# Matrix Multiplication¶

API Reference

## General¶

The matrix multiplication (MatMul) primitive computes the product of two 2D tensors with optional bias addition (the variable names follow the standard Naming Conventions):

$\dst(m, n) = \sum_{k=0}^{K - 1} \left( \src(m, k) \cdot \weights(k, n) \right) + \bias(m, n)$

The MatMul primitive also supports batching multiple independent matrix multiplication operations, in which case the tensors can be up to 12D:

$\dst(bs_0, bs_1, \ldots, m, n) = \sum_{k=0}^{K - 1} \left( \src(bs_0, bs_1, \ldots, m, k) \cdot \weights(bs_0, bs_1, \ldots, k, n) \right) + \bias(bs_0, bs_1, \ldots, m, n)$

MatMul also supports implicit broadcast semantics i.e., $$\src$$ can be broadcasted into $$\weights$$ if the corresponding dimension in $$\src$$ is 1 (and vice versa). However, all tensors (including $$\bias$$, if it exists) must have the same number of dimensions.

The shape of $$\dst$$ only depends on $$\src$$ and $$\weights$$ tensors. The $$\bias$$ cannot change the dimensions of $$\dst$$ by broadcasting. In other words, for every dimension, the following constraint must hold true: dimension(bias) == dimension(dst) || dimension(bias) == 1.

## 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

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

DNNL_ARG_ATTR_MULTIPLE_POST_OP(binary_post_op_position) | DNNL_ARG_SRC_1

## Implementation Details¶

### General Notes¶

1. The MatMul primitive supports input and output tensors with run-time specified shapes and memory formats. The run-time specified dimensions or strides are specified using the DNNL_RUNTIME_DIM_VAL wildcard value during the primitive initialization and creation stage. At the execution stage, the user must pass fully specified memory objects so that the primitive is able to perform the computations. Note that the less information about shapes or format is available at the creation stage, the less performant execution will be. In particular, if the shape is not known at creation stage, one cannot use the special format tag dnnl::memory::format_tag::any to enable an implementation to choose the most appropriate memory format for the corresponding input or output shapes. On the other hand, run-time specified shapes enable users to create a primitive once and use it in different situations.

2. Inconsistency with dimensions being “primitive-creation-time-defined” vs “runtime-defined” is invalid. For example, $$\src$$ and $$\weights$$ with dimensions set to {3, 4, 4} and {DNNL_RUNTIME_DIM_VAL, 4, 4} respectively is invalid.

3. The broadcasting shape consistency check is not done for the dimensions with DNNL_RUNTIME_DIM_VAL. It is user responsibility to make sure the dimensions for the tensors are valid.

4. Multiple batch dimensions and broadcasting of batch dimensions of src and weights are supported for both CPU and GPU engines.

Please check tutorials below to see DNNL_RUNTIME_DIM_VAL support in use.

### Data Types¶

The MatMul primitive supports the following combinations of data types for source, destination, weights, and bias tensors:

Source

Weights

Destination

Bias

f32

f32

f32

f32

f16

f16

f16, u8, s8

f16

bf16

bf16

f32, bf16

bf16, f32

u8, s8

u8, s8

u8, s8, s32, f32, bf16

u8, s8, s32, f32, bf16

### Data Representation¶

The MatMul primitive expects the following tensors:

Dims

Source

Weights

Destination

Bias

2D

M $$\times$$ K

K $$\times$$ N

M $$\times$$ N

None or $$(M \text{ or } 1) \times (N \text{ or } 1)$$

ND

S $$\times$$ M $$\times$$ K

W $$\times$$ K $$\times$$ N

D $$\times$$ M $$\times$$ N

None or B

where for the sake of notational convenience, we have

$\begin{split}S = \prod_{i = 0}^{ND - 3} \mathrm{src\_dims}[i], \; W = \prod_{i = 0}^{ND - 3} \mathrm{weights\_dims}[i] \\ D = \prod_{i = 0}^{ND - 3} \mathrm{\dst\_dims}[i], \; B = \prod_{i = 0}^{ND - 1} \left( \mathrm{\dst\_dims}[i] \mbox{ or } 1 \right)\end{split}$

The MatMul primitive is generally optimized for the case in which memory objects use plain memory formats. Additionally, the $$\src$$ and $$\weights$$ must have at least one of the axes m or k and n or k contiguous (i.e., stride=1) respectively. However, it is recommended to use the placeholder memory format dnnl::memory::format_tag::any if an input tensor is reused across multiple executions. In this case, the primitive will set the most appropriate memory format for the corresponding input tensor.

The memory format of the destination tensor should always be plain with n axis contiguous. For example, dnnl::memory::format_tag::ab for the 2D case and dnnl::memory::format_tag::abc or dnnl::memory::format_tag::bac for the 3D one.

### Attributes and Post-ops¶

Attributes and post-ops enable modifying the behavior of the MatMul primitive. The following attributes and post-ops are supported:

Type

Operation

Description

Restrictions

Attribute

Output scales

Scales the result by given scale factor(s)

Attribute

Zero points

Sets zero point(s) for the corresponding tensors

Int8 computations only

Post-op

Eltwise

Applies an Eltwise operation to the result

Post-op

Sum

Adds the operation result to the destination tensor instead of overwriting it

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.

Similarly to run-time output scales, the primitive supports run-time zero points. The wildcard value for zero points is DNNL_RUNTIME_S32_VAL. The following masks are supported by the primitive:

• 0, which applies one zero point value to an entire tensor, and

• 2, which applies a zero point value per each element in a k or n dimension for DNNL_ARG_SRC or DNNL_ARG_DST arguments respectively.

During the execution stage, the corresponding memory object needs to be passed in the argument with index set to (DNNL_ARG_ATTR_ZERO_POINTS | DNNL_ARG_\${MEMORY_INDEX}).

• For instance, source tensor zero points memory argument would be passed with index (DNNL_ARG_ATTR_ZERO_POINTS | DNNL_ARG_SRC).

Note

Please check tutorials below to see run-time attributes in use.

## Implementation Limitations¶

1. Check Data Types.

2. The CPU engine does not support u8 data type for weights.

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

4. GPU implementation is limited to 6D and plain memory formats.

## Performance Tips¶

• Use dnnl::memory::format_tag::any for either of the input tensors if and only if the shape of the corresponding tensor is fully known at creation time and it is possible to cache reordered tensors across multiple primitive executions. For instance, a good candidate for reuse are the weights tensors during inference: their shapes and data types are known in advance; thus they can be reordered during the first inference pass and can be reused during the subsequent passes. However, if any of the input tensors cannot be reused, it is best to force the primitive to use the same format as that used by the tensors.

## Examples¶

### matmul_example_cpp - CPU/GPU¶

This C++ API example demonstrates how to create and execute a MatMul primitive.

Key optimizations included in this example:

• Primitive attributes with fused post-ops.

### cpu_sgemm_and_matmul_cpp - CPU¶

C++ API example demonstrating MatMul as a replacement for SGEMM functions.

Concepts:

### inference_int8_matmul_cpp - CPU/GPU¶

C++ API example demonstrating how one can use MatMul fused with ReLU in INT8 inference.

Concepts:

### cpu_matmul_quantization_cpp - CPU¶

C++ API example demonstrating how one can perform reduced precision matrix-matrix multiplication using MatMul and the accuracy of the result compared to the floating point computations.

Concepts:

• Static and dynamic quantization

• Asymmetric quantization