A Score-Based Density Formula, with Applications in Diffusion Generative Models

## Abstract

Abstract Score-based generative models (SGMs) have revolutionized the field
of generative modeling, achieving unprecedented success in generating
realistic and diverse content. Despite empirical advances, the theoretical basis for why optimizing the evidence lower bound (ELBO) on the log-likelihood is effective for training diffusion generative models, such as DDPMs, remains largely unexplored.
In this paper, we address this question by establishing a density formula for a continuous-time diffusion process, which can be viewed as the continuous-time limit of the forward process in an SGM. This formula
reveals the connection between the target density and the score function
associated with each step of the forward process. Building on this, we demonstrate that the minimizer of the optimization objective for training DDPMs nearly coincides with that of the true objective, providing a theoretical foundation for optimizing DDPMs using the ELBO.
Furthermore, we offer new insights into the role of score-matching
regularization in training GANs, the use of ELBO in diffusion classifiers,
and the recently proposed diffusion loss.

## 1 Introduction

Score-based generative models (SGMs) represent a groundbreaking advancement
in the realm of generative models, significantly impacting machine
learning and artificial intelligence by their ability to synthesize
high-fidelity data instances, including images, audio, and text (Sohl-Dickstein et al.,, ; Ho et al.,, ; ; Song and Ermon,, ; Dhariwal and Nichol,, ; ).
These models operate by progressively refining noisy data into samples
that resemble the target distribution. Due to their innovative approach,
SGMs have achieved unprecedented success, setting new standards in
generative AI and demonstrating extraordinary proficiency in generating
realistic and diverse content across various domains, from image synthesis
and super-resolution to audio generation and molecular design (Ramesh et al.,, ; Rombach et al.,, ; Saharia et al.,, ; Croitoru et al.,, ; Yang et al.,, ).

The foundation of SGMs is rooted in the principles of stochastic processes,
especially stochastic differential equations (SDEs). These models
utilize a forward process, which involves the gradual corruption of
an initial data sample with Gaussian noise over several time steps.
This forward process can be described as:

$$ $X_{0}\overset{\textsf{add noise}}{\longrightarrow}X_{1}\overset{\textsf{add noise}}{\longrightarrow}\cdots\overset{\textsf{add noise}}{\longrightarrow}X_{ T},$ (1.1) $$

where $X_{0}\sim p_{\mathsf{data}}$ is the original data sample,
and $X_{T}$ is a sample close to pure Gaussian noise. The ingenuity
of SGMs lies in constructing a reverse denoising process that iteratively
removes the noise, thereby reconstructing the data distribution. This
reverse process starts from a Gaussian sample $Y_{T}$ and moves backward
as:

$$ $Y_{T}\overset{\textsf{denoise}}{\longrightarrow}Y_{T-1}\overset{\textsf{ denoise}}{\longrightarrow}\cdots\overset{\textsf{denoise}}{\longrightarrow}Y_{0}$ (1.2) $$

ensuring that $Y_{t}\overset{\mathrm{d}}{\approx}X_{t}$ at each step
$t$. The final output $Y_{0}$ is a new sample that closely mimics
the distribution of the initial data $p_{\mathsf{data}}$.

Inspired by the classical results on time-reversal of SDEs (Anderson,, ; Haussmann and Pardoux,, ),
SGMs construct the reverse process guided by score functions $\nabla\log p_{X_{t}}$
associated with each step of the forward process. Although these score
functions are unknown, they are approximated by neural networks trained
through score-matching techniques (Hyvärinen,, , ; Vincent,, ; Song and Ermon,, ).
This leads to two popular models: denoising diffusion probabilistic
models (DDPMs) (Ho et al.,, ; Nichol and Dhariwal,, ) and denoising
diffusion implicit models (DDIMs) (). While
the theoretical results in this paper do not depend on the specific
construction of the reverse process, we will use the DDPM framework
to discuss their implications for diffusion generative models.

However, despite empirical advances, there remains a lack of theoretical
understanding for diffusion generative models. For instance, the optimization
target of DDPM is derived from a variational lower bound on
the log-likelihood (Ho et al.,, ), which is also
referred to as the evidence lower bound (ELBO) (Luo,, ).
It is not yet clear, from a theoretical standpoint, why optimizing
a lower bound of the true objective is still a valid approach. More
surprisingly, recent research suggests incorporating the ELBO
of a pre-trained DDPM into other generative or learning frameworks
to leverage the strengths of multiple architectures, effectively using
it as a proxy for the negative log-likelihood of the data distribution.
This approach has shown empirical success in areas such as GAN training,
classification, and inverse problems (Xia et al.,, ; ; Graikos et al.,, ; Mardani et al.,, ).
While it is conceivable that the ELBO is a reasonable optimization
target for training DDPMs (as similar idea is utilized in e.g., the
majorize-minimization algorithm), it is more mysterious why it serves
as a good proxy for the negative log-likelihood in these applications.

In this paper, we take a step towards addressing the aforementioned
question. On the theoretical side, we establish a density formula
for a diffusion process $(X_{t})_{0\leq t<1}$ defined by the following
SDE:

$$ $\mathrm{d}X_{t}=-\frac{1}{2(1-t)}X_{t}\mathrm{d}t+\frac{1}{\sqrt{1-t}}\mathrm{ d}B_{t}\quad(0\leq t<1),\qquad X_{0}\sim p_{\mathsf{data}},$ $$

which can be viewed as a continuous-time limit of the forward process
(1.1). Under some regularity conditions, this formula
expresses the density of $X_{0}$ with the score function along this
process, having the form

$$ $\log p_{X_{0}}(x)=-\frac{1+\log(2\pi)}{2}d-\int_{0}^{1}\left[\frac{1}{2(1-t)} \mathbb{E}\Big{[}\Big{\|}\frac{X_{t}-\sqrt{1-t}X_{0}}{t}+\nabla\log p_{X_{t}}( X_{t})\Big{\|}_{2}^{2}\,|\,X_{0}=x\Big{]}-\frac{d}{2t}\right]\mathrm{d}t,$ $$

where $p_{X_{t}}(\cdot)$ is the density of $X_{t}$. By time-discretization,
this reveals the connection between the target density $p_{\mathsf{data}}$
and the score function associated with each step of the forward process
(1.1). These theoretical results will be presented
in Section 3.

Finally, using this density formula, we demonstrate that the minimizer of the optimization target for training DDPMs (derived from the ELBO) also nearly minimizes the true target—the KL divergence between the target distribution and the generator distribution. This finding provides a theoretical foundation for optimizing DDPMs using the ELBO. Additionally, we use this
formula to offer new insights into the role of score-matching regularization
in training GANs (Xia et al.,, ), the use of ELBO in diffusion
classifiers (), and the recently proposed diffusion
loss (Li et al.,, ). These implications will be discussed
in Section 4.

## 2 Problem set-up

In this section, we formally introduce the Denoising Diffusion Probabilistic
Model (DDPM) and the stochastic differential equation (SDE) that describes
the continuous-time limit of the forward process of DDPM.

### 2.1 Denoising diffusion probabilistic model

Consider the following forward Markov process in discrete time:

$$ $X_{t}=\sqrt{1-\beta_{t}}X_{t-1}+\sqrt{\beta_{t}}W_{t}\quad(t=1,\ldots,T), \qquad X_{0}\sim p_{\mathsf{data}},$ (2.1) $$

where $W_{1},\ldots,W_{T}\overset{\text{i.i.d.}}{\sim}\mathcal{N}(0,I_{d})$
and the learning rates $\beta_{t}\in(0,1)$. Since our main results
do not depend on the specific choice of $\beta_{t}$, we will specify
them as needed in later discussions. For each $t\in[T]$, let $q_{t}$
be the law or density function of $X_{t}$, and let $\alpha_{t}\coloneqq 1-\beta_{t}$
and $\overline{\alpha}_{t}\coloneqq\prod_{i=1}^{t}\alpha_{i}$. A
simple calculation shows that:

$$ $X_{t}=\sqrt{\overline{\alpha}_{t}}X_{0}+\sqrt{1-\overline{\alpha}_{t}}\, \overline{W}_{t}\qquad\text{where}\qquad\overline{W}_{t}\sim\mathcal{N}(0,I_{d }).$ (2.2) $$

We will choose the learning rates $\beta_{t}$ to ensure that $\overline{\alpha}_{T}$
is sufficiently small, such that $q_{T}$ is close to the standard
Gaussian distribution.

The key components for constructing the reverse process in the context
of DDPM are the score functions $s_{t}^{\star}:\mathbb{R}^{d}\to\mathbb{R}^{d}$
associated with each $q_{t}$, defined as the gradient of their log
density:

$$ $s_{t}^{\star}(x)\coloneqq\nabla\log q_{t}(x)\quad(t=1,\ldots,T).$ $$

While these score functions are not explicitly known, in practice,
noise-prediction networks $\varepsilon_{t}(x)$ are trained to predict

$$ $\varepsilon_{t}^{\star}(x)\coloneqq-\sqrt{1-\overline{\alpha}_{t}}s_{t}^{\star }(x),$ $$

which are often referred to as epsilon predictors. To construct the
reverse process, we use:

$$ $Y_{t-1}=\frac{1}{\sqrt{\alpha_{t}}}\big{(}Y_{t}+\eta_{t}s_{t}\left(Y_{t}\right )+\sigma_{t}Z_{t}\big{)}\quad(t=T,\ldots,1),\qquad Y_{T}\sim\mathcal{N}(0,I_{d})$ (2.3) $$

where $Z_{1},\ldots,Z_{T}\overset{\text{i.i.d.}}{\sim}\mathcal{N}(0,I_{d})$,
and $s_{t}(\cdot)\coloneqq-\varepsilon_{t}(\cdot)/\sqrt{1-\overline{\alpha}_{t}}$
is the estimate of the score function $s_{t}^{\star}(\cdot)$. Here
$\eta_{t},\sigma_{t}>0$ are the coefficients that influence the performance
of the DDPM sampler, and we will specify them as needed in later discussion.
For each $t\in[T]$, we use $p_{t}$ to denote the law or density
of $Y_{t}$.

### 2.2 A continuous-time SDE for the forward process

In this paper, we build our theoretical results on the continuous-time
limit of the aforementioned forward process, described by the diffusion
process:

$$ $\mathrm{d}X_{t}=-\frac{1}{2(1-t)}X_{t}\mathrm{d}t+\frac{1}{\sqrt{1-t}}\mathrm{ d}B_{t}\quad(0\leq t<1),\qquad X_{0}\sim p_{\mathsf{data}},$ (2.4) $$

where $(B_{t})_{t\geq 0}$ is a standard Brownian motion. The solution
to this stochastic differential equation (SDE) has the closed-form
expression:

$$ $X_{t}=\sqrt{1-t}X_{0}+\sqrt{t}\,\overline{Z}_{t}\qquad\text{where}\qquad \overline{Z}_{t}=\sqrt{\frac{1-t}{t}}\int_{0}^{t}\frac{1}{1-s}\mathrm{d}B_{s} \sim\mathcal{N}(0,I_{d}).$ (2.5) $$

It is important to note that the process $X_{t}$ is not defined at
$t=1$, although it is straightforward to see from the above equation
that $X_{t}$ converges to a Gaussian variable as $t\to 1$.

To demonstrate the connection between this diffusion process and the
forward process (2.1) of the diffusion model,
we evaluate the diffusion process at times $t_{i}=\sqrt{1-\overline{\alpha}_{i}}$
for $1\leq i\leq T$. It is straightforward to check that the marginal
distribution of the resulting discrete-time process $\{X_{t_{i}}:1\leq i\leq T\}$
is identical to that of the forward process (2.1).
Therefore the diffusion process (2.4) can be viewed
as a continuous-time limit of the forward process. In the next section,
we will establish theoretical results for the diffusion process (2.4).
Through time discretization, our theory will provide insights for
the DDPM.

We use the notation $X_{t}$ for both the discrete-time process $\{X_{t}:t\in[T]\}$
in (2.1) and the continuous-time diffusion process
$(X_{t})_{0\leq t<1}$ in (2.4) to maintain consistency
with standard literature. The context will clarify which process is
being referred to.

## 3 The score-based density formula

### 3.1 Main results

Our main results are based on the continuous-time diffusion process
$(X_{t})_{0\leq t<1}$ defined in (2.4). While
$X_{0}$ might not have a density, for any $t\in(0,1)$, the random
variable $X_{t}$ has a smooth density, denoted by $\rho_{t}(\cdot)$.
Our main result characterizes the evolution of the conditional mean
of $\log\rho_{t}(X_{t})$ given $X_{0}$, as stated below.

###### Theorem 1 .

Consider the diffusion process
$(X_{t})_{0\leq t<1}$ defined in (2.4), and let
$\rho_{t}$ be the density of $X_{t}$. For any $0<t_{1}<t_{2}<1$,
we have

$$ $\mathbb{E}\left[\log\rho_{t_{2}}(X_{t_{2}})-\log\rho_{t_{1}}(X_{t_{1}})\,|\,X_ {0}\right]=\int_{t_{1}}^{t_{2}}\bigg{(}\frac{1}{2(1-t)}\mathbb{E}\Big{[}\Big{ \|}\frac{X_{t}-\sqrt{1-t}X_{0}}{t}+\nabla\log\rho_{t}(X_{t})\Big{\|}_{2}^{2}\, |\,X_{0}\Big{]}-\frac{d}{2t}\bigg{)}\mathrm{d}t.$ $$

The proof of this theorem is deferred to Appendix 5.
A few remarks are as follows. First, it is worth mentioning that this
formula does not describe the evolution of the (conditional) differential
entropy of the process, because $\rho_{t}(\cdot)$ represents the
unconditional density of $X_{t}$, while the expectation is taken
conditional on $X_{0}$. Second, without further assumptions, we cannot
set $t_{1}=0$ or $t_{2}=1$ because $X_{0}$ might not have a density
(hence $\rho_{0}$ is not well-defined), and $X_{t}$ is only defined
for $t<1$. By assuming that $X_{0}$ has a finite second moment,
the following proposition characterizes the limit of $\mathbb{E}[\log\rho_{t}(X_{t})\,|\,X_{0}]$
as $t\to 1$.

