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RPC-Bench

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 which we refer to as the discrete-time ELBO (DT-ELBO). The reconstruction error of the stochastic layer can be seen as some form of finite difference approximation to differentiation, which gives rise to $\nabla \cdot \boldsymbol { \mu }$ in the CT-ELBO in the infinitesimal limit (as $\Delta t$ approaches 0). The regularization of $| | a | | ^ { 2 }$ pops up when we compare the difference between $\tilde { \mu } _ { i }$ and $\hat { \mu } _ { i }$ using the Gaussian reparameterization to compute the reconstruction error. We formalize this idea in the following theorem.
-Theorem 5 (Consistency). Assume $\mu$ , , $\sigma ^ { - 2 }$ , $^ { a , }$ , $| | a | | ^ { 2 }$ and their derivatives up to the fourth order are all bounded and continuous, and that $\sigma$ is non-singular. Then $\mathcal { E } ^ { L } \to \mathcal { E } ^ { \infty }$ as $L \to \infty$ .
 This theorem tells us that the CT-ELBO we derive for continuous-time diffusion models is not that different from the traditional ELBO, and that maximizing the CT-ELBO can be seen as training an infinitely deep hierarchical VAE. We present the proof in Appendix F, which formalizes the above intuition, using Taylor’s theorem to control the polynomial approximation error, which will go to 0 as the step size $\Delta t$ vanishes when the number of layers $L$ increases to infinity.

 which we refer to as the discrete-time ELBO (DT-ELBO). The reconstruction error of the stochastic layer can be seen as some form of finite difference approximation to differentiation, which gives rise to $\nabla \cdot \boldsymbol { \mu }$ in the CT-ELBO in the infinitesimal limit (as $\Delta t$ approaches 0). The regularization of $| | a | | ^ { 2 }$ pops up when we compare the difference between $\tilde { \mu } _ { i }$ and $\hat { \mu } _ { i }$ using the Gaussian reparameterization to compute the reconstruction error. We formalize this idea in the following theorem.
+Theorem 5 (Consistency). Assume $\mu$ , , $\sigma ^ { - 2 }$ , $^ { a , }$ , $| | a | | ^ { 2 }$ and their derivatives up to the fourth order are all bounded and continuous, and that $\sigma$ is non-singular. Then $\mathcal { E } ^ { L } \to \mathcal { E } ^ { \infty }$ as $L \to \infty$ .
 This theorem tells us that the CT-ELBO we derive for continuous-time diffusion models is not that different from the traditional ELBO, and that maximizing the CT-ELBO can be seen as training an infinitely deep hierarchical VAE. We present the proof in Appendix F, which formalizes the above intuition, using Taylor’s theorem to control the polynomial approximation error, which will go to 0 as the step size $\Delta t$ vanishes when the number of layers $L$ increases to infinity.