먼저, 앞에 보이는 수식(사진)은 DDPM 논문의 3번 수식이고, − log p θ ( x 0 ) -\log p_\theta(x_0) − log p θ ( x 0 ) log p θ ( x 0 ) \log p_\theta(x_0) log p θ ( x 0 )
DDPM에서는 관측 데이터 x 0 x_0 x 0 log p θ ( x 0 ) \log p_\theta(x_0) log p θ ( x 0 ) x 1 : T x_{1:T} x 1 : T
log p θ ( x 0 ) = log ∫ p θ ( x 0 , x 1 : T ) d x 1 : T . \log p_\theta(x_0) = \log \int p_\theta(x_0, x_{1:T}) \, dx_{1:T}. log p θ ( x 0 ) = log ∫ p θ ( x 0 , x 1 : T ) d x 1 : T . 이 적분은 직접 계산하기 어려우므로, 변분 추론(Variational Inference)의 표준 기법을 적용합니다. 여기서는 분자와 분모에 동일한 항, 즉 q ( x 1 : T ∣ x 0 ) q ( x 1 : T ∣ x 0 ) \frac{q(x_{1:T}\mid x_0)}{q(x_{1:T}\mid x_0)} q ( x 1 : T ∣ x 0 ) q ( x 1 : T ∣ x 0 )
log p θ ( x 0 ) = log ∫ p θ ( x 0 , x 1 : T ) q ( x 1 : T ∣ x 0 ) q ( x 1 : T ∣ x 0 ) d x 1 : T . \log p_\theta(x_0) = \log \int p_\theta(x_0, x_{1:T}) \, \frac{q(x_{1:T}\mid x_0)}{q(x_{1:T}\mid x_0)} \, dx_{1:T}. log p θ ( x 0 ) = log ∫ p θ ( x 0 , x 1 : T ) q ( x 1 : T ∣ x 0 ) q ( x 1 : T ∣ x 0 ) d x 1 : T . 분수 형태를 기댓값으로 표현하면 다음과 같이 쓸 수 있습니다:
log p θ ( x 0 ) = log ( E x 1 : T ∼ q ( ⋅ ∣ x 0 ) [ p θ ( x 0 , x 1 : T ) q ( x 1 : T ∣ x 0 ) ] ) . \log p_\theta(x_0) = \log \Bigl( \mathbb{E}_{x_{1:T}\sim q(\cdot\mid x_0)} \Bigl[\frac{p_\theta(x_0, x_{1:T})}{q(x_{1:T}\mid x_0)}\Bigr] \Bigr). log p θ ( x 0 ) = log ( E x 1 : T ∼ q ( ⋅ ∣ x 0 ) [ q ( x 1 : T ∣ x 0 ) p θ ( x 0 , x 1 : T ) ] ) . 여기서 log \log log log E [ X ] ≥ E [ log X ] \log \mathbb{E}[X] \ge \mathbb{E}[\log X] log E [ X ] ≥ E [ log X ]
log p θ ( x 0 ) = log ( E x 1 : T ∼ q ( ⋅ ∣ x 0 ) [ p θ ( x 0 , x 1 : T ) q ( x 1 : T ∣ x 0 ) ] ) ≥ E x 1 : T ∼ q ( ⋅ ∣ x 0 ) [ log p θ ( x 0 , x 1 : T ) q ( x 1 : T ∣ x 0 ) ] . \log p_\theta(x_0) = \log \Bigl( \mathbb{E}_{x_{1:T}\sim q(\cdot\mid x_0)} \Bigl[\frac{p_\theta(x_0, x_{1:T})}{q(x_{1:T}\mid x_0)}\Bigr] \Bigr) \ge \mathbb{E}_{x_{1:T}\sim q(\cdot\mid x_0)} \Bigl[ \log \frac{p_\theta(x_0, x_{1:T})}{q(x_{1:T}\mid x_0)} \Bigr]. log p θ ( x 0 ) = log ( E x 1 : T ∼ q ( ⋅ ∣ x 0 ) [ q ( x 1 : T ∣ x 0 ) p θ ( x 0 , x 1 : T ) ] ) ≥ E x 1 : T ∼ q ( ⋅ ∣ x 0 ) [ log q ( x 1 : T ∣ x 0 ) p θ ( x 0 , x 1 : T ) ] . 이 부등식은 log p θ ( x 0 ) \log p_\theta(x_0) log p θ ( x 0 )