###### Proposition 1 .

Suppose that $\mathbb{E}[\|X_{0}\|_{2}^{2}]<\infty$.
Then for any $x_{0}\in\mathbb{R}^{d}$, we have

$$ $\lim_{t\to 1-}\mathbb{E}\left[\log\rho_{t}\left(X_{t}\right)\,|\,X_{0}=x_{0} \right]=-\frac{1+\log\left(2\pi\right)}{2}d.$ $$

The proof of this proposition is deferred to Appendix A.
This result is not surprising, as it can be seen from (2.5)
that $X_{t}$ converges to a standard Gaussian variable as $t\to 1$
regardless of $x_{0}$, and we can check

$$ $\mathbb{E}[\log\phi(Z)]=-\frac{1+\log\left(2\pi\right)}{2}d$ $$

where $Z\sim\mathcal{N}(0,I_{d})$ and $\phi(\cdot)$ is its density
(we will use this notation throughout his section). The proof of Proposition 1
formalizes this intuitive analysis.

When $X_{0}$ has a smooth density $\rho_{0}(\cdot)$ with Lipschitz
continuous score function, we can show that $\mathbb{E}[\log\rho_{t}(X_{t})\,|\,X_{0}]\to\rho_{0}(x_{0})$
as $t\to 0$, as presented in the next proposition.

###### Proposition 2 .

Suppose that $X_{0}$ has density
$\rho_{0}(\cdot)$ and $\sup_{x}\|\nabla^{2}\log\rho_{0}(x)\|<\infty$.
Then for any $x_{0}\in\mathbb{R}^{d}$, we have

$$ $\lim_{t\to 0+}\mathbb{E}\left[\log\rho_{t}\left(X_{t}\right)\,|\,X_{0}=x_{0} \right]=\log\rho_{0}(x_{0}).$ $$

The proof of this proposition can be found in Appendix B.
With Propositions 1 and 2 in
place, we can take $t_{1}\to 0$ and $t_{2}\to 1$ in Theorem 1
to show that for any given point $x_{0}$,

$$ $\displaystyle\log\rho_{0}(x_{0})$ $\displaystyle=-\frac{1+\log(2\pi)}{2}d-\int_{0}^{1}D(t,x_{0})\mathrm{d}t$ (3.1a) where the function $D(x,t)$ is defined as $D(t,x)\coloneqq\frac{1}{2(1-t)}\mathbb{E}\Big{[}\Big{\|}\frac{X_{t}-\sqrt{1-t} X_{0}}{t}+\nabla\log\rho_{t}(X_{t})\Big{\|}_{2}^{2}\,|\,X_{0}=x\Big{]}-\frac{d }{2t}.$ (3.1b) In practice, we might not want to make smoothness assumptions on $X_{0}$ as in Proposition 2 . In that case, we can fix some sufficiently small $\delta>0$ and obtain a density formula $\displaystyle\mathbb{E}\left[\log\rho_{\delta}\left(X_{\delta}\right)\,|\,X_{0 }=x_{0}\right]$ $\displaystyle=-\frac{1+\log(2\pi)}{2}d-\int_{\delta}^{1}D(t,x_{0})\mathrm{d}t$ (3.1c) $$

for a smoothed approximation of $\log\rho_{0}(x_{0})$.
This kind of proximity is often used to circumvent non-smoothness
target distributions in diffusion model literature (e.g., ; Chen et al., (); ; Benton et al., ()).
We leave some more discussions to Appendix C.

###### Theorem 1 .

###### Proposition 1 .

###### Proposition 2 .

### 3.2 From continuous time to discrete time

In this section, to avoid ambiguity, we will use $(X_{t}^{\mathsf{sde}})_{0\leq t<1}$
to denote the continuous-time diffusion process (2.4)
studied in the previous section, while keep using $\{X_{t}:1\leq t\leq T\}$
to denote the forward process (2.1). The density
formula (3.1) is not readily implementable
because of its continuous-time nature. Consider time discretization
over the grid

$$ $0<t_{1}<t_{2}<\cdots<t_{T}<t_{T+1}=1\qquad\text{where}\qquad t_{i}\coloneqq 1- \overline{\alpha}_{i}\quad(1\leq i\leq T).$ $$

Recall that the forward process $X_{1},\ldots,X_{T}$ has the same
marginal distribution as $X_{t_{1}}^{\mathsf{sde}},\ldots,X_{t_{T}}^{\mathsf{sde}}$
snapshoted from the diffusion process (2.4). This
gives the following approximation of the density formula (3.1a):

$$ $\displaystyle\log\rho_{0}(x_{0})$ $\displaystyle\overset{\text{(i)}}{\approx}\mathbb{E}\left[\log\rho_{t_{1}}(X_{ t_{1}}^{\mathsf{sde}})\,|\,X_{0}^{\mathsf{sde}}=x_{0}\right]$ $\displaystyle\overset{\text{(ii)}}{\approx}-\frac{1+\log(2\pi t_{1})}{2}d-\sum _{i=1}^{T}\frac{t_{i+1}-t_{i}}{2(1-t_{i})}\mathbb{E}\Big{[}\Big{\|}\frac{X_{t_ {i}}^{\mathsf{sde}}-\sqrt{1-t_{i}}X_{0}^{\mathsf{sde}}}{t_{i}}+\nabla\log\rho_ {t_{i}}(X_{t}^{\mathsf{sde}})\Big{\|}_{2}^{2}\,|\,X_{0}^{\mathsf{sde}}=x_{0} \Big{]}$ $\displaystyle\overset{\text{(iii)}}{\approx}-\frac{1+\log\left(2\pi t_{1} \right)}{2}d-\sum_{i=1}^{T}\frac{t_{i+1}-t_{i}}{2t_{i}(1-t_{i})}\mathbb{E}_{ \varepsilon\sim\mathcal{N}(0,I_{d})}\Big{[}\big{\|}\varepsilon-\widehat{ \varepsilon}_{i}(\sqrt{1-t_{i}}x_{0}+\sqrt{t_{i}}\varepsilon)\big{\|}_{2}^{2} \Big{]}.$ $$

In step (i) we approximate $\log\rho_{0}(x_{0})$ with a smoothed
proxy; see the discussion around (3.1c)
for details; step (ii) applies (3.1c),
where we compute the integral $\int_{t_{1}}^{1}d/(2t)\mathrm{d}t=-(d/2)\log t_{1}$
in closed form and approximate the integral

$$ $\int_{t_{1}}^{1}\frac{1}{2(1-t)}\mathbb{E}\Big{[}\Big{\|}\frac{X_{t}^{\mathsf{ sde}}-\sqrt{1-t}X_{0}^{\mathsf{sde}}}{t}+\nabla\log\rho_{t}(X_{t}^{\mathsf{sde }})\Big{\|}_{2}^{2}\,|\,X_{0}^{\mathsf{sde}}=x_{0}\Big{]}\mathrm{d}t;$ $$

step (iii) follows from $X_{t_{i}}^{\mathsf{sde}}\overset{\text{d}}{=}\sqrt{1-t_{i}}x_{0}+\sqrt{t_{i}}\varepsilon$
for $\varepsilon\sim\mathcal{N}(0,I_{d})$ conditional on $X_{0}^{\mathsf{sde}}=x_{0}$,
and the relation

$$ $\nabla\log\rho_{t_{i}}=\nabla\log q_{i}=s_{i}^{\star}(x)=-\sqrt{t_{i}} \varepsilon_{i}^{\star}(x)\approx-\sqrt{t_{i}}\widehat{\varepsilon}_{i}(x).$ $$

In practice, we need to choose the learning rates $\{\beta_{t}:1\leq t\leq T\}$
such that the grid becomes finer as $T$ becomes large. More specifically,
we require

$$ $t_{i+1}-t_{i}=\overline{\alpha}_{i}-\overline{\alpha}_{i+1}=\overline{\alpha_{ i}}\beta_{i+1}\leq\beta_{i+1}\quad(1\leq i\leq T-1)$ $$

to be small (roughly of order $O(1/T)$), and $t_{1}=\beta_{1}$ and
$1-t_{T}=\overline{\alpha}_{T}$ to be vanishingly small (of order
$T^{-c}$ for some sufficiently large constant $c>0$); see e.g.,
; Benton et al., () for learning rate schedules
satisfying these properties. Finally, we replace the time steps $\{t_{i}:1\leq i\leq T\}$
with the learning rates for the forward process to achieve(^2^22Here we define $\alpha_{T+1}=0$ to accommodate the last term in the summation.)

$$ $\displaystyle\log\rho_{0}(x_{0})$ $\displaystyle\approx-\frac{1+\log\left(2\pi\beta_{1}\right)}{2}d-\sum_{t=1}^{T }\frac{1-\alpha_{t+1}}{2(1-\overline{\alpha}_{t})}\mathbb{E}_{\varepsilon\sim \mathcal{N}(0,I_{d})}\Big{[}\big{\|}\varepsilon-\widehat{\varepsilon}_{t}( \sqrt{\overline{\alpha}_{t}}x_{0}+\sqrt{1-\overline{\alpha}_{t}}\varepsilon) \big{\|}_{2}^{2}\Big{]},$ (3.2) $$

The density approximation (3.2) can
be evaluated with the trained epsilon predictors.

### 3.3 Comparison with other results

The density formulas (3.1) expresses the
density of $X_{0}$ using the score function along the continuous-time
limit of the forward process of the diffusion model. Other forms of
score-based density formulas can be derived using normalizing flows.
Notice that the probability flow ODE of the SDE (2.4)
is

$$ $\dot{x}_{t}=v_{t}(x_{t})\qquad\text{where}\qquad v_{t}(x)=-\frac{x-\nabla\log \rho_{t}(x)}{2(1-t)};$ (3.3) $$

namely, if we draw a particle $x_{0}\sim\rho_{0}$ and evolve it according
to the ODE (3.3) to get the trajectory $t\to x_{t}$
for $t\in[0,1)$, then $x_{t}\sim\rho_{t}$. See e.g.,
for the derivation of this result.

Under some smoothness condition, we can use the results developed
in Grathwohl et al., (); Albergo et al., () to show that
for any given $x_{0}$

$$ $\log\rho_{t}(x_{t})-\log\rho_{0}(x_{0})=-\int_{0}^{t}\mathsf{Tr}\left(\frac{ \partial}{\partial x}v_{s}(x_{s})\right)\mathrm{d}s=\int_{0}^{t}\frac{d- \mathsf{tr}\big{(}\nabla^{2}\log\rho_{s}(x_{s})\big{)}}{2(1-s)}\mathrm{d}s.$ (3.4) $$

Here $t\to x_{t}$ is the solution to the ODE (3.3)
with initial condition $x_{0}$. Since the ODE system (3.3)
is based on the score functions (hence $x_{t}$ can be numerically
solved), and the integral in (3.4)
is based on the Jacobian of the score functions, we may take $t\to 1$
and use the fact that $\rho_{t}(\cdot)\to\phi(\cdot)$ to obtain a
score-based density formula

$$ $\log\rho_{0}(x_{0})=-\frac{d}{2}\log(2\pi)-\frac{1}{2}\left\|x_{1}\right\|_{2} ^{2}-\int_{0}^{1}\frac{d-\mathsf{tr}\big{(}\nabla^{2}\log\rho_{s}(x_{s})\big{) }}{2(1-s)}\mathrm{d}s.$ (3.5) $$

However, numerically, this formula is more difficult to compute than
our formula (3.1) for the following reasons.
First, (3.5) involves the Jacobian
of the score functions, which are more challenging to estimate than
the score functions themselves. In fact, existing convergence guarantees
for DDPM do not depend on the accurate estimation of the Jacobian
of the score functions (Benton et al.,, ; ; Chen et al.,, ; Li and Yan,, ).
Second, using this density formula requires solving the ODE (3.3)
accurately to obtain $x_{1}$, which might not be numerically stable,
especially when the score function is not accurately estimated at
early stages, due to error propagation. In contrast, computing (3.1)
only requires evaluating a few Gaussian integrals (which can be efficiently
approximated by the Monte Carlo method) and is more stable to score
estimation error.

## 4 Implications

In the previous section, we established a density formula

$$ $\displaystyle\log q_{0}(x)$ $\displaystyle\approx\underbrace{-\frac{1+\log\left(2\pi\beta_{1}\right)}{2}d}_ {\eqqcolon C_{0}^{\star}}-\sum_{t=1}^{T}\underbrace{\frac{1-\alpha_{t+1}}{2(1- \overline{\alpha}_{t})}\mathbb{E}_{\varepsilon\sim\mathcal{N}(0,I_{d})}\Big{[} \big{\|}\varepsilon-\varepsilon_{t}^{\star}(\sqrt{\overline{\alpha}_{t}}x+ \sqrt{1-\overline{\alpha}_{t}}\varepsilon)\big{\|}_{2}^{2}\Big{]}}_{\eqqcolon L _{t-1}^{\star}(x)}$ (4.1) $$

up to discretization error (which vanishes as $T$ becomes large)
and score estimation error. In this section, we will discuss the implications
of this formula in various generative and learning frameworks.

