分数函数
设 q ( x ) q(x) q ( x )
s q ( x ) = ∇ x log q ( x ) s_q(x) = \nabla_x \log q(x)
s q ( x ) = ∇ x log q ( x )
直接使用 ∇ x q ( x ) \nabla_x q(x) ∇ x q ( x ) ∇ x log q ( x ) = ∇ x q ( x ) / q ( x ) \nabla_x\log q(x)=\nabla_xq(x)/q(x) ∇ x log q ( x ) = ∇ x q ( x ) / q ( x ) q ( x ) = N ( μ , σ 2 I ) q(x)=\mathcal N(\mu,\sigma^2I) q ( x ) = N ( μ , σ 2 I )
s q ( x ) = − x − μ σ 2 , s_q(x)=-\frac{x-\mu}{\sigma^2},
s q ( x ) = − σ 2 x − μ ,
偏离均值越远，指回中心的作用越强；方差越小，分布越集中，相同偏移对应的校正也越大。
若密度写为 p ( x ) = p ~ ( x ) / Z p(x)=\tilde p(x)/Z p ( x ) = p ~ ( x ) / Z Z = ∫ p ~ ( x ) d x Z=\int \tilde p(x)dx Z = ∫ p ~ ( x ) d x Z Z Z x x x
∇ x log p ( x ) = ∇ x log p ~ ( x ) \nabla_x\log p(x)=\nabla_x\log\tilde p(x)
∇ x log p ( x ) = ∇ x log p ~ ( x )
高维生成模型难以计算 Z Z Z
流匹配和扩散模型都使用一条从初始分布 p i n i t p_{init} p ini t p d a t a p_{data} p d a t a z ∈ R d z\in\mathbb R^d z ∈ R d p t ( x ∣ z ) p_t(x\mid z) p t ( x ∣ z )
p 0 ( ⋅ ∣ z ) = p i n i t , p 1 ( ⋅ ∣ z ) = δ z p_0(\cdot\mid z)=p_{init},\qquad p_1(\cdot\mid z)=\delta_z
p 0 ( ⋅ ∣ z ) = p ini t , p 1 ( ⋅ ∣ z ) = δ z
这里的时间方向是“噪声到数据”。将条件路径对数据分布边缘化，得到边缘路径：
p t ( x ) = ∫ p t ( x ∣ z ) p d a t a ( z ) d z p_t(x)=\int p_t(x\mid z)p_{data}(z)dz
p t ( x ) = ∫ p t ( x ∣ z ) p d a t a ( z ) d z
训练时固定 z z z ∇ x log p t ( x ∣ z ) \nabla_x\log p_t(x\mid z) ∇ x log p t ( x ∣ z ) ∇ x log p t ( x ) \nabla_x\log p_t(x) ∇ x log p t ( x )
∇ x log p t ( x ) = ∫ ∇ x p t ( x ∣ z ) p d a t a ( z ) d z p t ( x ) = ∫ ∇ x log p t ( x ∣ z ) p t ( x ∣ z ) p d a t a ( z ) p t ( x ) d z = E z ∼ p 1 ∣ t ( ⋅ ∣ x ) [ ∇ x log p t ( x ∣ z ) ] . \begin{aligned}
\nabla_x\log p_t(x)
&=\frac{\int \nabla_x p_t(x\mid z)p_{data}(z)dz}{p_t(x)}\\
&=\int \nabla_x\log p_t(x\mid z)
\frac{p_t(x\mid z)p_{data}(z)}{p_t(x)}dz\\
&=\mathbb E_{z\sim p_{1\mid t}(\cdot\mid x)}\left[\nabla_x\log p_t(x\mid z)\right].
\end{aligned} ∇ x log p t ( x ) = p t ( x ) ∫ ∇ x p t ( x ∣ z ) p d a t a ( z ) d z = ∫ ∇ x log p t ( x ∣ z ) p t ( x ) p t ( x ∣ z ) p d a t a ( z ) d z = E z ∼ p 1 ∣ t ( ⋅ ∣ x ) [ ∇ x log p t ( x ∣ z ) ] .
