最近被双线性搞迷糊了，看pytorch源码和数学感觉有点迷糊，所以在这里记录一下探究的结果。
Pytorch源码
class Bilinear(Module):
r"""Applies a bilinear transformation to the incoming data:
:math:`y = x_1^T A x_2 + b`
Args:
in1_features: size of each first input sample
in2_features: size of each second input sample
out_features: size of each output sample
bias: If set to False, the layer will not learn an additive bias.
Default: ``True``
Shape:
- Input1: :math:`(N, *, H_{in1})` where :math:`H_{in1}=\text{in1\_features}` and
:math:`*` means any number of additional dimensions. All but the last dimension
of the inputs should be the same.
- Input2: :math:`(N, *, H_{in2})` where :math:`H_{in2}=\text{in2\_features}`.
- Output: :math:`(N, *, H_{out})` where :math:`H_{out}=\text{out\_features}`
and all but the last dimension are the same shape as the input.
Attributes:
weight: the learnable weights of the module of shape
:math:`(\text{out\_features}, \text{in1\_features}, \text{in2\_features})`.
The values are initialized from :math:`\mathcal{U}(-\sqrt{k}, \sqrt{k})`, where
:math:`k = \frac{1}{\text{in1\_features}}`
bias: the learnable bias of the module of shape :math:`(\text{out\_features})`.
If :attr:`bias` is ``True``, the values are initialized from
:math:`\mathcal{U}(-\sqrt{k}, \sqrt{k})`, where
:math:`k = \frac{1}{\text{in1\_features}}`
Examples::
>>> m = nn.Bilinear(20, 30, 40)
>>> input1 = torch.randn(128, 20)
>>> input2 = torch.randn(128, 30)
>>> output = m(input1, input2)
>>> print(output.size())
torch.Size([128, 40])
"""
__constants__ = ['in1_features', 'in2_features', 'out_features']
in1_features: int
in2_features: int
out_features: int
weight: Tensor
def __init__(self, in1_features: int, in2_features: int, out_features: int, bias: bool = True,
device=None, dtype=None) -> None:
factory_kwargs = {'device': device, 'dtype': dtype}
super(Bilinear, self).__init__()
self.in1_features = in1_features
self.in2_features = in2_features
self.out_features = out_features
self.weight = Parameter(torch.empty((out_features, in1_features, in2_features), **factory_kwargs))
if bias:
self.bias = Parameter(torch.empty(out_features, **factory_kwargs))
else:
self.register_parameter('bias', None)
self.reset_parameters()
def reset_parameters(self) -> None:
bound = 1 / math.sqrt(self.weight.size(1))
init.uniform_(self.weight, -bound, bound)
if self.bias is not None:
init.uniform_(self.bias, -bound, bound)
def forward(self, input1: Tensor, input2: Tensor) -> Tensor:
return F.bilinear(input1, input2, self.weight, self.bias)
def extra_repr(self) -> str:
return 'in1_features={}, in2_features={}, out_features={}, bias={}'.format(
self.in1_features, self.in2_features, self.out_features, self.bias is not None
)
这里虽然说了\(x_1\) 要转置，但是没说怎么转置，而且如果直接实验的话，会发现按照数学公式进行两次乘法是会报错的，
所以我们从forward入手，看看torch.nn.functional.bilinear的定义:
def bilinear(input1: Tensor, input2: Tensor, weight: Tensor, bias: Optional[Tensor] = None) -> Tensor:
r"""
Applies a bilinear transformation to the incoming data:
:math:`y = x_1^T A x_2 + b`
Shape:
- input1: :math:`(N, *, H_{in1})` where :math:`H_{in1}=\text{in1\_features}`
and :math:`*` means any number of additional dimensions.
All but the last dimension of the inputs should be the same.
- input2: :math:`(N, *, H_{in2})` where :math:`H_{in2}=\text{in2\_features}`
- weight: :math:`(\text{out\_features}, \text{in1\_features},
\text{in2\_features})`
- bias: :math:`(\text{out\_features})`
- output: :math:`(N, *, H_{out})` where :math:`H_{out}=\text{out\_features}`
and all but the last dimension are the same shape as the input.
"""
if has_torch_function_variadic(input1, input2, weight):
return handle_torch_function(
bilinear,
(input1, input2, weight),
input1, input2, weight,
bias=bias
)
return torch.bilinear(input1, input2, weight, bias)
这里其实没有matmul的操作，后续也难以追踪了，而且官方文档写的不是特别清楚。所以这里我们来做做实验。
实验
输入为二维
输入为三维
输入为四维
通过实验可以确定：
- bilinear的实现应该不是直接使用matmul，猜测应该是weight有broadcast之后才能正常计算乘法
- 只要前面有不一样的维度就会报错，除了最后一维，必须全部一致才行