### 4.1 Certifying the validity of optimizing ELBO in DDPM

The seminal work (Ho et al.,, ) established the variational
lower bound (VLB), also known as the evidence lower bound (ELBO),
of the log-likelihood

$$ $\displaystyle\log p_{0}(x)$ $\displaystyle\geq-\sum_{t=2}^{T}\underbrace{\mathbb{E}_{x_{t}\sim p_{X_{t}|X_{ 0}}(\cdot\,|\,x)}\mathsf{KL}\left(p_{X_{t-1}|X_{t},X_{0}}(\cdot\,|\,x_{t},x)\, \|\,p_{Y_{t-1}|Y_{t}}(\cdot\,|\,x_{t})\right)}_{\eqqcolon L_{t-1}(x)}$ $\displaystyle\qquad-\underbrace{\mathsf{KL}\left(p_{Y_{T}}(\cdot)\,\|\,p_{X_{T }|X_{0}}(\cdot\,|\,x)\right)}_{\eqqcolon L_{T}(x)}+\underbrace{\mathbb{E}_{x_{ 1}\sim p_{X_{1}|X_{0}}(\cdot\,|\,x)}\left[\log p_{Y_{0}|Y_{1}}(x\,|\,x_{1}) \right]}_{\eqqcolon C_{0}(x)},$ (4.2) $$

where the reverse process $(Y_{t})_{0\leq t\leq T}$ was defined in
Section 2.1, and $p_{0}$ is the density of $Y_{0}$.
Under the coefficient design recommended by Li and Yan, ()
(other reasonable designs also lead to similar conclusions)

$$ $\eta_{t}=1-\alpha_{t}\qquad\text{and}\qquad\sigma_{t}^{2}=\frac{\left(1-\alpha _{t}\right)\left(\alpha_{t}-\overline{\alpha}_{t}\right)}{1-\overline{\alpha}_ {t}},$ (4.3) $$

it can be computed that for each $2\leq t\leq T$:

$$ $L_{t-1}(x)=\frac{1-\alpha_{t}}{2(\alpha_{t}-\overline{\alpha}_{t})}\mathbb{E}_ {\varepsilon\sim\mathcal{N}(0,I_{d})}\left[\big{\|}\varepsilon-\varepsilon_{t} (\sqrt{\overline{\alpha}_{t}}x+\sqrt{1-\overline{\alpha}_{t}}\varepsilon)\big{ \|}_{2}^{2}\right].$ $$

We can verify that (i) for each $2\leq t\leq T$, the coefficients
in $L_{t-1}$ from (4.2) and $L_{t-1}^{\star}$ from (4.1)
are identical up to higher-order error; (ii) when $T$ is large, $L_{T}$
becomes vanishingly small; and (iii) the function

$$ $C_{0}(x)=-\frac{1+\log\left(2\pi\beta_{1}\right)}{2}d+O(\beta_{1})=C_{0}^{ \star}+O(\beta_{1})$ $$

is nearly a constant. See Appendix D.1 for details.
It is worth highlighting that as far as we know, existing literature
haven’t pointed out that $C_{0}(x)$ is nearly a constant. For instance,
Ho et al., () discretize this term to obtain discrete log-likelihood
(see Section 3.3 therein), which is unnecessary in view of our observation.
Additionally, some later works falsely claim that $C_{0}(x)$ is negligible,
as we will discuss in the following sections.

Now we discuss the validity of optimizing the variational bound for training DDPMs.
Our discussion shows that

$$ $\displaystyle\underbrace{\mathsf{KL}(q_{0}\parallel p_{0})}_{\eqqcolon\mathcal {L}(\varepsilon_{1},\ldots,\varepsilon_{T})}$ $\displaystyle=-\mathbb{E}_{x\sim q_{0}}[\log p_{0}(x)]-H(q_{0})\leq\underbrace {\mathbb{E}_{x\sim q_{0}}[L(x)]-C_{0}^{\star}-H(q_{0})+o(1)}_{\eqqcolon \mathcal{L}_{\mathsf{vb}}(\varepsilon_{1},\ldots,\varepsilon_{T})},$ (4.4) $$

where $H(q_{0})=-\int\log q_{0}(x)\mathrm{d}q_{0}$ is the entropy
of $q_{0}$, and $L(x)$ denotes the widely used (negative) ELBO(^3^33We follow the convention in existing literature to remove the last two terms $L_{T}(x)$ and $C_{0}(x)$ from (4.2) in the ELBO.)

$$ $L(x)\coloneqq\sum_{t=1}^{T}\frac{1-\alpha_{t+1}}{2(1-\overline{\alpha}_{t})} \mathbb{E}_{\varepsilon\sim\mathcal{N}(0,I_{d})}\Big{[}\big{\|}\varepsilon- \varepsilon_{t}(\sqrt{\overline{\alpha}_{t}}x+\sqrt{1-\overline{\alpha}_{t}} \varepsilon)\big{\|}_{2}^{2}\Big{]}.$ $$

The true objective of DDPM is to learn the epsilon predictors $\varepsilon_{1},\ldots,\varepsilon_{T}$
that minimizes $\mathcal{L}$ in (4.4),
while in practice, the optimization target is the variational bound $\mathcal{L}_{\mathsf{vb}}$.
It is known that the global minimizer for

$$ $\mathbb{E}_{x\sim q_{0}}\left[L(x)\right]=\sum_{t=1}^{T}\frac{1-\alpha_{t+1}}{ 2(1-\overline{\alpha}_{t})}\mathbb{E}_{x\sim q_{0},\varepsilon\sim\mathcal{N}( 0,I_{d})}\Big{[}\big{\|}\varepsilon-\varepsilon_{t}(\sqrt{\overline{\alpha}_{t }}x+\sqrt{1-\overline{\alpha}_{t}}\varepsilon)\big{\|}_{2}^{2}\Big{]}$ (4.5) $$

is exactly $\widehat{\varepsilon}_{t}(\cdot)\equiv\varepsilon_{t}^{\star}(\cdot)$
for each $1\leq t\leq T$ (see Appendix D.1). Although
in practice the optimization is based on samples from the target
distribution $q_{0}$ (instead of the population level expectation
over $q_{0}$) and may not find the exact global minimizer, we consider the ideal scenario where the learned epsilon predictors $\widehat{\varepsilon}_{t}$
equal $\varepsilon_{t}^{\star}$ to facilitate discussion. When
$\varepsilon_{t}=\varepsilon_{t}^{\star}$ for each $t$, according
to (4.1), we have

$$ $L(x)\approx-\log q_{0}(x)+C_{0}^{\star}.$ (4.6) $$

Taking (4.4) and (4.6) together gives

$$ $0\leq\mathcal{L}(\widehat{\varepsilon}_{1},\ldots,\widehat{\varepsilon}_{T}) \leq\mathcal{L}_{\mathsf{vb}}(\widehat{\varepsilon}_{1},\ldots,\widehat{ \varepsilon}_{T})\approx-\mathbb{E}_{x\sim q_{0}}[\log q(x)]+C_{0}^{\star}-C_{ 0}^{\star}-H(q_{0})=0,$ (4.7) $$

namely the minimizer for $\mathcal{L}_{\mathsf{vb}}$ approximately minimizes $\mathcal{L}$, and the optimal value is asymptotically zero when the number of steps $T$ becomes large. This suggests that by minimizing the variational bound $\mathcal{L}_{\mathsf{vb}}$, the resulting generator distribution $p_{0}$ is guaranteed to be close to the target distribution $q_{0}$ in KL divergence.

Some experimental evidence suggests that using reweighted coefficients
can marginally improve empirical performance. For example, Ho et al., ()
suggests that in practice, it might be better to use uniform coefficients
in the ELBO

$$ $\displaystyle L_{\mathsf{simple}}(x)$ $\displaystyle\coloneqq\frac{1}{T}\sum_{i=1}^{T}\mathbb{E}_{\varepsilon\sim \mathcal{N}(0,I_{d})}\Big{[}\big{\|}\varepsilon-\widehat{\varepsilon}_{t_{i}}( \sqrt{\overline{\alpha}_{t}}x+\sqrt{1-\overline{\alpha}_{t}}\varepsilon)\big{ \|}_{2}^{2}\Big{]}$ (4.8) $$

when trainging DDPM to improve sampling quality.(^4^44Note that the optimal epsilon predictors $\widehat{\varepsilon}_{t}$ for $L$ and $L_{\mathsf{simple}}$ are the same, but in practice, we may not find the optimal predictors. This practical strategy is beyond the scope of our theoretical result, and implies that the influence of terms from different steps needs more careful investigation. We conjecture that this is mainly because the estimation error for terms when $t$ is close to zero is larger, hence smaller coefficients for these terms can improve performance.) This strategy has been adopted by many later works. In the following
sections, we will discuss the role of using the ELBO in different
applications. While the original literature might use the modified
ELBO (4.8), in our discussion we will stick
to the original ELBO (4.6) to gain intuition from our theoretical
findings.

### 4.2 Understanding the role of regularization in GAN

Generative Adversarial Networks (GANs) are a powerful and flexible
framework for learning the unknown probability distribution $p_{\mathsf{data}}$
that generates a collection of training data (Goodfellow et al.,, ).
GANs operate on a game between a generator $G$ and a discriminator
$D$, typically implemented using neural networks. The generator $G$
takes a random noise vector $z$ sampled from a simple distribution
$p_{\mathsf{noise}}$ (e.g., Gaussian) and maps it to a data sample
resembling the training data, aiming for the distribution of $G(z)$
to be close to $p_{\mathsf{data}}$. Meanwhile, the discriminator
$D$ determines whether a sample $x$ is real (i.e., drawn from $p_{\mathsf{data}}$)
or fake (i.e., produced by the generator), outputting the probability
$D(x)$ of the former. The two networks engage in a zero-sum game:

$$ $\min_{G}\max_{D}V(G,D)\coloneqq\mathbb{E}_{x\sim p_{\mathsf{data}}}[\log D(x)] +\mathbb{E}_{z\sim p_{\mathsf{noise}}}[\log(1-D(G(z)))],$ $$

with the generator striving to produce realistic data while the discriminator
tries to distinguish real data from fake. The generator and discriminator
are trained iteratively(^5^55While the most natural update rule for the generator is $G\leftarrow\arg\min\,\mathbb{E}_{z\sim p_{\mathsf{noise}}}[\log(1-D(G(z)))]$, both schemes are used in practice and have similar performance. Our choice is for consistency with Xia et al., (), and our analysis can be extended to the other choice.)

$$ $\displaystyle D$ $\displaystyle\leftarrow\arg\min\,-\mathbb{E}_{x\sim p_{\mathsf{data}}}[\log D( x)]-\mathbb{E}_{z\sim p_{\mathsf{noise}}}[\log(1-D(G(z)))],$ $\displaystyle G$ $\displaystyle\leftarrow\arg\min\,-\mathbb{E}_{z\sim p_{\mathsf{noise}}}[\log D (G(z))]$ $$

to approach the Nash equilibrium $(G^{\star},D^{\star})$, where the
distribution of $G^{\star}(z)$ with $z\sim p_{\mathsf{noise}}$ matches
the target distribution $p_{\mathsf{data}}$, and $D(x)=1/2$ for
all $x$.

It is believed that adding a regularization term to make the generated
samples fit the VLB can improve the sampling quality of the generative
model. For example, Xia et al., () proposed adding the VLB
$L(x)$ as a regularization term to the objective function, where
$\{\widehat{\varepsilon}_{t_{i}}(\cdot):1\leq i\leq T\}$ are the
learned epsilon predictors for $p_{\mathsf{data}}$. The training
procedure then becomes

$$ $\displaystyle D$ $\displaystyle\leftarrow\arg\min\,-\mathbb{E}_{x\sim p_{\mathsf{data}}}[\log D( x)]-\mathbb{E}_{z\sim p_{\mathsf{noise}}}[\log(1-D(G(z)))],$ $\displaystyle G$ $\displaystyle\leftarrow\arg\min\,-\mathbb{E}_{z\sim p_{\mathsf{noise}}}[\log D (G(z))]+\lambda\mathbb{E}_{z\sim p_{\mathsf{noise}}}[L(G(z))],$ $$

where $\lambda>0$ is some tuning parameter. However, it remains unclear
what exactly is optimized through the above objective. According to
our theory, $L(x)\approx-\log p_{\mathsf{data}}(x)+C_{0}^{\star}.$
Assuming that this approximation is exact for intuitive understanding,
the unique Nash equilibrium $(G_{\lambda},D_{\lambda})$ satisfies

$$ $\displaystyle p_{G_{\lambda}}(x)$ $\displaystyle=\big{(}zp_{\mathsf{data}}(x)^{\lambda}-1\big{)}_{+}p_{\mathsf{ data}}(x)$ $$

for some normalizing factor $z>0$, where $p_{G_{\lambda}}$ is the
density of $G_{\lambda}(z)$ with $z\sim p_{\mathsf{noise}}$. See
Appendix D.2 for details. This can be viewed as amplifying
the density $p_{\mathsf{data}}$ wherever it is not too small, while
zeroing out the density where $p_{\mathsf{data}}$ is vanishingly
small (which is difficult to estimated accurately), thus improving
the sampling quality.

### 4.3 Confirming the use of ELBO in diffusion classifier

Motivated by applications like image classification and text-to-image
diffusion model, we consider a joint underlying distribution $p_{0}(x,c)$,
where typically $x$ is the image data and the latent variable $c$
is the class index or text embedding, taking values in a finite set
$\mathcal{C}$. For each $c\in\mathcal{C}$, we train a diffusion
model for the conditional data distribution $p_{0}(x\,|\,c)$, which
provides a set of epsilon predictors $\big{\{}\widehat{\varepsilon}_{t}(x;c):1\leq t\leq T,c\in\mathcal{C}\big{\}}$.
Assuming a uniform prior over $\mathcal{C}$, we can use Bayes’ formula
to obtain:

$$ $p_{0}\left(c\,|\,x\right)=\frac{p_{0}\left(c\right)p_{0}\left(x\,|\,c_{i} \right)}{\sum_{j\in\mathcal{C}}p_{0}\left(c_{j}\right)p_{0}\left(x\,|\,c_{j} \right)}=\frac{p_{0}\left(x\,|\,c\right)}{\sum_{j\in\mathcal{C}}p_{0}\left(x\, |\,c_{j}\right)}.$ $$

for each $c\in\mathcal{C}$. Recent work () proposed
to use the ELBO(^6^66The original paper adopted uniform coefficients; see the last paragraph of Section 4.1 for discussion.)

$$ $-L(x;c)\coloneqq-\sum_{t=1}^{T}\frac{1-\alpha_{t+1}}{2(1-\overline{\alpha}_{t} )}\mathbb{E}_{\varepsilon\sim\mathcal{N}(0,I_{d})}\Big{[}\big{\|}\varepsilon- \widehat{\varepsilon}_{t}(\sqrt{\overline{\alpha}_{t}}x+\sqrt{1-\overline{ \alpha}_{t}}\varepsilon;c)\big{\|}_{2}^{2}\Big{]}$ $$

as an approximate class-conditional log-likelihood $\log p_{0}(x\,|\,c)$
for each $c\in\mathcal{C}$, which allows them to obtain a posterior
distribution

$$ $\widehat{p}_{0}\left(c\,|\,x\right)=\frac{\exp\left(-L(x;c)\right)}{\sum_{j\in \mathcal{C}}\exp\left(-L(x;c_{j})\right)}.$ (4.9) $$

Our theory suggests that $-L(x;c)\approx\log p_{0}(x\,|\,c)-C_{0}^{\star}$,
where $C_{0}^{\star}=-[1+\log(2\pi\beta_{1})]d/2$ is a universal
constant that does not depend on $p_{0}$ and $c$. This implies that

$$ $\widehat{p}_{0}\left(c\,|\,x\right)\approx\frac{\exp\left(\log p_{0}(x\,|\,c)- C_{0}^{\star}\right)}{\sum_{j\in\mathcal{C}}\exp\left(\log p_{0}(x\,|\,c_{j})- C_{0}^{\star}\right)}=\frac{p_{0}\left(x\,|\,c\right)}{\sum_{j\in\mathcal{C}}p _{0}\left(x\,|\,c_{j}\right)}=p_{0}\left(c\,|\,x\right)$ $$

providing theoretical justification for using the computed posterior
$\widehat{p}_{0}$ in classification tasks.