其中 p 1 ∣ t ( z ∣ x ) = p t ( x ∣ z ) p d a t a ( z ) / p t ( x ) p_{1\mid t}(z\mid x)=p_t(x\mid z)p_{data}(z)/p_t(x) p 1 ∣ t ( z ∣ x ) = p t ( x ∣ z ) p d a t a ( z ) / p t ( x ) x x x z z z x x x
高斯路径
常用的高斯条件路径写为
p t ( x ∣ z ) = N ( x ; α t z , β t 2 I d ) p_t(x\mid z)=\mathcal N(x;\alpha_tz,\beta_t^2I_d)
p t ( x ∣ z ) = N ( x ; α t z , β t 2 I d )
其中 α t \alpha_t α t β t \beta_t β t α t / β t \alpha_t/\beta_t α t / β t α 0 = 0 , β 0 = 1 \alpha_0=0,\beta_0=1 α 0 = 0 , β 0 = 1 α 1 = 1 , β 1 = 0 \alpha_1=1,\beta_1=0 α 1 = 1 , β 1 = 0 0 < t < 1 0<t<1 0 < t < 1 α t > 0 \alpha_t>0 α t > 0 β t > 0 \beta_t>0 β t > 0
z ∼ p d a t a , ϵ ∼ N ( 0 , I d ) , x t = α t z + β t ϵ ∼ p t ( ⋅ ∣ z ) z\sim p_{data},\qquad \epsilon\sim\mathcal N(0,I_d),\qquad x_t=\alpha_tz+\beta_t\epsilon\sim p_t(\cdot\mid z)
z ∼ p d a t a , ϵ ∼ N ( 0 , I d ) , x t = α t z + β t ϵ ∼ p t ( ⋅ ∣ z )
重参数化把“从随 z , t z,t z , t x x x
∇ x log p t ( x ∣ z ) = − x − α t z β t 2 = − ϵ β t \nabla_x\log p_t(x\mid z)=-\frac{x-\alpha_tz}{\beta_t^2}=-\frac{\epsilon}{\beta_t}
∇ x log p t ( x ∣ z ) = − β t 2 x − α t z = − β t ϵ
式中的 x − α t z x-\alpha_tz x − α t z 1 / β t 2 1/\beta_t^2 1/ β t 2 β t ϵ \beta_t\epsilon β t ϵ − ϵ / β t -\epsilon/\beta_t − ϵ / β t β t → 0 \beta_t\to0 β t → 0
高斯路径下，条件目标向量场为
u t t a r g e t ( x ∣ z ) = ( α ˙ t − β ˙ t β t α t ) z + β ˙ t β t x u_t^{target}(x\mid z)=\left(\dot\alpha_t-\frac{\dot\beta_t}{\beta_t}\alpha_t\right)z+\frac{\dot\beta_t}{\beta_t}x
u t t a r g e t ( x ∣ z ) = ( α ˙ t − β t β ˙ t α t ) z + β t β ˙ t x
向量场要求粒子的分布按日程改变，分数则描述同一时刻的密度几何。把上式反解为 z = ( x + β t 2 ∇ x log p t ( x ∣ z ) ) / α t z=(x+\beta_t^2\nabla_x\log p_t(x\mid z))/\alpha_t z = ( x + β t 2 ∇ x log p t ( x ∣ z )) / α t
a t = β t 2 α ˙ t α t − β ˙ t β t , b t = α ˙ t α t a_t=\beta_t^2\frac{\dot\alpha_t}{\alpha_t}-\dot\beta_t\beta_t,\qquad b_t=\frac{\dot\alpha_t}{\alpha_t}
a t = β t 2 α t α ˙ t − β ˙ t β t , b t = α t α ˙ t
则在 0 < t < 1 0<t<1 0 < t < 1
u t t a r g e t ( x ∣ z ) = a t ∇ x log p t ( x ∣ z ) + b t x u_t^{target}(x\mid z)=a_t\nabla_x\log p_t(x\mid z)+b_tx
u t t a r g e t ( x ∣ z ) = a t ∇ x log p t ( x ∣ z ) + b t x
其中 b t x b_tx b t x a t ∇ x log p t a_t\nabla_x\log p_t a t ∇ x log p t z z z z z z
u t t a r g e t ( x ) = a t ∇ x log p t ( x ) + b t x u_t^{target}(x)=a_t\nabla_x\log p_t(x)+b_tx
u t t a r g e t ( x ) = a t ∇ x log p t ( x ) + b t x
因此，满足上述条件的高斯路径中，边缘向量场与边缘分数函数只是同一概率路径的两种局部描述：前者直接给出样本速度，后者给出密度几何，并通过由噪声日程确定的 a t , b t a_t,b_t a t , b t α t \alpha_t α t β t \beta_t β t
去噪器使用另一种等价参数化，定义为干净数据的后验均值：
