Performance library for Deep Learning
1.96.0
Inner Product

API Reference

## General

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.

### Forward

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).

#### Difference Between Forward Training and Forward Inference

There is no difference between the dnnl::prop_kind::forward_training and dnnl::prop_kind::forward_inference propagation kinds.

### Backward

The backward propagation computes $$\diffsrc$$ based on $$\diffdst$$ and $$\weights$$.

The weights update computes $$\diffweights$$ and $$\diffbias$$ based on $$\diffdst$$ and $$\src$$.

Note
The optimized memory formats $$\src$$ and $$\weights$$ might be different on forward propagation, backward propagation, and weights update.

## Execution Arguments

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
$$binary post-op$$ DNNL_ARG_ATTR_MULTIPLE_POST_OP(binary_post_op_position) | DNNL_ARG_SRC_1

## Implementation Details

N/A.

### Data Types

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

### Data Representation

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 oneDNN 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

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 Description Restrictions
forward attribute Output scale Scales the result of inner product by given scale factor(s) int8 inner products only
forward post-op Eltwise Applies an Eltwise operation to the result
forward post-op Sum Adds the operation result to the destination tensor instead of overwriting it
forward post-op Binary Applies a Binary operation to the result General binary post-op restrictions

## Implementation Limitations

1. Check Data Types.

## Performance Tips

• Use dnnl::memory::format_tag::any for source, weights, and destinations memory format tags when create an inner product primitive to allow the library to choose the most appropriate memory format.

## Examples

Engine Name Com
CPU/GPU Inner Product Primitive Example

This C++ API example demonstrates how to create and execute an Inner Product primitive.

Key optimizations included in this example:

• Primitive attributes with fused post-ops;
• Creation of optimized memory format from the primitive descriptor.