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Convolution {#dev_guide_op_convolution}
=======================================
## General
Convolution operation performs the convolution between src tensor and weight
tensor, which is defined as by the following formulas. Variable names follow the
standard @ref dev_guide_conventions.
Let \src, \weights and \dst tensors have shape \f$N \times IC \times IH \times
IW\f$, \f$OC \times IC \times KH \times KW\f$, and \f$N \times OC \times OH
\times OW\f$ respectively.
Furthermore, let the remaining convolution parameters be:
| Parameter | Depth | Height | Width | Comment |
|:---------------------------------|:-----------|:-----------|:-----------|:---------------------------------------------------------------------------------------|
| Paddings: Front, top, and left | \f$PD_L\f$ | \f$PH_L\f$ | \f$PW_L\f$ | In the attributes we use `pads_begin` to indicate the corresponding vector of paddings |
| Padding: Back, bottom, and right | \f$PD_R\f$ | \f$PH_R\f$ | \f$PW_R\f$ | In the attributes we use `pads_end` to indicate the corresponding vector of paddings |
| Stride | \f$SD\f$ | \f$SH\f$ | \f$SW\f$ | In the attributes we use `strides` to indicate the corresponding vector of strides |
| Dilation | \f$DD\f$ | \f$DH\f$ | \f$DW\f$ | In the attributes we use `dilations` to indicate the corresponding vector of dilations |
To further simplify the formulas, we assume that the attribute `data_format` and
`weights_format` are set to `NCX` and `OIX` respectively. `NCX` means the fist
axis represents batch dimension, the second axis represents channel dimension
and the rest represents spatial dimensions. `OIX` means the first axis
represents output channel dimension, the second axis represents input channel
dimension and the rest represents weights spatial dimensions.
### Regular Convolution
This is the same as the formula in
[Convolution primitive](@ref dev_guide_convolution).
\f[\dst(n, oc, oh, ow) = \bias(oc) \\
+ \sum_{ic=0}^{IC-1}\sum_{kh=0}^{KH-1}\sum_{kw=0}^{KW-1}
\src(n, ic, oh \cdot SH + kh - PH_L, ow \cdot SW + kw - PW_L)
\cdot
\weights(oc, ic, kh, kw).\f]
Here:
- \f$OH = \left\lfloor{\frac{IH - KH + PH_L + PH_R}{SH}} \right\rfloor + 1,\f$
- \f$OW = \left\lfloor{\frac{IW - KW + PW_L + PW_R}{SW}} \right\rfloor + 1.\f$
### Convolution with Groups
The attribute `groups` is set to \f$>1\f$.
\f[
\dst(n, g \cdot OC_G + oc_g, oh, ow) =
\bias(g \cdot OC_G + oc_g) \\
+
\sum_{ic_g=0}^{IC_G-1}\sum_{kh=0}^{KH-1}\sum_{kw=0}^{KW-1}
\src(n, g \cdot IC_G + ic_g, oh \cdot SH + kh - PH_L,
ow \cdot SW + kw - PW_L)
\cdot
\weights(g, oc_g, ic_g, kh, kw),
\f]
where
- \f$IC_G = \frac{IC}{G}\f$,
- \f$OC_G = \frac{OC}{G}\f$, and
- \f$oc_g \in [0, OC_G).\f$
### Convolution with Dilation
The attribute `dilation` contains the element which is \f$>1\f$.
\f[
\dst(n, oc, oh, ow) =
\bias(oc) \\
+
\sum_{ic=0}^{IC-1}\sum_{kh=0}^{KH-1}\sum_{kw=0}^{KW-1}
\src(n, ic, oh \cdot SH + kh \cdot DH - PH_L,
ow \cdot SW + kw \cdot DW - PW_L)
\cdot
\weights(oc, ic, kh, kw).
\f]
Here:
- \f$OH = \left\lfloor{\frac{IH - DKH + PH_L + PH_R}{SH}}
\right\rfloor + 1,\f$ where \f$DKH = 1 + (KH - 1) \cdot DH\f$, and
- \f$OW = \left\lfloor{\frac{IW - DKW + PW_L + PW_R}{SW}}
\right\rfloor + 1,\f$ where \f$DKW = 1 + (KW - 1) \cdot DW\f$.
## Operation attributes
| Attribute Name | Description | Value Type | Supported Values | Required or Optional |
|:-------------------------------------------------------------|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|:-----------|:---------------------------------------------------------------------|:---------------------|
| [strides](@ref dnnl::graph::op::attr::strides) | Controls the strides the weights tensor is moved when computing convolution | s64 | A s64 list containing positive values | Required |
| [pads_begin](@ref dnnl::graph::op::attr::pads_begin) | Controls number of zeros to be add to the front/top/left of spatial dimensions | s64 | A s64 list containing non-negative values | Required |
| [pads_end](@ref dnnl::graph::op::attr::pads_end) | Controls number of zeros to be add to the back/bottom/right of spatial dimensions | s64 | A s64 list containing non-negative values | Required |
| [dilations](@ref dnnl::graph::op::attr::dilations) | Controls the amount of stretching the kernel before convolution ([visualization link](https://github.com/vdumoulin/conv_arithmetic/blob/master/README.md#dilated-convolution-animations)) | s64 | A s64 list containing positive values (>1 means dilated convolution) | Required |
| [auto_pad](@ref dnnl::graph::op::attr::auto_pad) | Controls how the padding is calculated | string | `none` (default), `same_upper`, `same_lower`, `valid` | Optional |
| [groups](@ref dnnl::graph::op::attr::groups) | Controls how input channels and output channels are divided into | s64 | A positive s64 value, `1` by default | Optional |
| [data_format](@ref dnnl::graph::op::attr::data_format) | Controls how to interpret the shape of `src` and `dst`. | string | `NCX`, `NXC` (default) | Optional |
| [weights_format](@ref dnnl::graph::op::attr::weights_format) | Controls how to interpret the shape of `weights` | string | `OIX`, `XIO` (default) | Optional |
## Execution arguments
The inputs and outputs must be provided according to below index order when
constructing an operation.
### Inputs
| Index | Argument Name | Required or Optional |
|:------|:--------------|:---------------------|
| 0 | `src` | Required |
| 1 | `weights` | Required |
| 2 | `bias` | Optional |
@note
The shape of \weights is
\f$(out\_channels, in\_channels / groups, spatial\_shape)\f$ for `OIX` format or
\f$(spatial\_shape, in\_channels / groups, out\_channels)\f$ for `XIO` format.
Both \f$in\_channels\f$ and \f$out\_channels\f$ must be divisible by *groups*
attribute.
### Outputs
| Index | Argument Name | Required or Optional |
|:------|:--------------|:---------------------|
| 0 | `dst` | Required |
## Supported data types
Convolution operation supports the following data type combinations.
| Src | Weights | Bias | Dst |
|:-----|:--------|:-----|:-----|
| f32 | f32 | f32 | f32 |
| bf16 | bf16 | bf16 | bf16 |
| f16 | f16 | f16 | f16 |
|
