the number of channels.^[ref] With no intervening spatial operation, pointwise layers and pointwise activation functions alone do not enlarge the spatial neural network receptive field.