It is worth mentioning that, although this framework was proposed
in the literature (), it remains a heuristic method
before our work. For example, in general, replacing the intractable
log-likelihood with a lower bound does not guarantee good performance,
as they might not be close. Additionally, recall that there is a term
$C_{0}(x)$ in the ELBO (4.2). claimed
that “*Since $T=1000$ is large and $\log p_{\theta}(x_{0}\,|\,x_{1},c)$
is typically small, we choose to drop this term*”. However this
argument is not correct, as we already computed in Section 4.1
that this term

$$ $C_{0}(x)=-\frac{1+\log\left(2\pi\beta_{1}\right)}{2}d+O(\beta_{1})$ $$

can be very large since $\beta_{1}$ is typically very close to 0 0.
In view of our results, the reason why this term can be dropped is
that it equals a universal constant that does not depend on the image
data $x$ and the class index $c$, thus it does not affect the posterior
(4.9).

### 4.4 Demystifying the diffusion loss in autoregressive models

Finally, we use our results to study a class of diffusion loss recently
introduced in Li et al., (), in the context of autoregressive
image generation. Let $x^{k}$ denote the next token to be predicted,
and $z$ be the condition parameterized by an autoregressive network
$z=f(x^{1},\ldots,x^{k-1})$ based on previous tokens as input. The goal is to train the network $z=f(\cdot)$ together with a diffusion
model $\{\varepsilon_{t}(\cdot\,;z):1\leq t\leq T\}$ such
that $\widehat{p}(x\,|\,z)$ (induced by the diffusion model)
with $z=f(x^{1},\ldots,x^{k-1})$ can predict the next token $x^{k}$.

The diffusion loss is defined as follows: for some weights $w_{t}\geq 0$,
let

$$ $\displaystyle L(z,x)$ $\displaystyle=\sum_{t=1}^{T}w_{t}\mathbb{E}_{\varepsilon\sim\mathcal{N}(0,I_{d })}\Big{[}\big{\|}\varepsilon-\varepsilon_{t}(\sqrt{\overline{\alpha}_{t}}x+ \sqrt{1-\overline{\alpha}_{t}}\varepsilon;z)\big{\|}_{2}^{2}\Big{]}.$ (4.10) $$

With training data $\{(x_{i}^{1},\ldots,x_{i}^{k}):1\leq i\leq n\}$,
we can train the autoregressive network $f(\cdot)$ and the diffusion model by minimizing
the following empirical risk:

$$ $\mathop{\arg\min}_{f,\varepsilon_{1},\ldots,\varepsilon_{T}}\frac{1}{n}\sum_{i =1}^{n}L\left(f(x_{i}^{1},\ldots,x_{i}^{k-1}),x_{i}^{k}\right).$ (4.11) $$

To gain intuition from our theoretical results, we take the weights
in the diffusion loss (4.10) to be the coefficients
in the ELBO (4.6), and for each $z$, suppose that the
learned diffusion model for $p(x^{k}\,|\,z)$ is already good enough, which
returns the set of epsilon predictors $\{\widehat{\varepsilon}_{t}(\cdot\,;z):1\leq t\leq T\}$
for the probability distribution of $x^{k}$ conditioned on $z$.
Under this special case, our approximation result (4.6)
shows that

$$ $L(z,x)\approx-\log p(x\,|\,z)+C_{0}^{\star},$ $$

which suggests that the training objective for the network $f$ in (4.11)
can be viewed as approximate MLE, as the loss function

$$ $\frac{1}{n}\sum_{i=1}^{n}L\left(f(x_{i}^{1},\ldots,x_{i}^{k-1}),x_{i}^{k} \right)\approx-\frac{1}{n}\sum_{i=1}^{n}\log p(x_{i}^{k}\,|\,f(x_{i}^{1}, \ldots,x_{i}^{k-1}))+C_{0}^{\star}$ $$

represents the negative log-likelihood function (up to an additive
constant) of the observed $x_{1}^{k},\ldots,x_{n}^{k}$ in terms of
$f$.

## 5 Proof of Theorem 1

Recall the definition of the stochastic process $(X_{t})_{0\leq t\leq 1}$

$$ $\mathrm{d}X_{t}=-\frac{1}{2(1-t)}X_{t}\mathrm{d}t+\frac{1}{\sqrt{1-t}}\mathrm{ d}B_{t}.$ $$

Define $Y_{t}\coloneqq X_{t}/\sqrt{1-t}$ for any $0\leq t<1$, and
let $f(t,x)=x/\sqrt{1-t}$, we can use Itô’s formula to show that

$$ $\displaystyle\mathrm{d}Y_{t}$ $\displaystyle=\mathrm{d}f\left(t,X_{t}\right)=\frac{\partial f}{\partial t} \left(t,X_{t}\right)\mathrm{d}t+\nabla_{x}f\left(t,X_{t}\right)^{\top}\mathrm{ d}X_{t}+\frac{1}{2}\mathrm{d}X_{t}^{\top}\nabla_{x}^{2}f\left(t,X_{t}\right) \mathrm{d}X_{t}$ $\displaystyle=\frac{X_{t}}{2(1-t)^{3/2}}\mathrm{d}t+\frac{1}{\sqrt{1-t}}\left( -\frac{1}{2(1-t)}X_{t}\mathrm{d}t+\frac{1}{\sqrt{1-t}}\mathrm{d}B_{t}\right)= \frac{\mathrm{d}B_{t}}{1-t}.$ (5.1) $$

Therefore the Itô process $Y_{t}$ is a martingale, which is easier
to handle. Let $g(t,y)=\log\rho_{t}(\sqrt{1-t}y)$, and we can express
$\log\rho_{t}(x)=g(t,x/\sqrt{1-t})$. In view of Itô’s formula, we
have

$$ $\displaystyle\mathrm{d}\log\rho_{t}(X_{t})$ $\displaystyle=\mathrm{d}g\left(t,Y_{t}\right)\overset{\text{(i)}}{=}\frac{ \partial g}{\partial t}\left(t,Y_{t}\right)\mathrm{d}t+\nabla_{y}g\left(t,Y_{t }\right)^{\top}\mathrm{d}Y_{t}+\frac{1}{2}\mathrm{d}Y_{t}^{\top}\nabla_{y}^{2} g\left(t,Y_{t}\right)\mathrm{d}Y_{t}$ $\displaystyle\overset{\text{(ii)}}{=}\frac{\partial g}{\partial t}\left(t,Y_{t }\right)\mathrm{d}t+\frac{1}{1-t}\nabla_{y}g\left(t,Y_{t}\right)^{\top}\mathrm {d}B_{t}+\frac{1}{2\left(1-t\right)^{2}}\mathrm{d}B_{t}^{\top}\nabla_{y}^{2}g \left(t,Y_{t}\right)\mathrm{d}B_{t}$ $\displaystyle\overset{\text{(iii)}}{=}\frac{\partial g}{\partial t}\left(t,Y_{ t}\right)\mathrm{d}t+\frac{1}{1-t}\nabla_{y}g\left(t,Y_{t}\right)^{\top} \mathrm{d}B_{t}+\frac{1}{2\left(1-t\right)^{2}}\mathsf{tr}\left(\nabla_{y}^{2} g\left(t,Y_{t}\right)\right)\mathrm{d}t.$ (5.2) $$

Here step (i) follows from the Itô rule, step (ii) utilizes (5.1),
while step (iii) can be derived from the Itô calculus. Then we investigate
the three terms above. Notice that

$$ $\displaystyle\nabla_{y}g\left(t,y\right)\,|\,_{y=Y_{t}}$ $\displaystyle=\frac{\nabla_{y}\rho_{t}(\sqrt{1-t}y)}{\rho_{t}(\sqrt{1-t}Y_{t}) }\,|\,_{y=Y_{t}}=\frac{\nabla_{x}\rho_{t}(X_{t})\sqrt{1-t}}{\rho_{t}(X_{t})}= \sqrt{1-t}\nabla\log\rho_{t}\left(X_{t}\right),$ (5.3) $$

and similarly, we have

$$ $\nabla_{y}^{2}g\left(t,y\right)\,|\,_{y=Y_{t}}=\left(1-t\right)\nabla^{2}\log \rho_{t}\left(X_{t}\right).$ (5.4) $$

Substituting (5.3) and (5.4)
back into (5.2) gives

$$ $\displaystyle\mathrm{d}\log\rho_{t}(X_{t})$ $\displaystyle=\frac{\partial g}{\partial t}\left(t,Y_{t}\right)\mathrm{d}t+ \frac{1}{\sqrt{1-t}}\nabla\log\rho_{t}\left(X_{t}\right)^{\top}\mathrm{d}B_{t} +\frac{1}{2\left(1-t\right)}\mathsf{tr}\left(\nabla^{2}\log\rho_{t}\left(X_{t} \right)\right)\mathrm{d}t.$ $$

or equivalently, for any given $0<t_{1}<t_{2}<1$, we have

$$ $\log\rho_{t}\left(X_{t}\right)\Big{|}_{t_{1}}^{t_{2}}=\int_{t_{1}}^{t_{2}}\Big {[}\frac{\partial g}{\partial t}\left(t,Y_{t}\right)+\frac{\mathsf{tr}\left( \nabla^{2}\log\rho_{t}\left(X_{t}\right)\right)}{2\left(1-t\right)}\Big{]} \mathrm{d}t+\int_{t_{1}}^{t_{2}}\frac{1}{\sqrt{1-t}}\nabla\log\rho_{t}\left(X_ {t}\right)^{\top}\mathrm{d}B_{t}.$ (5.5) $$

Conditional on $X_{0}$, we take expectation on both sides of (5.5)
to achieve

$$ $\mathbb{E}\left[\log\rho_{t_{2}}\left(X_{t_{2}}\right)-\log\rho_{t_{1}}\left(X _{t_{1}}\right)\,|\,X_{0}\right]=\mathbb{E}\left[\int_{t_{1}}^{t_{2}}\left( \frac{\partial g}{\partial t}\left(t,Y_{t}\right)+\frac{1}{2\left(1-t\right)} \mathsf{tr}\left(\nabla^{2}\log\rho_{t}\left(X_{t}\right)\right)\right)\mathrm {d}t\,|\,X_{0}\right].$ (5.6) $$

We need the following lemmas, whose proof can be found at the end
of this section.

###### Claim 1 .

For any $0<t<1$ and any $y\in\mathbb{R}^{d}$,
we have

$$ $\frac{\partial g}{\partial t}\left(t,y\right)=-\frac{d}{2t}+\frac{1}{2t^{2}} \int_{x_{0}}\rho_{X_{0}|X_{t}}\left(x_{0}\,|\,\sqrt{1-t}y\right)\|y-x_{0}\|_{2 }^{2}\mathrm{d}x_{0}.$ $$

###### Claim 2 .

For any $0<t<1$ and any $x\in\mathbb{R}^{d}$,
we have

$$ $\displaystyle\mathsf{tr}\left(\nabla^{2}\log\rho_{t}(x)\right)$ $\displaystyle=-\frac{d}{t}-\big{\|}\nabla\log\rho_{t}(x)\big{\|}_{2}^{2}+\frac {1}{t^{2}}\int\big{\|}x-\sqrt{1-t}x_{0}\big{\|}_{2}^{2}\rho_{X_{0}|X_{t}}\left (x_{0}\,|\,x\right)\mathrm{d}x_{0}.$ $$

It also admits the lower bound

$$ $\mathsf{tr}\left(\nabla^{2}\log\rho_{t}(x)\right)\geq-\frac{d}{t}.$ $$

Therefore for any $x$ and $y=x/\sqrt{1-t}$, we know that

$$ $\displaystyle\frac{\partial g}{\partial t}\left(t,y\right)+\frac{1}{2\left(1-t \right)}\mathsf{tr}\left(\nabla^{2}\log\rho_{t}\left(x\right)\right)$ $\displaystyle\geq-\frac{d}{2t}-\frac{d}{2\left(1-t\right)t}\geq-\frac{d}{\left (1-t\right)t}.$ (5.7) $$

Hence we have

$$ $\displaystyle\mathbb{E}\left[\log\rho_{t_{2}}\left(X_{t_{2}}\right)-\log\rho_{ t_{1}}\left(X_{t_{1}}\right)\,|\,X_{0}\right]$ $\displaystyle\qquad\overset{\text{(i)}}{=}\mathbb{E}\left[\int_{t_{1}}^{t_{2}} \left(\frac{\partial g}{\partial t}\left(t,Y_{t}\right)+\frac{1}{2\left(1-t \right)}\mathsf{tr}\left(\nabla^{2}\log\rho_{t}\left(X_{t}\right)\right)+\frac {d}{\left(1-t\right)t}\right)\mathrm{d}t\,|\,X_{0}\right]-\int_{t_{1}}^{t_{2}} \frac{d}{\left(1-t\right)t}\mathrm{d}t$ $\displaystyle\qquad\overset{\text{(ii)}}{=}\int_{t_{1}}^{t_{2}}\mathbb{E}\left [\left(\frac{\partial g}{\partial t}\left(t,Y_{t}\right)+\frac{1}{2\left(1-t \right)}\mathsf{tr}\left(\nabla^{2}\log\rho_{t}\left(X_{t}\right)\right)+\frac {d}{\left(1-t\right)t}\right)\,|\,X_{0}\right]\mathrm{d}t-\int_{t_{1}}^{t_{2}} \frac{d}{\left(1-t\right)t}\mathrm{d}t$ $\displaystyle\qquad=\int_{t_{1}}^{t_{2}}\mathbb{E}\left[\left(\frac{\partial g }{\partial t}\left(t,Y_{t}\right)+\frac{1}{2\left(1-t\right)}\mathsf{tr}\left( \nabla^{2}\log\rho_{t}\left(X_{t}\right)\right)\right)\,|\,X_{0}\right]\mathrm {d}t.$ (5.8) $$

