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 \(N \times IC\), \(OC \times IC\), \(OC\), and \(N \times OC\) tensors, respectively (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 cases where the \(\src\) and \(\weights\) tensors have spatial dimensions, they are flattened to 2D. For example, if they are 4D \(N \times IC' \times IH \times IW\) and \(OC \times IC' \times KH \times KW\) tensors, then the formula above is applied with \(IC = IC' \cdot IH \cdot IW\). In such cases, the \(\src\) and \(\weights\) tensors must have equal spatial dimensions (e.g. \(KH = IH\) and \(KW = IW\) for 4D tensors).
There is no difference between the dnnl::prop_kind::forward_training and dnnl::prop_kind::forward_inference propagation kinds.
The backward propagation computes \(\diffsrc\) based on \(\diffdst\) and \(\weights\).
The weights update computes \(\diffweights\) and \(\diffbias\) based on \(\diffdst\) and \(\src\).
When executed, the inputs and outputs should be mapped to an execution argument index as specified by the following table.
| Primitive input/output | Execution argument index |
|---|---|
| \(\src\) | DNNL_ARG_SRC |
| \(\weights\) | DNNL_ARG_WEIGHTS |
| \(\bias\) | DNNL_ARG_BIAS |
| \(\dst\) | DNNL_ARG_DST |
| \(\diffsrc\) | DNNL_ARG_DIFF_SRC |
| \(\diffweights\) | DNNL_ARG_DIFF_WEIGHTS |
| \(\diffbias\) | DNNL_ARG_DIFF_BIAS |
| \(\diffdst\) | DNNL_ARG_DIFF_DST |
N/A.
Inner product primitive supports the following combination of data types for source, destination, weights, and bias:
| Propagation | Source | Weights | Destination | Bias |
|---|---|---|---|---|
| forward / backward | f32 | f32 | f32 | f32 |
| forward | f16 | f16 | f16 | f16 |
| forward | u8, s8 | s8 | u8, s8, s32, f32 | u8, s8, s32, f32 |
| forward | bf16 | bf16 | f32, bf16 | f32, bf16 |
| backward | f32, bf16 | bf16 | bf16 | |
| weights update | bf16 | f32, bf16 | bf16 | f32, bf16 |
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 DNNL programming model, inner product primitive is one of the few primitives that support the placeholder format dnnl::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 dnnl::memory::format_tag::nc (dnnl::memory::format_tag::ab).
| Spatial | Source / Weights logical tensor | Imp |
|---|---|---|
| 0D | NC / OI | dnnl_nc (dnnl_ab) / dnnl_oi (dnnl_ab) |
| 0D | NC / OI | dnnl_nc (dnnl_ab) / dnnl_io (dnnl_ba) |
| 1D | NCW / OIW | dnnl_ncw (dnnl_abc) / dnnl_oiw (dnnl_abc) |
| 1D | NCW / OIW | dnnl_nwc (dnnl_acb) / dnnl_wio (dnnl_cba) |
| 2D | NCHW / OIHW | dnnl_nchw (dnnl_abcd) / dnnl_oihw (dnnl_abcd) |
| 2D | NCHW / OIHW | dnnl_nhwc (dnnl_acdb) / dnnl_hwio (dnnl_cdba) |
| 3D | NCDHW / OIDHW | dnnl_ncdhw (dnnl_abcde) / dnnl_oidhw (dnnl_abcde) |
| 3D | NCDHW / OIDHW | dnnl_ndhwc (dnnl_acdeb) / dnnl_dhwio (dnnl_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 |