log p θ ( x 0 ) ≥ E x 1 : T ∼ q ( ⋅ ∣ x 0 ) [ log p θ ( x 0 , x 1 : T ) − log q ( x 1 : T ∣ x 0 ) ] . \log p_\theta(x_0) \ge \mathbb{E}_{x_{1:T}\sim q(\cdot\mid x_0)} \Bigl[ \log p_\theta(x_0, x_{1:T}) - \log q(x_{1:T}\mid x_0) \Bigr]. log p θ ( x 0 ) ≥ E x 1 : T ∼ q ( ⋅ ∣ x 0 ) [ log p θ ( x 0 , x 1 : T ) − log q ( x 1 : T ∣ x 0 ) ] . 실제 학습에서는 최대화 대신 − log p θ ( x 0 ) -\log p_\theta(x_0) − log p θ ( x 0 )
− log p θ ( x 0 ) ≤ E x 1 : T ∼ q ( ⋅ ∣ x 0 ) [ − log p θ ( x 0 , x 1 : T ) + log q ( x 1 : T ∣ x 0 ) ] . -\log p_\theta(x_0) \le \mathbb{E}_{x_{1:T}\sim q(\cdot\mid x_0)} \Bigl[ -\log p_\theta(x_0, x_{1:T}) + \log q(x_{1:T}\mid x_0) \Bigr]. − log p θ ( x 0 ) ≤ E x 1 : T ∼ q ( ⋅ ∣ x 0 ) [ − log p θ ( x 0 , x 1 : T ) + log q ( x 1 : T ∣ x 0 ) ] . DDPM 논문에서는 이 ELBO 식을 체인 룰을 사용하여 p θ ( x 0 , x 1 : T ) p_\theta(x_0, x_{1:T}) p θ ( x 0 , x 1 : T ) q ( x 1 : T ∣ x 0 ) q(x_{1:T}\mid x_0) q ( x 1 : T ∣ x 0 )
p θ ( x 0 , x 1 : T ) = p θ ( x 0 : T ) = p θ ( x T ) ∏ t = 1 T p θ ( x t − 1 ∣ x t ) , p_\theta(x_0, x_{1:T}) = p_\theta(x_{0:T}) = p_\theta(x_T) \prod_{t=1}^T p_\theta\bigl(x_{t-1}\mid x_t\bigr), p θ ( x 0 , x 1 : T ) = p θ ( x 0 : T ) = p θ ( x T ) t = 1 ∏ T p θ ( x t − 1 ∣ x t ) , 로 나타낼 수 있고, Forward process는
q ( x 1 : T ∣ x 0 ) = ∏ t = 1 T q ( x t ∣ x t − 1 ) q(x_{1:T}\mid x_0) = \prod_{t=1}^T q(x_t\mid x_{t-1}) q ( x 1 : T ∣ x 0 ) = t = 1 ∏ T q ( x t ∣ x t − 1 ) 와 같이 인수분해됩니다.
이를 이용해 체인 룰로 p θ ( x 0 , x 1 : T ) p_\theta(x_0, x_{1:T}) p θ ( x 0 , x 1 : T ) q ( x 1 : T ∣ x 0 ) q(x_{1:T}\mid x_0) q ( x 1 : T ∣ x 0 )
− log p θ ( x 0 ) ≤ E x 1 : T ∼ q ( ⋅ ∣ x 0 ) [ − log p θ ( x 0 , x 1 : T ) + log q ( x 1 : T ∣ x 0 ) ] -\log p_\theta(x_0) \;\;\le\;\; \mathbb{E}_{x_{1:T}\sim q(\cdot\mid x_0)} \Bigl[ -\log p_\theta\bigl(x_0, x_{1:T}\bigr) \;+\; \log q\bigl(x_{1:T}\mid x_0\bigr) \Bigr] − log p θ ( x 0 ) ≤ E x 1 : T ∼ q ( ⋅ ∣ x 0 ) [ − log p θ ( x 0 , x 1 : T ) + log q ( x 1 : T ∣ x 0 ) ] 에서, p θ ( x 0 , x 1 : T ) p_\theta(x_0, x_{1:T}) p θ ( x 0 , x 1 : T ) q ( x 1 : T ∣ x 0 ) q(x_{1:T}\mid x_0) q ( x 1 : T ∣ x 0 )
이제 우변의 기댓값 안을 구체적으로 써 보면,
