Mamba-minimal Mamba的最小限度实现 （一）

johnma2006/mamba-minimal: Simple, minimal implementation of the Mamba SSM in one file of PyTorch. (github.com)

manba的简单最小限度实现，和原始论文实现state-spaces/mamba (github.com)](https://github.com/state-spaces/mamba/tree/main)相比，为了可读性对参数没有很好的初始化，原论文用CUDA写了并行扫描，所以速度会快。
这里介绍Mamba Block的实现

参数和数据尺寸约定

之后的数据尺寸以(b, l, d_in) 或者(b, l, d_model, d_state)简单表示

class MambaBlock

def forward

根据forward简单梳理MambaBlock的结构

def forward(self, x):
  
        (b, l, d) = x.shape
        
        x_and_res = self.in_proj(x)  # shape (b, l, 2 * d_in)
        (x, res) = x_and_res.split(split_size=[self.args.d_inner, self.args.d_inner], dim=-1)

        x = rearrange(x, 'b l d_in -> b d_in l')
        x = self.conv1d(x)[:, :, :l]
        x = rearrange(x, 'b d_in l -> b l d_in')
        
        x = F.silu(x)

        y = self.ssm(x)
        
        y = y * F.silu(res)
        
        output = self.out_proj(y)

        return output
    
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def __ int__

初始化主要初始了几个部分

组件定义

ssm初始化

def __init__(self, args: ModelArgs):

        super().__init__()
        self.args = args

        self.in_proj = nn.Linear(args.d_model, args.d_inner * 2, bias=args.bias)

        self.conv1d = nn.Conv1d(
            in_channels=args.d_inner,
            out_channels=args.d_inner,
            bias=args.conv_bias,
            kernel_size=args.d_conv,
            groups=args.d_inner,
            padding=args.d_conv - 1,
        )
        
         # ssm模型的初始化部分
         # x_proj takes in `x` and outputs the input-specific Δ, B, C
        self.x_proj = nn.Linear(args.d_inner, args.dt_rank + args.d_state * 2, bias=False)
        
        # dt_proj projects Δ from dt_rank to d_in
        self.dt_proj = nn.Linear(args.dt_rank, args.d_inner, bias=True)

        A = repeat(torch.arange(1, args.d_state + 1), 'n -> d n', d=args.d_inner)
        self.A_log = nn.Parameter(torch.log(A))
        self.D = nn.Parameter(torch.ones(args.d_inner))
        self.out_proj = nn.Linear(args.d_inner, args.d_model, bias=args.bias)
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def ssm

这是我们数据处理流水线的搭建，这一部分是ssm模型参数定义，是ssm模型中相对于数据“不变”的部分。

其中一部分变量初始化于class MambaBlock的初始化部分

 def ssm(self, x):

        (d_in, n) = self.A_log.shape
        
        A = -torch.exp(self.A_log.float())  # shape (d_in, n)
        D = self.D.float()

        x_dbl = self.x_proj(x)  # (b, l, dt_rank + 2*n)
        
        (delta, B, C) = x_dbl.split(split_size=[self.args.dt_rank, n, n], dim=-1)  # delta: (b, l, dt_rank). B, C: (b, l, n)
        delta = F.softplus(self.dt_proj(delta))  # (b, l, d_in)
        
        y = self.selective_scan(x, delta, A, B, C, D)  # This is similar to run_SSM(A, B, C, u) in The Annotated S4 [2]
        
        return y
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def selective_scan

我们的数据流水线搭建好以后，接下来就要让它动起来，这一部分是数据处理的动态或者动力。

在这里，$A$使用ZOH零阶保持离散化，$B$则简化为欧拉离散化

前向欧拉离散化
x k = ( I + Δ k A ) x k − 1 + Δ k B ⋅ u k x ( t + Δ ) = ( I + Δ A ) x ( t ) + Δ B ⋅ u ( t )

$$\begin{aligned} x_{k}& \begin{aligned}=(\boldsymbol{I}+\Delta_{k}\boldsymbol{A})x_{k-1}+\Delta_{k}\boldsymbol{B}\cdot u_{k}\end{aligned}$$ \\ x(t+\Delta)& =(\boldsymbol{I}+\Delta\boldsymbol{A})x(t)+\Delta\boldsymbol{B}\cdot u(t) \end{aligned} xk​x(t+Δ)​=(I+Δk​A)xk−1​+Δk​B⋅uk​​=(I+ΔA)x(t)+ΔB⋅u(t)​

零阶保持离散化
x k = e Δ k A x k − 1 + ( Δ k A ) − 1 ( e Δ k A − I ) ⋅ Δ k B ⋅ u k x ( t + Δ ) = e Δ A x ( t ) + ( Δ A ) − 1 ( e Δ A − I ) ⋅ Δ B ⋅ u ( t )

$$\begin{aligned} x_{k}& =e^{\Delta_{k}\boldsymbol A}x_{k-1}+(\Delta_{k}\boldsymbol A)^{-1}(e^{\Delta_{k}\boldsymbol A}-\boldsymbol{I})\cdot\Delta_{k}\boldsymbol B\cdot u_{k} \\ x(t+\Delta)& =e^{\Delta \boldsymbol A}x(t)+(\Delta \boldsymbol A)^{-1}(e^{\Delta \boldsymbol A}-\boldsymbol{I})\cdot\Delta \boldsymbol B\cdot u(t) \end{aligned}$$ xk​x(t+Δ)​=eΔk​Axk−1​+(Δk​A)−1(eΔk​A−I)⋅Δk​B⋅uk​=eΔAx(t)+(ΔA)−1(eΔA−I)⋅ΔB⋅u(t)​

这里selective_scan是顺序形式，因此与原论文CUDA编写的并行感知算法相比要慢

def selective_scan(self, u, delta, A, B, C, D):

        (b, l, d_in) = u.shape
        n = A.shape[1]
        

        deltaA = torch.exp(einsum(delta, A, 'b l d_in, d_in n -> b l d_in n'))
        deltaB_u = einsum(delta, B, u, 'b l d_in, b l n, b l d_in -> b l d_in n')
        
        # Perform selective scan (see scan_SSM() in The Annotated S4 [2])

        x = torch.zeros((b, d_in, n), device=deltaA.device)
        ys = []    
        for i in range(l):
            x = deltaA[:, i] * x + deltaB_u[:, i]
            y = einsum(x, C[:, i, :], 'b d_in n, b n -> b d_in')
            ys.append(y)
        y = torch.stack(ys, dim=1)  # shape (b, l, d_in)
        
        y = y + u * D
    
        return y

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