The inner product primitive (sometimes called fully connected) treats each activation in the minibatch as a vector and computes its product with a weights 2D tensor producing a 2D tensor as an output.
More precisely, let \(src\), \(weights\), \(bias\) and \(dst\) be \(N \times IC\), \(OC \times IC\), \(OC\), \(N \times OC\) tensors (the variable names follow the standard Naming Conventions). Then:
\[dst(n, oc) = bias(oc) + \sum_{ic=0}^{IC-1} src(n, ic) \cdot weights(oc, ic)\]
In case when the \(src\) tensor has spatial dimension it is flattened to 2D. For example, if it is a 4D \(N \times IC' \times IH \times IW\) tensor, then the formula above is applied with \(IC = IC' \cdot IH \cdot IW\).
There is no difference between the mkldnn::forward_training and mkldnn::forward_inference propagation kinds.
The backward propagation computes \(diff\_src\) based on \(diff\_dst\) and \(weights\).
The weights update computes \(diff\_weights\) and \(diff\_bias\) based on \(diff\_dst\) and \(src\).
N/A.
Inner product primitive supports the following combination of data types for source, destination, weights, and bias:
| Propagation | Source | Weights | Destination | Bia |
|---|---|---|---|---|
| forward / backward | f32 | f32 | f32 | f32 |
| forward | f16 | f16 | f16 | f16 |
| forward | u8, s8 | s8 | u8, s8, s32, f32 | u8, s8, s32, f32 |
Like other CNN primitives, the inner product primitive expects the following tensors:
| Spatial | Source | Destination | Wei |
|---|---|---|---|
| 1D | \(N \times C \times W\) | \(N \times C\) | \(OC \times IC \times KW\) |
| 2D | \(N \times C \times H \times W\) | \(N \times C\) | \(OC \times IC \times KH \times KW\) |
| 3D | \(N \times C \times D \times H \times W\) | \(N \times C\) | \(OC \times IC \times KD \times KH \times KW\) |
Memory format of data and weights memory objects is critical for inner product primitive performance. In the Intel MKL-DNN programming model, inner product primitive is one of the few primitives that support the placeholder format mkldnn::memory::format_tag::any (shortened to any from now on) and can define data and weight memory objects formats based on the primitive parameters. When using any it is necessary to first create an inner product primitive descriptor and then query it for the actual data and weight memory objects formats.
The table below shows the combinations for which plain memory formats the inner product primitive is optimized for. For the destination tensor (which is always \(N \times C\)) the memory format is always mkldnn::memory::format_tag::nc (mkldnn::memory::format_tag::ab).
| Spatial | Source / Weights logical tensor | Imp |
|---|---|---|
| 0D | NC / OI | mkldnn_nc (mkldnn_ab) / mkldnn_oi (mkldnn_ab) |
| 0D | NC / OI | mkldnn_nc (mkldnn_ab) / mkldnn_io (mkldnn_ba) |
| 1D | NCW / OIW | mkldnn_ncw (mkldnn_abc) / mkldnn_oiw (mkldnn_abc) |
| 1D | NCW / OIW | mkldnn_nwc (mkldnn_acb) / mkldnn_wio (mkldnn_cba) |
| 2D | NCHW / OIHW | mkldnn_nchw (mkldnn_abcd) / mkldnn_oihw (mkldnn_abcd) |
| 2D | NCHW / OIHW | mkldnn_nhwc (mkldnn_acdb) / mkldnn_hwio (mkldnn_cdba) |
| 3D | NCDHW / OIDHW | mkldnn_ncdhw (mkldnn_abcde) / mkldnn_oidhw (mkldnn_abcde) |
| 3D | NCDHW / OIDHW | mkldnn_ndhwc (mkldnn_acdeb) / mkldnn_dhwio (mkldnn_cdeba) |
Post-ops and attributes enable you to modify the behavior of the inner product primitive by chaining certain operations after the inner product operation. The following post-ops are supported by inner product primitives:
| Propagation | Type | Operation | Des |
|---|---|---|---|
| forward | post-op | eltwise | Applies an Eltwise operation to the result (currently only mkldnn_eltwise_relu algorithm is supported) |