− log p θ ( x 0 , x 1 : T ) = − log p θ ( x T ) − ∑ t = 1 T log p θ ( x t − 1 ∣ x t ) , -\log p_\theta\bigl(x_0, x_{1:T}\bigr) = -\log p_\theta(x_T) \;-\; \sum_{t=1}^T \log p_\theta\bigl(x_{t-1}\mid x_t\bigr), − log p θ ( x 0 , x 1 : T ) = − log p θ ( x T ) − t = 1 ∑ T log p θ ( x t − 1 ∣ x t ) , log q ( x 1 : T ∣ x 0 ) = ∑ t = 1 T log q ( x t ∣ x t − 1 ) . \log q\bigl(x_{1:T}\mid x_0\bigr) = \sum_{t=1}^T \log q\bigl(x_t\mid x_{t-1}\bigr). log q ( x 1 : T ∣ x 0 ) = t = 1 ∑ T log q ( x t ∣ x t − 1 ) . 두 식을 합치면,
− log p θ ( x T ) − ∑ t = 1 T log p θ ( x t − 1 ∣ x t ) + ∑ t = 1 T log q ( x t ∣ x t − 1 ) . -\log p_\theta(x_T) \;-\; \sum_{t=1}^T \log p_\theta\bigl(x_{t-1}\mid x_t\bigr) \;+\; \sum_{t=1}^T \log q\bigl(x_t\mid x_{t-1}\bigr). − log p θ ( x T ) − t = 1 ∑ T log p θ ( x t − 1 ∣ x t ) + t = 1 ∑ T log q ( x t ∣ x t − 1 ) . 이를 기댓값 E q ( ⋅ ∣ x 0 ) [ ⋯ ] \mathbb{E}_{q(\cdot\mid x_0)}[\cdots] E q ( ⋅ ∣ x 0 ) [ ⋯ ]
log q ( x t ∣ x t − 1 ) − log p θ ( x t − 1 ∣ x t ) \log q\bigl(x_t\mid x_{t-1}\bigr) \;-\; \log p_\theta\bigl(x_{t-1}\mid x_t\bigr) log q ( x t ∣ x t − 1 ) − log p θ ( x t − 1 ∣ x t ) 라는 항이 등장합니다. 이것을
K L ( q ( x t − 1 ∣ x t , x 0 ) ∥ p θ ( x t − 1 ∣ x t ) ) \mathrm{KL}\Bigl( q(x_{t-1}\mid x_t, x_0) \;\|\; p_\theta(x_{t-1}\mid x_t) \Bigr) K L ( q ( x t − 1 ∣ x t , x 0 ) ∥ p θ ( x t − 1 ∣ x t ) ) 형태로 묶을 수 있으며, 그 합산이 곧 DDPM 논문에서 말하는 최종 Variational Bound가 됩니다.
따라서
− log p θ ( x 0 ) ≤ E q ( ⋅ ∣ x 0 ) [ − log p θ ( x T ) − ∑ t = 1 T log p θ ( x t − 1 ∣ x t ) + ∑ t = 1 T log q ( x t ∣ x t − 1 ) ] ⏟ 논문 식 (3)의 우변 , -\log p_\theta(x_0) \;\;\le\;\; \underbrace{ \mathbb{E}_{q(\cdot\mid x_0)}\Bigl[ -\log p_\theta(x_T) \;-\; \sum_{t=1}^T \log p_\theta(x_{t-1}\mid x_t) \;+\; \sum_{t=1}^T \log q(x_t\mid x_{t-1}) \Bigr] }_{\text{논문 식 (3)의 우변}}, − log p θ ( x 0 ) ≤ 논문 식 (3) 의 우변 E q ( ⋅ ∣ x 0 ) [ − log p θ ( x T ) − t = 1 ∑ T log p θ ( x t − 1 ∣ x t ) + t = 1 ∑ T log q ( x t ∣ x t − 1 ) ] , 이/가 되고, 이를 단계별로 재배열하면 각 t t t − log p θ ( x 0 ) -\log p_\theta(x_0) − log p θ ( x 0 ) log p θ ( x 0 ) \log p_\theta(x_0) log p θ ( x 0 ) p θ ( x t − 1 ∣ x t ) p_\theta(x_{t-1}\mid x_t) p θ ( x t − 1 ∣ x t )