D t ( x ) = E z ∼ p 1 ∣ t ( ⋅ ∣ x ) [ z ] D_t(x)=\mathbb E_{z\sim p_{1\mid t}(\cdot\mid x)}[z]
D t ( x ) = E z ∼ p 1 ∣ t ( ⋅ ∣ x ) [ z ]
含噪状态可能对应多个干净样本，平方误差下无法可靠恢复其中某个未知来源，后验均值却是均方误差最小的单值估计。它直接给出分数与向量场：
∇ x log p t ( x ) = α t D t ( x ) − x β t 2 \nabla_x\log p_t(x)=\frac{\alpha_tD_t(x)-x}{\beta_t^2}
∇ x log p t ( x ) = β t 2 α t D t ( x ) − x
u t t a r g e t ( x ) = ( α ˙ t − β ˙ t β t α t ) D t ( x ) + β ˙ t β t x u_t^{target}(x)=\left(\dot\alpha_t-\frac{\dot\beta_t}{\beta_t}\alpha_t\right)D_t(x)+\frac{\dot\beta_t}{\beta_t}x
u t t a r g e t ( x ) = ( α ˙ t − β t β ˙ t α t ) D t ( x ) + β t β ˙ t x
第一式也表明，α t D t ( x ) − x \alpha_tD_t(x)-x α t D t ( x ) − x D t ( x ) D_t(x) D t ( x ) s t s_t s t ϵ t \epsilon_t ϵ t D t D_t D t
ϵ t ( x ) = − β t s t ( x ) , D t ( x ) = x + β t 2 s t ( x ) α t , \epsilon_t(x)=-\beta_ts_t(x),
\qquad
D_t(x)=\frac{x+\beta_t^2s_t(x)}{\alpha_t},
ϵ t ( x ) = − β t s t ( x ) , D t ( x ) = α t x + β t 2 s t ( x ) ,
因而表达能力相同，但训练标签和网络输出的尺度不同。噪声预测把标准目标保持在 O ( 1 ) O(1) O ( 1 ) α t \alpha_t α t β t \beta_t β t
SDE 采样
设 u t t a r g e t u_t^{target} u t t a r g e t p t p_t p t p t p_t p t σ t ≥ 0 \sigma_t\ge0 σ t ≥ 0
X 0 ∼ p i n i t , d X t = [ u t t a r g e t ( X t ) + σ t 2 2 ∇ x log p t ( X t ) ] d t + σ t d W t X_0\sim p_{init},\qquad dX_t=\left[u_t^{target}(X_t)+\frac{\sigma_t^2}{2}\nabla_x\log p_t(X_t)\right]dt+\sigma_t dW_t
X 0 ∼ p ini t , d X t = [ u t t a r g e t ( X t ) + 2 σ t 2 ∇ x log p t ( X t ) ] d t + σ t d W t
这就是 SDE Extension Trick。σ t d W t \sigma_t\mathrm dW_t σ t d W t σ t 2 \sigma_t^2 σ t 2 σ t 2 / 2 \sigma_t^2/2 σ t 2 /2 p t p_t p t
这一结论可由 Fokker-Planck 方程验证。对于
d X t = μ t ( X t ) d t + σ t d W t dX_t=\mu_t(X_t)dt+\sigma_tdW_t
d X t = μ t ( X t ) d t + σ t d W t
其密度满足
∂ t p t ( x ) = − ∇ ⋅ ( p t ( x ) μ t ( x ) ) + σ t 2 2 Δ p t ( x ) \partial_tp_t(x)=-\nabla\cdot\bigl(p_t(x)\mu_t(x)\bigr)+\frac{\sigma_t^2}{2}\Delta p_t(x)
∂ t p t ( x ) = − ∇ ⋅ ( p t ( x ) μ t ( x ) ) + 2 σ t 2 Δ p t ( x )
将
μ t ( x ) = u t t a r g e t ( x ) + σ t 2 2 ∇ x log p t ( x ) \mu_t(x)=u_t^{target}(x)+\frac{\sigma_t^2}{2}\nabla_x\log p_t(x)
μ t ( x ) = u t t a r g e t ( x ) + 2 σ t 2 ∇ x log p t ( x )
代入，并使用 p t ∇ x log p t = ∇ x p t p_t\nabla_x\log p_t=\nabla_xp_t p t ∇ x log p t = ∇ x p t
∂ t p t = − ∇ ⋅ ( p t u t t a r g e t ) − σ t 2 2 Δ p t + σ t 2 2 Δ p t = − ∇ ⋅ ( p t u t t a r g e t ) . \begin{aligned}
\partial_t p_t
&=-\nabla\cdot(p_tu_t^{target})-\frac{\sigma_t^2}{2}\Delta p_t+\frac{\sigma_t^2}{2}\Delta p_t\\
&=-\nabla\cdot(p_tu_t^{target}).