Here step (i) follows from (5.6), and its validity
is guaranteed by

$$ $\int_{t_{1}}^{t_{2}}\frac{d}{t\left(1-t\right)}\mathrm{d}t=\log\frac{t_{2} \left(1-t_{1}\right)}{t_{1}\left(1-t_{2}\right)}<+\infty,$ $$

while step (ii) utilizes Tonelli’s Theorem, and the nonnegativity
of the integrand is ensured by (5.7). Taking Claims 1 and 2
collectively, we know that for any $x$ and $y=x/\sqrt{1-t}$,

$$ $\displaystyle\frac{\partial g}{\partial t}\left(t,y\right)-\frac{\mathsf{tr} \left(\nabla^{2}\log\rho_{t}\left(x\right)\right)}{2\left(1-t\right)}$ $\displaystyle=\frac{d+\big{\|}\nabla\log\rho_{t}(x)\big{\|}_{2}^{2}}{2\left(1- t\right)}+\frac{1}{2t^{2}}\int_{x_{0}}\rho_{X_{0}|X_{t}}\left(x_{0}\,|\,\sqrt{ 1-t}y\right)\|y-x_{0}\|_{2}^{2}\mathrm{d}x_{0}$ $\displaystyle\mathbb{\qquad}-\frac{1}{2\left(1-t\right)}\frac{1}{t^{2}}\int \big{\|}x-\sqrt{1-t}x_{0}\big{\|}_{2}^{2}\rho_{X_{0}|X_{t}}\left(x_{0}\,|\,x \right)\mathrm{d}x_{0}$ $\displaystyle=\frac{d+\big{\|}\nabla\log\rho_{t}(x)\big{\|}_{2}^{2}}{2\left(1- t\right)}.$ (5.9) $$

Putting (5.8) and (5.9) together,
we arrive at

$$ $\mathbb{E}\left[\log\rho_{t_{2}}\left(X_{t_{2}}\right)-\log\rho_{t_{1}}\left(X _{t_{1}}\right)\,|\,X_{0}\right]=\int_{t_{1}}^{t_{2}}\mathbb{E}\bigg{[}\frac{d +\big{\|}\nabla\log\rho_{t}(X_{t})\big{\|}_{2}^{2}}{2\left(1-t\right)}+\frac{1 }{1-t}\mathsf{tr}\left(\nabla^{2}\log\rho_{t}\left(X_{t}\right)\right)\,|\,X_{ 0}\bigg{]}\mathrm{d}t.$ (5.10) $$

Notice that conditional on $X_{0}$, we have $X_{t}\sim\mathcal{N}(\sqrt{1-t}X_{0},tI_{d})$.
Then we have

$$ $\displaystyle\mathbb{E}\left[\log\rho_{t_{2}}\left(X_{t_{2}}\right)-\log\rho_{ t_{1}}\left(X_{t_{1}}\right)\,|\,X_{0}\right]$ $\displaystyle\qquad\overset{\text{(i)}}{=}\int_{t_{1}}^{t_{2}}\mathbb{E}\bigg{ [}\frac{d+\big{\|}\nabla\log\rho_{t}(X_{t})\big{\|}_{2}^{2}}{2\left(1-t\right) }+\frac{1}{1-t}\nabla\log\rho_{t}(X_{t})^{\top}\frac{X_{t}-\sqrt{1-t}X_{0}}{t} \,|\,X_{0}\bigg{]}\mathrm{d}t$ $\displaystyle\qquad\overset{\text{(ii)}}{=}\int_{t_{1}}^{t_{2}}\bigg{(}\frac{1 }{2(1-t)}\mathbb{E}\Big{[}\Big{\|}\frac{X_{t}-\sqrt{1-t}X_{0}}{t}+\nabla\log \rho_{t}(X_{t})\Big{\|}_{2}^{2}\,|\,X_{0}\Big{]}-\frac{d}{2t}\bigg{)}\mathrm{d}t$ $$

Here step (i) follows from (5.10) and an application
of Stein’s lemma

$$ $\displaystyle\mathbb{E}\bigg{[}\nabla\log\rho_{t}(X_{t})^{\top}\left(X_{t}- \sqrt{1-t}X_{0}\right)\,|\,X_{0}\bigg{]}$ $\displaystyle=t\mathbb{E}\left[\mathsf{tr}\big{(}\nabla^{2}\log\rho_{t}(X_{t}) \big{)}\,|\,X_{0}\right],$ $$

while step (ii) holds since

$$ $\mathbb{E}\Big{[}\Big{\|}\frac{X_{t}-\sqrt{1-t}X_{0}}{t}\Big{\|}_{2}^{2}\Big{] }=\frac{d}{t}.$ $$

###### Claim 1 .

###### Claim 2 .

#### Proof of Claim 1 .

For any $t\in(0,1)$, since $X_{t}=\sqrt{1-t}X_{0}+\sqrt{t}Z$, we
have

$$ $\rho_{t}(\sqrt{1-t}y)=\int_{x_{0}}(2\pi t)^{-d/2}\exp\Big{(}-\frac{(1-t)\|y-x_ {0}\|_{2}^{2}}{2t}\Big{)}\rho_{0}(\mathrm{d}x_{0}).$ (5.11) $$

Note that here $\rho_{0}(\cdot)$ stands for the law of $X_{0}$.
Hence we have

$$ $\displaystyle\frac{\partial g}{\partial t}\left(t,y\right)$ $\displaystyle=\frac{\partial}{\partial t}\log\rho_{t}(\sqrt{1-t}y)=\text{$ \frac{1}{\rho_{t}(\sqrt{1-t}y)}$}\frac{\partial}{\partial t}\rho_{t}(\sqrt{1-t }y)$ $\displaystyle=\text{$\frac{1}{\rho_{t}(\sqrt{1-t}y)}$}\int_{x_{0}}(2\pi)^{-d/2 }\bigg{[}-\frac{d}{2}t^{-d/2-1}\exp\Big{(}-\frac{(1-t)\|y-x_{0}\|_{2}^{2}}{2t} \Big{)}$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+t^{-d/2}\exp\Big{(}- \frac{(1-t)\|y-x_{0}\|_{2}^{2}}{2t}\Big{)}\frac{\|y-x_{0}\|_{2}^{2}}{2t^{2}} \bigg{]}\rho_{0}(\mathrm{d}x_{0})$ $\displaystyle=\text{$\frac{1}{\rho_{t}(\sqrt{1-t}y)}$}\int_{x_{0}}\rho_{X_{t}| X_{0}}\left(\sqrt{1-t}y\,|\,x_{0}\right)\left[-\frac{d}{2t}+\frac{\|y-x_{0}\|_ {2}^{2}}{2t^{2}}\right]\rho_{0}(\mathrm{d}x_{0})$ $\displaystyle=\int_{x_{0}}\Big{(}-\frac{d}{2t}+\frac{\|y-x_{0}\|_{2}^{2}}{2t^{ 2}}\Big{)}\rho_{X_{0}|X_{t}}\left(\mathrm{d}x_{0}\,|\,\sqrt{1-t}y\right)$ $$

as claimed.

#### Proof of Claim 2 .

Notice that we can express

$$ $\displaystyle\nabla\log\rho_{t}(x)$ $\displaystyle=-\frac{1}{t}\mathbb{E}\left[X_{t}-\sqrt{1-t}X_{0}\,|\,X_{t}=x \right]=-\frac{1}{t}\int_{x_{0}}\left(x-\sqrt{1-t}x_{0}\right)\rho_{X_{0}|X_{t }}\left(\mathrm{d}x_{0}\,|\,x\right);$ $$

see Chen et al., () for the proof of this relationship. Then
we can compute

$$ $\displaystyle\nabla^{2}\log\rho_{t}(x)$ $\displaystyle=-\frac{1}{t}\Big{\{}I_{d}+\frac{1}{t}\mathbb{E}\left[X_{t}-\sqrt {1-t}X_{0}\,|\,X_{t}=x\right]\mathbb{E}\left[X_{t}-\sqrt{1-t}X_{0}\,|\,X_{t}=x \right]^{\top}$ $\displaystyle\qquad\qquad-\frac{1}{t}\mathbb{E}\left[\left(X_{t}-\sqrt{1-t}X_{ 0}\right)\left(X_{t}-\sqrt{1-t}X_{0}\right)^{\top}\,|\,X_{t}=x\right]\Big{\}}$ $\displaystyle=-\frac{1}{t}\Big{\{}I_{d}+\frac{1}{t}\Big{[}\int\left(x-\sqrt{1- t}x_{0}\right)\rho_{X_{0}|X_{t}}\left(\mathrm{d}x_{0}\,|\,x\right)\Big{]}\Big{ [}\int\left(x-\sqrt{1-t}x_{0}\right)\rho_{X_{0}|X_{t}}\left(\mathrm{d}x_{0}\,| \,x\right)\Big{]}^{\top}$ $\displaystyle\qquad\qquad-\frac{1}{t}\int\left(x-\sqrt{1-t}x_{0}\right)\left(x -\sqrt{1-t}x_{0}\right)^{\top}\rho_{X_{0}|X_{t}}\left(\mathrm{d}x_{0}\,|\,x \right)\Big{\}}.$ $$

Hence we have

$$ $\displaystyle\mathsf{tr}\left(\nabla^{2}\log\rho_{t}(x)\right)$ $\displaystyle=-\frac{1}{t}\bigg{\{}d+\frac{1}{t}\big{\|}\int\left(x-\sqrt{1-t} x_{0}\right)\rho_{X_{0}|X_{t}}\left(\mathrm{d}x_{0}\,|\,x\right)\big{\|}_{2}^{ 2}-\frac{1}{t}\int\big{\|}x-\sqrt{1-t}x_{0}\big{\|}_{2}^{2}\rho_{X_{0}|X_{t}} \left(\mathrm{d}x_{0}\,|\,x\right)\bigg{\}}$ $\displaystyle=-\frac{d}{t}-\frac{1}{t^{2}}\big{\|}\nabla\log\rho_{t}(x)\big{\| }_{2}^{2}+\frac{1}{t^{2}}\int\big{\|}x-\sqrt{1-t}x_{0}\big{\|}_{2}^{2}\rho_{X_ {0}|X_{t}}\left(x_{0}\,|\,x\right)\mathrm{d}x_{0}.$ $$

By Jensen’s inequality, we know that

$$ $\mathsf{tr}\left(\nabla^{2}\log\rho_{t}(x)\right)\geq-\frac{d}{t}.$ $$

## 6 Discussion

This paper develops a score-based density formula that expresses the
density function of a target distribution using the score function
along a continuous-time diffusion process that bridges this distribution
and standard Gaussian. By connecting this diffusion process with the
forward process of score-based diffusion models, our results provide
theoretical support for training DDPMs by optimizing the ELBO,
and offer novel insights into several applications of diffusion
models, including GAN training and diffusion classifiers.

Our work opens several directions for future research.
First, our theoretical results are established for the continuous-time
diffusion process. It is crucial to carefully analyze the error induced
by time discretization, which could inform the number of steps required
for the results in this paper to be valid in practice. Additionally,
while our results provide theoretical justification for using the
ELBO (4.6) as a proxy for the negative log-likelihood of
the target distribution, they do not cover other practical variants
of ELBO with modified weights (e.g., the simplified ELBO (4.8)).
Extending our analysis to other diffusion processes might yield new
density formulas incorporating these modified weights. Lastly, further
investigation is needed into other applications of this score-based
density formula, including density estimation and inverse problems.

## Acknowledgements

G. Li is supported in part by the Chinese University of Hong Kong
Direct Grant for Research. Y. Yan was supported in part by a Norbert
Wiener Postdoctoral Fellowship from MIT.

## Appendix A Proof of Proposition 1

We establish the desired result by sandwiching $\mathbb{E}[\log\rho_{t}(X_{t})\,|\,X_{0}=x_{0}]$
and find its limit as $t\to 1$ . We first record that the density
of $X_{t}$ can be expressed as

$$ $\rho_{t}\left(x\right)=\mathbb{E}_{X_{0}}\bigg{[}(2\pi t)^{-d/2}\exp\bigg{(}- \frac{\|x-\sqrt{1-t}X_{0}\|_{2}^{2}}{2t}\bigg{)}\bigg{]},$ (A.1) $$

since $X_{t}\overset{\text{d}}{=}\sqrt{1-t}X_{0}+\sqrt{t}Z$ for an
independent variable $Z\sim\mathcal{N}(0,I_{d})$.

#### Lower bounding 𝔼 ⁢ [ log ⁡ ρ t ⁢ ( X t ) | X 0 = x 0 ] 𝔼 delimited-[] conditional subscript 𝜌 𝑡 subscript 𝑋 𝑡 subscript 𝑋 0 subscript 𝑥 0 \mathbb{E}[\log\rho_{t}(X_{t})\,|\,X_{0}=x_{0}] blackboard_E [ roman_log italic_ρ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) | italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] .

