The matrix multiplication (MatMul) primitive computes the product of two 2D tensors with optional bias addition (the variable names follow the standard Naming Conventions):
\[ \dst(m, n) = \sum_{k=0}^{K} \left( \src(m, k) \cdot \weights(k, n) \right) + \bias(m, n) \]
The MatMul primitive also supports batching multiple independent matrix multiplication operations, in which case the tensors must be 3D:
\[ \dst(mb, m, n) = \sum_{k=0}^{K} \left( \src(mb, m, k) \cdot \weights(mb, k, n) \right) + \bias(mb, m, n) \]
The bias tensor is optional and supports implicit broadcast semantics: any of its dimensions can be 1 and the same value would be used across the corresponding dimension. However, \(\bias\) must have the same number of dimensions as the \(\dst\).
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 |
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.
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 | f16 |
| bf16 | bf16 | bf16 | bf16, f32 |
| u8, s8 | s8, u8 | u8, s8, s32, f32 | u8, s8, s32, f32 |
The MatMul primitive expects the following tensors:
| Dims | Source | Weights | Destination | Bias |
|---|---|---|---|---|
| 2D | \(M \times K\) | \(K \times N\) | \(M \times N\) | None or \((M \text{ or } 1) \times (N \text{ or } 1)\) |
| 3D | \(MB \times M \times K\) | \(MB \times K \times N\) | \(MB \times M \times N\) | None or \((MB \text{ or } 1) \times (M \text{ or } 1) \times (N \text{ or } 1)\) |
The MatMul primitive is generally optimized for the case in which memory objects use plain memory formats (with some restrictions; see the table below). 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 table below shows the combinations of memory formats for which the MatMul primitive is optimized. The memory format of the destination tensor should always be dnnl::memory::format_tag::ab for the 2D case and dnnl::memory::format_tag::abc for the 3D one.
| Dims | Logical tensors | Mat |
|---|---|---|
| 2D | Source: \(M \times K\) Weights: \(K \times N\) | Source: dnnl_ab or dnnl_ba Weights: dnnl_ab or dnnl_ba |
| 3D | Source: \(MB \times M \times K\) Weights: \(MB \times K \times N\) | Source: dnnl_abc or dnnl_acb Weights: dnnl_abc or dnnl_acb |
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 |
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. 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}).
DNNL_ARG_ATTR_ZERO_POINTS | DNNL_ARG_SRC).u8 data type for weights.| Engine | Name | Com |
|---|---|---|
| CPU | MatMul Tutorial: Comparison with SGEMM | C++ API example demonstrating MatMul as a replacement for SGEMM functions. Concepts:
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| CPU/GPU | MatMul Tutorial: INT8 Inference | C++ API example demonstrating how one can use MatMul fused with ReLU in INT8 inference. Concepts:
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| CPU | MatMul Tutorial: Quantization | 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:
|