\end{aligned} ∂ t p t = − ∇ ⋅ ( p t u t t a r g e t ) − 2 σ t 2 Δ p t + 2 σ t 2 Δ p t = − ∇ ⋅ ( p t u t t a r g e t ) .
右侧正是目标 ODE 的连续性方程：布朗运动产生的 + ( σ t 2 / 2 ) Δ p t +(\sigma_t^2/2)\Delta p_t + ( σ t 2 /2 ) Δ p t σ t = 0 \sigma_t=0 σ t = 0 σ t \sigma_t σ t σ t \sigma_t σ t
用步长 h = 1 / n h=1/n h = 1/ n t k = k h t_k=kh t k = kh ξ k ∼ N ( 0 , I d ) \xi_k\sim\mathcal N(0,I_d) ξ k ∼ N ( 0 , I d )
X t k + 1 = X t k + h [ u t k θ ( X t k ) + σ t k 2 2 s t k θ ( X t k ) ] + σ t k h ξ k X_{t_{k+1}}=X_{t_k}+h\left[u_{t_k}^\theta(X_{t_k})+\frac{\sigma_{t_k}^2}{2}s_{t_k}^\theta(X_{t_k})\right]+\sigma_{t_k}\sqrt h\,\xi_k
X t k + 1 = X t k + h [ u t k θ ( X t k ) + 2 σ t k 2 s t k θ ( X t k ) ] + σ t k h ξ k
从 X 0 ∼ p i n i t X_0\sim p_{init} X 0 ∼ p ini t t = 1 t=1 t = 1 X 1 X_1 X 1 h h h h \sqrt h h ξ k \xi_k ξ k ϵ t θ \epsilon_t^\theta ϵ t θ s t θ ( x ) = − ϵ t θ ( x ) / β t s_t^\theta(x)=-\epsilon_t^\theta(x)/\beta_t s t θ ( x ) = − ϵ t θ ( x ) / β t
当概率路径恒为 p t = p p_t=p p t = p u t t a r g e t = 0 u_t^{target}=0 u t t a r g e t = 0
d X t = σ t 2 2 ∇ x log p ( X t ) d t + σ t d W t dX_t=\frac{\sigma_t^2}{2}\nabla_x\log p(X_t)dt+\sigma_tdW_t
d X t = 2 σ t 2 ∇ x log p ( X t ) d t + σ t d W t
若 X 0 ∼ p X_0\sim p X 0 ∼ p p p p p p p
分数匹配
采样需要在任意噪声水平和状态查询边缘分数 ∇ x log p t ( x ) \nabla_x\log p_t(x) ∇ x log p t ( x ) x x x
s t θ : R d × [ 0 , 1 ] → R d s_t^\theta:\mathbb R^d\times[0,1]\rightarrow\mathbb R^d
s t θ : R d × [ 0 , 1 ] → R d
理想的边缘分数匹配损失是
L S M ( θ ) = E t ∼ U n i f [ 0 , 1 ] , x ∼ p t [ ∥ s t θ ( x ) − ∇ x log p t ( x ) ∥ 2 ] \mathcal L_{SM}(\theta)=\mathbb E_{t\sim\mathrm{Unif}[0,1],\,x\sim p_t}\left[\left\|s_t^\theta(x)-\nabla_x\log p_t(x)\right\|^2\right]
L SM ( θ ) = E t ∼ Unif [ 0 , 1 ] , x ∼ p t [ s t θ ( x ) − ∇ x log p t ( x ) 2 ]
这个目标准确表达了需求，却不可直接计算。条件去噪分数匹配改用已知的条件分数：
L C S M ( θ ) = E t ∼ U n i f [ 0 , 1 ] , z ∼ p d a t a x ∼ p t ( ⋅ ∣ z ) [ ∥ s t θ ( x ) − ∇ x log p t ( x ∣ z ) ∥ 2 ] \mathcal L_{CSM}(\theta)=\mathbb E_{\substack{t\sim\mathrm{Unif}[0,1],\,z\sim p_{data}\\x\sim p_t(\cdot\mid z)}}\left[\left\|s_t^\theta(x)-\nabla_x\log p_t(x\mid z)\right\|^2\right]
L CSM ( θ ) = E t ∼ Unif [ 0 , 1 ] , z ∼ p d a t a x ∼ p t ( ⋅ ∣ z ) [ s t θ ( x ) − ∇ x log p t ( x ∣ z ) 2 ]
对固定 ( x , t ) (x,t) ( x , t )