Starting from (A.1), for any $x\in\mathbb{R}^{d}$
and any $0<t<1$,

$$ $\displaystyle\log\rho_{t}(x)$ $\displaystyle=\log\mathbb{E}_{X_{0}}\bigg{[}(2\pi t)^{-d/2}\exp\bigg{(}-\frac{ \|x-\sqrt{1-t}X_{0}\|_{2}^{2}}{2t}\bigg{)}\bigg{]}$ $\displaystyle\overset{\text{(i)}}{\geq}\log\bigg{\{}(2\pi t)^{-d/2}\exp\bigg{( }-\mathbb{E}_{X_{0}}\bigg{[}\frac{\|x-\sqrt{1-t}X_{0}\|_{2}^{2}}{2t}\bigg{]} \bigg{)}\bigg{\}}$ $\displaystyle=-\frac{d}{2}\log(2\pi t)-\mathbb{E}_{X_{0}}\bigg{[}\frac{\|x- \sqrt{1-t}X_{0}\|_{2}^{2}}{2t}\bigg{]}$ $\displaystyle=-\frac{d}{2}\log(2\pi t)-\frac{\|x\|_{2}^{2}}{2t}-\frac{1-t}{2t} \mathbb{E}[\|X_{0}\|_{2}^{2}]+\frac{\sqrt{1-t}}{t}\mathbb{E}[x^{\top}X_{0}]$ $\displaystyle\overset{\text{(ii)}}{=}-\frac{d}{2}\log(2\pi t)-\left(1+O(\sqrt{ 1-t})\right)\frac{\|x\|_{2}^{2}}{2t}+O(\sqrt{1-t})\mathbb{E}[\|X_{0}\|_{2}^{2}].$ $$

Here step (i) follows from Jensen’s inequality and the fact that $e^{-x}$
is a convex function, while step (ii) follows from elementary inequalities

$$ $\left|\mathbb{E}[x^{\top}X_{0}]\right|\leq\mathbb{E}\big{[}\|x\|\|X_{0}\|_{2} \big{]}\leq\frac{1}{2}\mathbb{E}\big{[}\|x\|_{2}^{2}+\|X_{0}\|_{2}^{2}\big{]}.$ $$

This immediately gives, for any given $x_{0}\in\mathbb{R}^{d}$ and
any $0<t<1$,

$$ $\mathbb{E}[\log\rho_{t}(X_{t})\,|\,X_{0}=x_{0}]\geq\underbrace{-\frac{d}{2} \log(2\pi t)-\frac{1+O(\sqrt{1-t})}{2t}\mathbb{E}\big{[}\|X_{t}\|_{2}^{2}\,|\, X_{0}=x_{0}\big{]}+O(\sqrt{1-t})\mathbb{E}[\|X_{0}\|_{2}^{2}]}_{\eqqcolon f_{x _{0}}(t)}.$ (A.2a) Since $\mathbb{E}[\|X_{0}\|_{2}^{2}]<\infty$ , it is straightforward to check that $\displaystyle\lim_{t\to 1-}f_{x_{0}}(t)$ $\displaystyle=-\frac{d}{2}\log(2\pi)-\lim_{t\to 1-}\frac{1}{2}\mathbb{E}\left[ \|\sqrt{1-t}x_{0}+\sqrt{t}Z\|_{2}^{2}\right]\qquad\text{for }Z\sim\mathcal{N}( 0,I_{d})$ $\displaystyle=-\frac{d}{2}\log(2\pi)-\frac{d}{2}.$ (A.2b) $$

#### Upper bounding 𝔼 ⁢ [ log ⁡ ρ t ⁢ ( X t ) | X 0 = x 0 ] 𝔼 delimited-[] conditional subscript 𝜌 𝑡 subscript 𝑋 𝑡 subscript 𝑋 0 subscript 𝑥 0 \mathbb{E}[\log\rho_{t}(X_{t})\,|\,X_{0}=x_{0}] blackboard_E [ roman_log italic_ρ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) | italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] .

Towards that, we need to obtain point-wise upper bound for $\log\rho_{t}(x)$.
Since the desired result only depends on the limiting behavior when
$t\to 1$, from now on we only consider $t>0.9$, under which

$$ $(1-t)^{1/4}<\frac{1}{2}\sqrt{\log\frac{1}{1-t}}$ $$

holds. It would be helpful to develop the upper bound for the following
two cases separately.

- •
For any $(1-t)^{1/4}<\|x\|_{2}<0.5\sqrt{\log 1/(1-t)}$, we
have
$\displaystyle\log\rho_{t}(x)$
$\displaystyle\overset{\text{(a)}}{\leq}\log\mathbb{E}_{X_{0}}\bigg{[}(2\pi t)^
{-d/2}\exp\bigg{(}-\frac{(\|x\|_{2}-(1-t)^{1/4})^{2}}{2t}\bigg{)}+
\operatorname{\mathds{1}}\big{(}\|X_{0}\|_{2}>(1-t)^{-1/4}\big{)}\bigg{]}$
$\displaystyle\overset{\text{(b)}}{\leq}-\frac{d}{2}\log(2\pi t)-\frac{(\|x\|_{
2}-(1-t)^{1/4})^{2}}{2t}+\exp\Big{(}\frac{(\|x\|_{2}-(1-t)^{1/4})^{2}}{2t}\Big
{)}\mathbb{P}\big{(}\|X_{0}\|_{2}>(1-t)^{-1/4}\big{)}$
$\displaystyle\overset{\text{(c)}}{\leq}-\frac{d}{2}\log(2\pi t)-\frac{(\|x\|_{
2}-(1-t)^{1/4})^{2}}{2t}+\exp\Big{(}\frac{\|x\|_{2}^{2}}{2t}\Big{)}\mathbb{E}[
\|X_{0}\|_{2}^{2}](1-t)^{1/2}$
$\displaystyle\overset{\text{(d)}}{\leq}-\frac{d}{2}\log(2\pi t)-\frac{(\|x\|_{
2}-(1-t)^{1/4})^{2}}{2t}+\mathbb{E}[\|X_{0}\|_{2}^{2}](1-t)^{1/4}.$
(A.3)
Here step (a) follows from (A.1); step (b) holds
since $\log(x+y)\leq\log x+y/x$ holds for any $x>0$ and $y\geq 0$;
step (c) follows from $\|x\|_{2}>(1-t)^{1/4}$ and Chebyshev’s
inequality; while step (d) holds since $\|x\|_{2}<0.5\sqrt{\log 1/(1-t)}$.
- •
For $\|x\|_{2}\geq 0.5\sqrt{\log 1/(1-t)}$ or $\|x\|\leq(1-t)^{1/4}$,
we will use the naive upper bound
$\displaystyle\log\rho_{t}(x)$
$\displaystyle\leq-\frac{d}{2}\log(2\pi t)<0,$
(A.4)
where the first relation simply follows from (A.1)
and the second relation holds when $t>0.9$.

Then we have

$$ $\displaystyle\mathbb{E}[\log\rho_{t}(X_{t})\,|\,X_{0}=x_{0}]\overset{\text{(i) }}{\leq}\mathbb{E}[\log\rho_{t}(X_{t})\operatorname{\mathds{1}}\left\{(1-t)^{1 /4}<\|X_{t}\|_{2}<0.5\sqrt{\log 1/(1-t)}\right\}\,|\,X_{0}=x_{0}]$ $\displaystyle\qquad\overset{\text{(ii)}}{\leq}\mathbb{E}\bigg{[}\Big{(}-\frac{ d}{2}\log(2\pi t)-\frac{(\|x\|_{2}-(1-t)^{1/4})^{2}}{2t}+\mathbb{E}[\|X_{0}\|_ {2}^{2}](1-t)^{1/4}\Big{)}$ $\displaystyle\qquad\qquad\qquad\cdot\operatorname{\mathds{1}}\Big{\{}(1-t)^{1/ 4}<\|X_{t}\|_{2}<0.5\sqrt{\log 1/(1-t)}\Big{\}}\,|\,X_{0}=x_{0}\bigg{]}$ $\displaystyle\qquad=\underbrace{\Big{(}-\frac{d}{2}\log(2\pi t)+\mathbb{E}[\|X _{0}\|_{2}^{2}](1-t)^{1/4}\Big{)}\mathbb{P}\left((1-t)^{1/4}<\|X_{t}\|_{2}<0.5 \sqrt{\log 1/(1-t)}\right)}_{\eqqcolon\overline{g}_{x_{0}}(t)}$ $\displaystyle\qquad\qquad-\underbrace{\mathbb{E}\left[\frac{(\|X_{t}\|_{2}-(1- t)^{1/4})^{2}}{2t}\operatorname{\mathds{1}}\left\{(1-t)^{1/4}<\|X_{t}\|_{2}<0. 5\sqrt{\log 1/(1-t)}\right\}\,|\,X_{0}=x_{0}\right]}_{\eqqcolon\widetilde{g}_{ x_{0}}(t)}.$ $$

Here step (i) follows from (A.4), while step (ii)
utilizes (A.3). Since $X_{t}$ is a continuous random
variable for any $t\in(0,1)$, we have

$$ $\lim_{t\to 1-}\mathbb{P}\left((1-t)^{1/4}<\|X_{t}\|_{2}<0.5\sqrt{\log 1/(1-t)} \right)=1.$ $$

Therefore we know that

$$ $\lim_{t\to 1-}\overline{g}_{x_{0}}(t)=-\frac{d}{2}\log(2\pi).$ $$

Recall that $X_{t}\overset{\text{d}}{=}\sqrt{1-t}X_{0}+\sqrt{t}Z$
for a Gaussian variable $Z\sim\mathcal{N}(0,I_{d})$ independent of
$X_{0}$, we can express

$$ $\displaystyle\widetilde{g}_{x_{0}}(t)$ $\displaystyle=\mathbb{E}\left[\frac{(\|\sqrt{t}Z+\sqrt{1-t}x_{0}\|_{2}-(1-t)^{ 1/4})^{2}}{2t}\operatorname{\mathds{1}}\left\{(1-t)^{1/4}<\|\sqrt{t}Z+\sqrt{1- t}x_{0}\|_{2}<\frac{1}{2}\sqrt{\log 1/(1-t)}\right\}\right]$ $\displaystyle=\int\underbrace{\frac{(\|\sqrt{t}z+\sqrt{1-t}x_{0}\|_{2}-(1-t)^{ 1/4})^{2}}{2t}\operatorname{\mathds{1}}\bigg{\{}(1-t)^{1/4}<\|\sqrt{t}z+\sqrt{ 1-t}x_{0}\|_{2}<\frac{1}{2}\sqrt{\log\frac{1}{1-t}}\bigg{\}}\phi(z)}_{ \eqqcolon h_{t}(z)}\mathrm{d}z,$ $$

where $\phi(z)=(2\pi)^{-d/2}\exp(-\|z\|_{2}^{2}/2)$ is the
density function of $\mathcal{N}(0,I_{d})$. For any $t\in(0.9,1)$,
we have

$$ $h_{t}(z)\leq\|\sqrt{t}z+\sqrt{1-t}x_{0}\|_{2}^{2}\phi(z)\leq 2(\|z\|_{2}^{2}+ \|x_{0}\|_{2}^{2})\phi(z)\eqqcolon h(z),$ $$

and it is straightforward to check that

$$ $\int h(z)\mathrm{d}z=2d+2\|x_{0}\|_{2}^{2}<\infty.$ $$

By dominated convergence theorem, we know that

$$ $\lim_{t\to 1-}\widetilde{g}_{x_{0}}(t)=\lim_{t\to 1-}\int h_{t}(z)\mathrm{d}z= \int\lim_{t\to 1-}h_{t}(z)\mathrm{d}z=\int\frac{\|z\|_{2}^{2}}{2}\phi(z) \mathrm{d}z=\frac{d}{2}.$ $$

Therefore we have

$$ $\mathbb{E}[\log\rho_{t}(X_{t})\,|\,X_{0}=x_{0}]\leq g_{x_{0}}(t)\qquad\text{ where}\qquad g_{x_{0}}(t)\coloneqq\overline{g}_{x_{0}}(t)-\widetilde{g}_{x_{0} }(t),$ (A.5a) such that $\lim_{t\to 1-}g_{x_{0}}(t)=\lim_{t\to 1-}\overline{g}_{x_{0}}(t)-\lim_{t\to 1- }\widetilde{g}_{x_{0}}(t)=-\frac{d}{2}\log(2\pi)-\frac{d}{2}.$ (A.5b) $$

#### Conclusion.

By putting together (A.2) and (A.5), we know that
for any $t\in(0.9,1)$

$$ $f_{x_{0}}(t)\leq\mathbb{E}[\log\rho_{t}(X_{t})\,|\,X_{0}=x_{0}]\leq g_{x_{0}}( t)\qquad\text{and}\qquad\lim_{t\to 1-}f_{x_{0}}(t)=\lim_{t\to 1-}g_{x_{0}}(t)= -\frac{d}{2}\log(2\pi)-\frac{d}{2}.$ $$

By the sandwich theorem, we arrive at the desired result

$$ $\lim_{t\to 1-}\mathbb{E}[\log\rho_{t}(X_{t})\,|\,X_{0}=x_{0}]=-\frac{d}{2}\log (2\pi)-\frac{d}{2}.$ $$

## Appendix B Proof of Proposition 2

Suppose that $L\coloneqq\sup_{x}\|\nabla^{2}\log\rho_{0}(x)\|$.
The following claim will be useful in establishing the proposition,
whose proof is deferred to the end of this section.

###### Claim 3 .

There exists some $t_{0}>0$
such that

$$ $\sup_{x}\|\nabla^{2}\log\rho_{t}(x)\|\leq 4L.$ (B.1) $$

holds for any $0\leq t\leq t_{0}$.

