# 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

$$\text{binary post-op}$$

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, u8, s8

f16

forward

u8, s8

s8

u8, s8, s32, bf16, f32

u8, s8, s32, bf16, 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

Weights

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

Implementation optimized for memory formats

0D

NC / OI

0D

NC / OI

1D

NCW / OIW

1D

NCW / OIW

2D

NCHW / OIHW

2D

NCHW / OIHW

3D

NCDHW / OIDHW

3D

NCDHW / OIDHW

### 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

To facilitate dynamic quantization, the primitive supports run-time output scales. That means a user could configure attributes with output scales set to the DNNL_RUNTIME_F32_VAL wildcard value instead of the actual scales, if the scales are not known at the primitive descriptor creation stage. In this case, the user must provide the scales as an additional input memory object with argument DNNL_ARG_ATTR_OUTPUT_SCALES during the execution stage.

## Implementation Limitations¶

1. Check Data Types.

2. The CPU engine does not support u8 or s8 data type for dst with f16 src and weights.

## 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.

## Example¶

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.