L C S M ( θ ) = L S M ( θ ) + C \mathcal L_{CSM}(\theta)=\mathcal L_{SM}(\theta)+C
L CSM ( θ ) = L SM ( θ ) + C
其中 C = E ∥ ∇ x log p t ( x ∣ z ) − ∇ x log p t ( x ) ∥ 2 ≥ 0 C=\mathbb E\|\nabla_x\log p_t(x\mid z)-\nabla_x\log p_t(x)\|^2\ge0 C = E ∥ ∇ x log p t ( x ∣ z ) − ∇ x log p t ( x ) ∥ 2 ≥ 0 θ \theta θ x x x C C C
对高斯路径，将 x t = α t z + β t ϵ x_t=\alpha_tz+\beta_t\epsilon x t = α t z + β t ϵ − ϵ / β t -\epsilon/\beta_t − ϵ / β t
L C S M ( θ ) = E t , z , ϵ [ ∥ s t θ ( x t ) + ϵ β t ∥ 2 ] = E t , z , ϵ [ 1 β t 2 ∥ ϵ t θ ( x t ) − ϵ ∥ 2 ] , \begin{aligned}
\mathcal L_{CSM}(\theta)
&=\mathbb E_{t,z,\epsilon}\left[\left\|s_t^\theta(x_t)+\frac{\epsilon}{\beta_t}\right\|^2\right]\\
&=\mathbb E_{t,z,\epsilon}\left[\frac{1}{\beta_t^2}\left\|\epsilon_t^\theta(x_t)-\epsilon\right\|^2\right],
\end{aligned} L CSM ( θ ) = E t , z , ϵ [ s t θ ( x t ) + β t ϵ 2 ] = E t , z , ϵ [ β t 2 1 ϵ t θ ( x t ) − ϵ 2 ] ,
其中 ϵ t θ ( x ) = − β t s t θ ( x ) \epsilon_t^\theta(x)=-\beta_ts_t^\theta(x) ϵ t θ ( x ) = − β t s t θ ( x ) − ϵ / β t -\epsilon/\beta_t − ϵ / β t ϵ \epsilon ϵ 1 / β t 2 1/\beta_t^2 1/ β t 2 β t \beta_t β t
L D D P M ( θ ) = E t ∼ U n i f [ 0 , 1 ] , z ∼ p d a t a ϵ ∼ N ( 0 , I d ) [ ∥ ϵ t θ ( α t z + β t ϵ ) − ϵ ∥ 2 ] \mathcal L_{DDPM}(\theta)=\mathbb E_{\substack{t\sim\mathrm{Unif}[0,1],\,z\sim p_{data}\\\epsilon\sim\mathcal N(0,I_d)}}\left[\left\|\epsilon_t^\theta(\alpha_tz+\beta_t\epsilon)-\epsilon\right\|^2\right]
L DD PM ( θ ) = E t ∼ Unif [ 0 , 1 ] , z ∼ p d a t a ϵ ∼ N ( 0 , I d ) [ ϵ t θ ( α t z + β t ϵ ) − ϵ 2 ]
去掉权重后，不同时间的噪声标签都处在近似统一的数值尺度，优化通常更稳定，也不会让少量低噪声样本主导梯度；代价是各噪声水平的相对重要性被重新定义。在网络容量无限且能对每个 ( x , t ) (x,t) ( x , t )
每个小批量的训练步骤如下：
从数据集采样 z z z t ∼ U n i f [ 0 , 1 ] t\sim\mathrm{Unif}[0,1] t ∼ Unif [ 0 , 1 ] ϵ ∼ N ( 0 , I d ) \epsilon\sim\mathcal N(0,I_d) ϵ ∼ N ( 0 , I d ) β t \beta_t β t
用闭式加噪公式构造 x t = α t z + β t ϵ x_t=\alpha_tz+\beta_t\epsilon x t = α t z + β t ϵ t t t
若训练分数网络，使用目标 − ϵ / β t -\epsilon/\beta_t − ϵ / β t ϵ \epsilon ϵ x t , t x_t,t x t , t
对损失反向传播并更新参数。各样本可独立选择时间和噪声，因此训练能够并行；生成时则必须根据预测的分数或向量场连续执行多步更新，训练的一步监督并没有消除采样的积分成本。
训练完成后，噪声预测、分数预测或去噪器预测都可转换为采样所需的分数或向量场，再通过 ODE 或 SDE 数值积分生成样本。选择 ODE 会在给定初值后产生确定轨迹，选择具有补偿漂移的 SDE 会在保持理想边缘路径的同时继续注入随机性；参数化、噪声日程与求解器共同决定数值稳定性和所需网络调用次数。