Equipped with Claim 3, we know that for any
$t\leq t_{0}$,

$$ $\displaystyle\mathbb{E}\big{[}\log\rho_{t}(X_{t})\,|\,X_{0}=x_{0}\big{]}= \mathbb{E}\big{[}\log\rho_{t}(\sqrt{1-t}x_{0}+\sqrt{t}Z)\big{]}$ $\displaystyle\qquad\overset{\text{(i)}}{=}\mathbb{E}\big{[}\log\rho_{t}(\sqrt{ 1-t}x_{0})+\sqrt{t}Z^{\top}\nabla\log\rho_{t}(\sqrt{1-t}x_{0})+O(Lt)\|Z\|_{2}^ {2}\big{]}$ $\displaystyle\qquad\overset{\text{(ii)}}{=}\log\rho_{t}(\sqrt{1-t}x_{0})+O(Ldt)$ $\displaystyle\qquad\overset{\text{(iii)}}{=}\log\int_{x}\rho_{0}(x)(2\pi t)^{- d/2}\exp\Big{(}-\frac{(1-t)\|x-x_{0}\|_{2}^{2}}{2t}\Big{)}\mathrm{d}x+O(L\sqrt {dt})$ $\displaystyle\qquad=(1-t)^{-d/2}\log\int_{x}\rho_{0}(x)\left(\frac{2\pi t}{1-t }\right)^{-d/2}\exp\Big{(}-\frac{(1-t)\|x-x_{0}\|_{2}^{2}}{2t}\Big{)}\mathrm{d }x+O(L\sqrt{dt}),$ (B.2) $$

where $Z\sim\mathcal{N}(0,I_{d})$. Here step (i) follows from (B.1)
in Claim 3; step (ii) holds since $\mathbb{E}[Z]=0$
and $\mathbb{E}[\|Z\|_{2}^{2}]=d$; while step (iii) follows
from (5.11). It is straightforward to check
that

$$ $\int_{x}\rho_{0}(x)\left(\frac{2\pi t}{1-t}\right)^{-d/2}\exp\Big{(}-\frac{(1- t)\|x-x_{0}\|_{2}^{2}}{2t}\Big{)}\mathrm{d}x$ $$

is the density of $\rho_{0}*\mathcal{N}(0,t/(1-t))$ evaluated at
$x_{0}$, which taken collectively with the assumption that $\rho_{0}(\cdot)$
is continuous yields

$$ $\lim_{t\to 0+}\int_{x}\rho_{0}(x)\left(\frac{2\pi t}{1-t}\right)^{-d/2}\exp \Big{(}-\frac{(1-t)\|x-x_{0}\|_{2}^{2}}{2t}\Big{)}\mathrm{d}x=\rho_{0}(x_{0}).$ $$

Therefore we can take $t\to 0+$ in (B.2) to
achieve

$$ $\lim_{t\to 0+}\mathbb{E}\big{[}\log\rho_{t}(X_{t})\,|\,X_{0}=x_{0}\big{]}=\log \rho_{0}(x_{0})$ $$

as claimed.

###### Claim 3 .

#### Proof of Claim 3 .

The conditional density of $X_{0}$ given $X_{t}=x$ is

$$ $p_{X_{0}|X_{t}}(x_{0}\,|\,x)=\frac{p_{X_{0}}(x_{0})p_{X_{t}|X_{0}}(x\,|\,x_{0} )}{p_{X_{t}}(x)}=\frac{\rho_{0}(x_{0})}{\rho_{t}(x)}(2\pi t)^{-d/2}\exp\left(- \frac{\|x-\sqrt{1-t}x_{0}\|_{2}^{2}}{2t}\right),$ (B.3) $$

which leads to

$$ $\displaystyle-\nabla_{x_{0}}^{2}\log p_{X_{0}|X_{t}}(x_{0}\,|\,x)$ $\displaystyle=-\nabla_{x_{0}}^{2}\log\rho_{0}(x_{0})+\frac{1}{2t}\nabla_{x_{0} }^{2}\|x-\sqrt{1-t}x_{0}\|_{2}^{2}$ $\displaystyle=-\nabla_{x_{0}}^{2}\log\rho_{0}(x_{0})+\frac{1-t}{t}I_{d}\succeq \left(\frac{1-t}{t}-L\right)I_{d}.$ $$

Therefore we know that

$$ $-\nabla_{x_{0}}^{2}\log p_{X_{0}|X_{t}}(x_{0}\,|\,x)\succeq\frac{1}{2t}I_{d} \qquad\text{for}\qquad t\leq\frac{1}{2(L+1)},$ (B.4) $$

namely the conditional distribution of $X_{0}$ given $X_{t}=x$ is
$1/(2t)$-strongly log-concave for any $x$, when $t\leq 1/2(L+1)$.
By writting

$$ $\displaystyle\rho_{t}(x)$ $\displaystyle=p_{X_{t}}(x)=\int\phi(z)p_{\sqrt{1-t}X_{0}}\left(x-\sqrt{t}z \right)\mathrm{d}z=(1-t)^{-d/2}\int\phi(z)\rho_{0}\left(\frac{x-\sqrt{t}z}{ \sqrt{1-t}}\right)\mathrm{d}z,$ (B.5) $$

we can express the score function of $\rho_{t}$ as

$$ $\displaystyle\nabla\log\rho_{t}(x)$ $\displaystyle=\frac{\nabla\rho_{t}(x)}{\rho_{t}(x)}=(1-t)^{-\frac{d+1}{2}} \frac{1}{\rho_{t}(x)}\int\phi(z)\nabla\rho_{0}\left(\frac{x-\sqrt{t}z}{\sqrt{1 -t}}\right)\mathrm{d}z$ $\displaystyle=(1-t)^{-\frac{d+1}{2}}\frac{1}{\rho_{t}(x)}\int\phi(z)\rho_{0} \left(\frac{x-\sqrt{t}z}{\sqrt{1-t}}\right)\nabla\log\rho_{0}\left(\frac{x- \sqrt{t}z}{\sqrt{1-t}}\right)\mathrm{d}z$ (B.6) $\displaystyle\overset{\text{(i)}}{=}(1-t)^{-\frac{d+1}{2}}\left(\frac{1-t}{t} \right)^{d/2}\frac{1}{\rho_{t}(x)}\int\phi\left(\frac{x-\sqrt{1-t}x_{0}}{\sqrt {t}}\right)\rho_{0}\left(x_{0}\right)\nabla\log\rho_{0}\left(x_{0}\right) \mathrm{d}x_{0}$ $\displaystyle\overset{\text{(ii)}}{=}\frac{1}{\sqrt{1-t}}\int p_{X_{0}|X_{t}}( x_{0}\,|\,x)\nabla\log\rho_{0}\left(x_{0}\right)\mathrm{d}x_{0}=\frac{1}{\sqrt {1-t}}\mathbb{E}\left[\nabla\log\rho_{0}\left(X_{0}\right)\,|\,X_{t}=x\right].$ (B.7) $$

Here step (i) uses the change of variable $x_{0}=(x-\sqrt{t}z)/\sqrt{1-t}$,
while step (ii) follows from (B.3). Starting
from (B.6), we take the derivative to achieve

$$ $\displaystyle\nabla^{2}\log\rho_{t}(x)$ $\displaystyle=\underbrace{(1-t)^{-\frac{d}{2}+1}\frac{1}{\rho_{t}(x)}\int\phi( z)\rho_{0}\left(\frac{x-\sqrt{t}z}{\sqrt{1-t}}\right)\nabla\log\rho_{0}\left( \frac{x-\sqrt{t}z}{\sqrt{1-t}}\right)\left[\nabla\log\rho_{0}\left(\frac{x- \sqrt{t}z}{\sqrt{1-t}}\right)\right]^{\top}\mathrm{d}z}_{\eqqcolon H_{1}(x)}$ $\displaystyle\qquad+\underbrace{(1-t)^{-\frac{d}{2}+1}\frac{1}{\rho_{t}(x)} \int\phi(z)\rho_{0}\left(\frac{x-\sqrt{t}z}{\sqrt{1-t}}\right)\nabla^{2}\log \rho_{0}\left(\frac{x-\sqrt{t}z}{\sqrt{1-t}}\right)\mathrm{d}z}_{\eqqcolon H_{ 2}(x)}$ $\displaystyle\qquad-\underbrace{(1-t)^{-\frac{d+1}{2}}\frac{1}{\rho_{t}^{2}(x) }\int\phi(z)\rho_{0}\left(\frac{x-\sqrt{t}z}{\sqrt{1-t}}\right)\nabla\log\rho_ {0}\left(\frac{x-\sqrt{t}z}{\sqrt{1-t}}\right)\mathrm{d}z\left[\nabla\rho_{t}( x)\right]^{\top}}_{\eqqcolon H_{3}(x)}.$ (B.8) $$

Then we investigate $H_{1}(x)$, $H_{2}(x)$ and $H_{3}(x)$ respectively.
Regarding $H_{1}(x)$, we have

$$ $\displaystyle H_{1}(x)$ $\displaystyle\overset{\text{(a1)}}{=}(1-t)^{-\frac{d}{2}+1}\left(\frac{1-t}{t} \right)^{d/2}\frac{1}{\rho_{t}(x)}\int\phi\left(\frac{x-\sqrt{1-t}x_{0}}{\sqrt {t}}\right)\rho_{0}\left(x_{0}\right)\nabla\log\rho_{0}\left(x_{0}\right)\left [\nabla\log\rho_{0}\left(x_{0}\right)\right]^{\top}\mathrm{d}z$ $\displaystyle\overset{\text{(b1)}}{=}\frac{1}{1-t}\int p_{X_{0}|X_{t}}(x_{0}\, |\,x)\nabla\log\rho_{0}\left(x_{0}\right)\left[\nabla\log\rho_{0}\left(x_{0} \right)\right]^{\top}\mathrm{d}x_{0}$ $\displaystyle=\frac{1}{1-t}\mathbb{E}\left[\nabla\log\rho_{0}\left(X_{0}\right )\left[\nabla\log\rho_{0}\left(X_{0}\right)\right]^{\top}\,|\,X_{t}=x\right];$ (B.9a) for $H_{2}(x)$ , we have $\displaystyle H_{2}(x)$ $\displaystyle\overset{\text{(a2)}}{=}(1-t)^{-\frac{d}{2}+1}\left(\frac{1-t}{t} \right)^{d/2}\frac{1}{\rho_{t}(x)}\int\phi\left(\frac{x-\sqrt{1-t}x_{0}}{\sqrt {t}}\right)\rho_{0}\left(x_{0}\right)\nabla^{2}\log\rho_{0}\left(\frac{x-\sqrt {t}z}{\sqrt{1-t}}\right)\mathrm{d}x_{0}$ $\displaystyle\overset{\text{(b2)}}{=}\frac{1}{1-t}\int p_{X_{0}|X_{t}}(x_{0}\, |\,x)\nabla^{2}\log\rho_{0}\left(x_{0}\right)\mathrm{d}x_{0}=\frac{1}{1-t} \mathbb{E}\left[\nabla^{2}\log\rho_{0}\left(X_{0}\right)\,|\,X_{t}=x\right];$ (B.9b) for the final term $H_{3}(x)$ , we have $\displaystyle H_{3}(x)$ $\displaystyle\overset{\text{(c)}}{=}-(1-t)^{-\frac{d+1}{2}}\frac{1}{\rho_{t}(x )}\left[\int\phi(z)\rho_{0}\left(\frac{x-\sqrt{t}z}{\sqrt{1-t}}\right)\nabla \log\rho_{0}\left(\frac{x-\sqrt{t}z}{\sqrt{1-t}}\right)\mathrm{d}z\right]\left [\nabla\log\rho_{t}(x)\right]^{\top}$ $\displaystyle\overset{\text{(a3)}}{=}-(1-t)^{-\frac{d+1}{2}}\left(\frac{1-t}{t }\right)^{d/2}\frac{1}{\rho_{t}(x)}\left[\int\phi\left(\frac{x-\sqrt{1-t}x_{0} }{\sqrt{t}}\right)\rho_{0}\left(x_{0}\right)\nabla\log\rho_{0}\left(x_{0} \right)\mathrm{d}x_{0}\right]\left[\nabla\log\rho_{t}(x)\right]^{\top}$ $\displaystyle\overset{\text{(b3)}}{=}-\frac{1}{\sqrt{1-t}}\int p_{X_{0}|X_{t}} (x_{0}\,|\,x)\nabla\log\rho_{0}\left(x_{0}\right)\mathrm{d}x_{0}\left[\nabla \log\rho_{t}(x)\right]^{\top}$ $\displaystyle\overset{\text{(d)}}{=}-\frac{1}{1-t}\mathbb{E}\left[\nabla\log \rho_{0}\left(X_{0}\right)\,|\,X_{t}=x\right]\mathbb{E}\left[\nabla\log\rho_{0 }\left(X_{0}\right)\,|\,X_{t}=x\right]^{\top}.$ (B.9c) $$

Here steps (a1), (a2) and (a3) follow from the
change of variable $x_{0}=(x-\sqrt{t}z)/\sqrt{1-t}$; steps (b1),
(b2) and (b3) utilize (B.3); step (c) follows
from $\nabla\log\rho_{t}(x)=\nabla\rho_{t}(x)/\rho_{t}(x)$; while
step (d) follows from (B.7). Substituting
(B.9) back into (B.8),
we have

$$ $\nabla^{2}\log\rho_{t}(x)=\frac{1}{1-t}\mathbb{E}\left[\nabla^{2}\log\rho_{0} \left(X_{0}\right)\,|\,X_{t}=x\right]+\frac{1}{1-t}\mathsf{cov}\left(\nabla \log\rho_{0}\left(X_{0}\right)\,|\,X_{t}=x\right).$ (B.10) $$

Notice that for any $t\leq 1/2(L+1)$, we have

$$ $\displaystyle\left\|\mathsf{cov}\left(\nabla\log\rho_{0}\left(X_{0}\right)\,| \,X_{t}=x\right)\right\|$ $\displaystyle=\sup_{u\in\mathbb{S}^{d-1}}\mathbb{E}\left[\left[u^{\top}\left( \nabla\log\rho_{0}\left(X_{0}\right)-\mathbb{E}\left[\nabla\log\rho_{0}\left(X _{0}\right)\,|\,X_{t}=x\right]\right)\right]^{2}\,|\,X_{t}=x\right]$ $\displaystyle\overset{\text{(i)}}{\leq}\sup_{u\in\mathbb{S}^{d-1}}\mathbb{E} \left[\left[u^{\top}\left(\nabla\log\rho_{0}\left(X_{0}\right)-\nabla\log\rho_ {0}\left(\mathbb{E}\left[X_{0}\,|\,X_{t}=x\right]\right)\right)\right]^{2}\,| \,X_{t}=x\right]$ $\displaystyle\leq\mathbb{E}\left[\left\|\nabla\log\rho_{0}\left(X_{0}\right)- \nabla\log\rho_{0}\left(\mathbb{E}\left[X_{0}\,|\,X_{t}=x\right]\right)\right \|_{2}^{2}\,|\,X_{t}=x\right]$ $\displaystyle\overset{\text{(ii)}}{\leq}\mathbb{E}\left[\left\|X_{0}-\mathbb{E }\left[X_{0}\,|\,X_{t}=x\right]\right\|_{2}^{2}\,|\,X_{t}=x\right]$ $\displaystyle\overset{\text{(iii)}}{\leq}2tL^{2}d,$ (B.11) $$

Here step (i) holds since for any random variable $X$, $\mathbb{E}[(X-c)^{2}]$
is minimized at $c=\mathbb{E}[X]$; step (ii) holds since the score
function $\nabla\log\rho_{0}(\cdot)$ is $L$-Lipschitz; step (iii)
follows from the Poincaré inequality for log-concave distribution,
and the fact that the conditional distribution of $X_{0}$ given $X_{t}=x$
is $1/2t$-strongly log-concave (cf. (B.4)).
We conclude that

$$ $\displaystyle\left\|\nabla^{2}\log\rho_{t}(x)\right\|$ $\displaystyle\overset{\text{(a)}}{\leq}\frac{1}{1-t}L+\frac{2tL^{2}d}{1-t} \overset{\text{(b)}}{\leq}4L.$ $$

Here step (a) follows from (B.10), (B.11),
and the assumption that $\sup_{x}\|\nabla^{2}\log\rho_{t}(x)\|\leq L$,
while step (b) holds provided that $t\leq\min\{1/2,1/(2Ld)\}$.

## Appendix C More discussions on the density formulas

Although the density formulas (3.1a) have
been rigorously established, it is helpful to inspect the limiting
behavior of the integrand $D(t,x_{0})$ at the boundary to understand
why the integral converges. Throughout the discussion, we let $\varepsilon\sim\mathcal{N}(0,I_{d})$.

- •
As $t\to 0$, we can compute
$\displaystyle D(t,x_{0})$
$\displaystyle\asymp\frac{\mathbb{E}\big{[}\|\varepsilon+\sqrt{t}\nabla\log\rho
_{t}(\sqrt{1-t}x_{0}+\sqrt{t}\varepsilon)\|_{2}^{2}\big{]}-d}{t}$
$\displaystyle\overset{\text{(i)}}{\asymp}\mathbb{E}\big{[}\|\nabla\log\rho_{t}
(\sqrt{1-t}x_{0}+\sqrt{t}\varepsilon)\|_{2}^{2}\big{]}+\frac{1}{\sqrt{t}}
\mathbb{E}\big{[}\varepsilon^{\top}\nabla\log\rho_{t}(\sqrt{1-t}x_{0}+\sqrt{t}
\varepsilon)\big{]}$
$\displaystyle\overset{\text{(ii)}}{\asymp}\mathbb{E}\big{[}\|\nabla\log\rho_{t
}(\sqrt{1-t}x_{0}+\sqrt{t}\varepsilon)\|_{2}^{2}\big{]}+\mathbb{E}\left[
\mathsf{tr}\left(\nabla^{2}\log\rho_{t}(\sqrt{1-t}x_{0}+\sqrt{t}\varepsilon)
\right)\right].$
Here step (i) holds since $\mathbb{E}[\|\varepsilon\|_{2}^{2}]=d$,
while step (ii) follows from Stein’s lemma. Therefore, when the score
functions are reasonably smooth as $t\to 0$, one may expect that the
integrand $D(t,x_{0})$ is of constant order, allowing the integral
to converge at $t=0$.
- •
As $t\to 1$, we can compute
$\displaystyle D(t,x_{0})$
$\displaystyle=\frac{1}{2(1-t)t}\mathbb{E}\big{[}\|\varepsilon+\sqrt{t}\nabla
\log\rho_{t}(\sqrt{1-t}x_{0}+\sqrt{t}\varepsilon)\|_{2}^{2}\big{]}-\frac{d}{2t}$
$\displaystyle\asymp\frac{1}{2(1-t)}\mathbb{E}\big{[}\|\varepsilon+\sqrt{t}
\nabla\log\rho_{t}(\sqrt{1-t}x_{0}+\sqrt{t}\varepsilon)\|_{2}^{2}\big{]}-\frac
{d}{2}.$
Since $\rho_{t}$ converges to $\phi$ as $t\to 1$ and $\nabla\log\phi(x)=-x$,
we have
$\lim_{t\to 1}\varepsilon+\sqrt{t}\nabla\log\rho_{t}(\sqrt{1-t}x_{0}+\sqrt{t}
\varepsilon)=0.$
Hence one may expect that $\mathbb{E}\big{[}\|\varepsilon+\sqrt{t}\nabla\log\rho_{t}(\sqrt{1-t}x_{0}+
\sqrt{t}\varepsilon)\|_{2}^{2}\big{]}$
converges to zero quickly, allowing the integral to converge at $t=1$.

## Appendix D Technical details in Section 4

### D.1 Technical details in Section 4.1

#### Computing L t − 1 ⁢ ( x 0 ) subscript 𝐿 𝑡 1 subscript 𝑥 0 L_{t-1}(x_{0}) italic_L start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) .

Conditional on $X_{t}=x_{t}$ and $X_{0}=x_{0}$, we have

$$ $X_{t-1}\,|\,X_{t}=x_{t},X_{0}=x_{0}\sim\mathcal{N}\left(\frac{\sqrt{\overline{ \alpha}_{t-1}}\beta_{t}}{1-\overline{\alpha}_{t}}x_{0}+\frac{\sqrt{\alpha_{t}} (1-\overline{\alpha}_{t-1})}{1-\overline{\alpha}_{t}}x_{t},\frac{1-\overline{ \alpha}_{t-1}}{1-\overline{\alpha}_{t}}\beta_{t}I_{d}\right),$ $$

and conditional on $Y_{t}=x_{t}$, we have

$$ $Y_{t-1}\,|\,Y_{t}=x_{t}\sim\mathcal{N}\left(\frac{x_{t}+\eta_{t}s_{t}\left(x_{ t}\right)}{\sqrt{\alpha_{t}}},\frac{\sigma_{t}^{2}}{\alpha_{t}}\right).$ $$

Recall that the KL divergence between two $d$-dimensional Gaussian
$\mathcal{N}(\mu_{1},\Sigma_{1})$ and $\mathcal{N}(\mu_{2},\Sigma_{2})$
admits the following closed-form expression:

$$ $\mathsf{KL}\left(\mathcal{N}(\mu_{1},\Sigma_{1})\,\|\,\mathcal{N}(\mu_{2}, \Sigma_{2})\right)=\frac{1}{2}\left[\mathsf{tr}\left(\Sigma_{2}^{-1}\Sigma_{1} \right)+\left(\mu_{2}-\mu_{1}\right)^{\top}\Sigma_{2}^{-1}\left(\mu_{2}-\mu_{1 }\right)-d+\log\det\Sigma_{2}-\log\det\Sigma_{1}\right].$ $$

Then we can check that for $2\leq t\leq T$,

$$ $\mathsf{KL}\big{(}p_{X_{t-1}|X_{t},X_{0}}(\cdot\,|\,x_{t},x_{0})\,\|\,p_{Y_{t- 1}|Y_{t}}(\cdot\,|\,x_{t})\big{)}=\frac{\alpha_{t}}{2\sigma_{t}^{2}}\left\| \frac{\sqrt{\overline{\alpha}_{t-1}}\beta_{t}}{1-\overline{\alpha}_{t}}x_{0}+ \frac{\alpha_{t}-1}{\sqrt{\alpha_{t}}(1-\overline{\alpha}_{t})}x_{t}-\frac{ \eta_{t}s_{t}(x_{t})}{\sqrt{\alpha_{t}}}\right\|_{2}^{2},$ $$

where we use the coefficient design (4.3). This
immediately gives

$$ $\displaystyle L_{t-1}(x_{0})$ $\displaystyle=\frac{\alpha_{t}}{2\sigma_{t}^{2}}\mathbb{E}_{x_{t}\sim p_{X_{t} |X_{0}}(\cdot\,|\,x_{0})}\left[\left\|\frac{\sqrt{\overline{\alpha}_{t-1}} \beta_{t}}{1-\overline{\alpha}_{t}}x_{0}+\frac{\alpha_{t}-1}{\sqrt{\alpha_{t}} (1-\overline{\alpha}_{t})}x_{t}-\frac{\eta_{t}s_{t}(x_{t})}{\sqrt{\alpha_{t}}} \right\|_{2}^{2}\right]$ $\displaystyle\overset{\text{(i)}}{=}\frac{\alpha_{t}}{2\sigma_{t}^{2}}\mathbb{ E}_{\varepsilon\sim\mathcal{N}(0,I_{d})}\left[\bigg{\|}\frac{\alpha_{t}-1}{ \sqrt{\alpha_{t}(1-\overline{\alpha}_{t})}}\varepsilon-\frac{1-\alpha_{t}}{ \sqrt{\alpha_{t}}}s_{t}(\sqrt{\overline{\alpha}_{t}}x_{0}+\sqrt{1-\overline{ \alpha}_{t}}\varepsilon)\bigg{\|}_{2}^{2}\right]$ $\displaystyle\overset{\text{(ii)}}{=}\frac{1-\alpha_{t}}{2(\alpha_{t}- \overline{\alpha}_{t})}\mathbb{E}_{\varepsilon\sim\mathcal{N}(0,I_{d})}\left[ \left\|\varepsilon-\varepsilon_{t}(\sqrt{\overline{\alpha}_{t}}x_{0}+\sqrt{1- \overline{\alpha}_{t}}\varepsilon)\right\|_{2}^{2}\right].$ $$

Here in step (i), we utilize the coefficient design (4.3)
and replace $x_{t}$ with $\sqrt{\overline{\alpha}_{t}}x_{0}+\sqrt{1-\overline{\alpha}_{t}}\varepsilon$,
which has the same distribution; while in step (ii), we replace the
score function $s_{t}(\cdot)$ with the epsilon predictor $\varepsilon_{t}(\cdot)\coloneqq-\sqrt{1-\overline{\alpha}_{t}}s_{t}(\cdot)$.
Comparing the coefficients in $L_{t-1}^{\star}$ and $L_{t-1}$, we
decompose

$$ $\left|\frac{1-\alpha_{t+1}}{2(1-\overline{\alpha}_{t})}-\frac{1-\alpha_{t}}{2( \alpha_{t}-\overline{\alpha}_{t})}\right|\leq\underbrace{\left|\frac{1-\alpha_ {t+1}}{2(1-\overline{\alpha}_{t})}-\frac{1-\alpha_{t+1}}{2(\alpha_{t}- \overline{\alpha}_{t})}\right|}_{\eqqcolon\gamma_{1}}+\underbrace{\left|\frac{ 1-\alpha_{t+1}}{2(\alpha_{t}-\overline{\alpha}_{t})}-\frac{1-\alpha_{t}}{2( \alpha_{t}-\overline{\alpha}_{t})}\right|}_{\eqqcolon\gamma_{2}}.$ $$

Consider the learning rate schedule in ; Li and Yan, ():

$$ $\beta_{1}=\frac{1}{T^{c_{0}}},\qquad\beta_{t+1}=\frac{c_{1}\log T}{T}\min\left \{\beta_{1}\left(1+\frac{c_{1}\log T}{T}\right)^{t},1\right\}\quad(t=1,\ldots, T-1)$ (D.1) $$

for sufficiently large constants $c_{0},c_{1}>0$. Then using the
properties in e.g., Li and Yan, (, Lemma 8), we can check
that

$$ $\displaystyle\gamma_{1}$ $\displaystyle=\left|\frac{(1-\alpha_{t+1})(\alpha_{t}-1)}{2(1-\overline{\alpha }_{t})(\alpha_{t}-\overline{\alpha}_{t})}\right|\leq\frac{8c_{1}\log T}{T} \left|\frac{1-\alpha_{t+1}}{2(1-\overline{\alpha}_{t})}\right|,$ $$

and

$$ $\displaystyle\gamma_{2}$ $\displaystyle=\left|\frac{\alpha_{t}-\alpha_{t+1}}{2(\alpha_{t}-\overline{ \alpha}_{t})}\right|=\left|\frac{\beta_{t}-\beta_{t+1}}{2(\alpha_{t}-\overline {\alpha}_{t})}\right|\leq\left|1-\frac{\beta_{t}}{\beta_{t+1}}\right|\left|1+ \frac{1-\alpha_{t}}{\alpha_{t}-\overline{\alpha}_{t}}\right|\left|\frac{1- \alpha_{t+1}}{2(1-\overline{\alpha}_{t})}\right|\leq\frac{8c_{1}\log T}{T} \left|\frac{1-\alpha_{t+1}}{2(1-\overline{\alpha}_{t})}\right|.$ $$

Hence the coefficients in $L_{t-1}^{\star}$ and $L_{t-1}$ are identical
up to higher-order error:

$$ $\left|\frac{1-\alpha_{t+1}}{2(1-\overline{\alpha}_{t})}-\frac{1-\alpha_{t}}{2( \alpha_{t}-\overline{\alpha}_{t})}\right|\leq\frac{16c_{1}\log T}{T}\left| \frac{1-\alpha_{t+1}}{2(1-\overline{\alpha}_{t})}\right|.$ $$

#### Computing L 0 ⁢ ( x 0 ) subscript 𝐿 0 subscript 𝑥 0 L_{0}(x_{0}) italic_L start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) .

By taking $\eta_{1}=\sigma_{1}^{2}=1-\alpha_{1}$ (notice that (4.3)
does not cover the case $t=1$), we have

$$ $\displaystyle p_{Y_{0}|Y_{1}}(x_{0}\,|\,x_{1})$ $\displaystyle=\left(\frac{2\pi\sigma_{1}^{2}}{\alpha_{1}}\right)^{-d/2}\exp \left(-\frac{\alpha_{1}}{2\sigma_{1}^{2}}\left\|x_{0}-\frac{x_{1}-\eta_{1}s_{1 }\left(x_{1}\right)}{\sqrt{\alpha_{1}}}\right\|_{2}^{2}\right)$ $\displaystyle=\left(\frac{2\pi\beta_{1}}{\alpha_{1}}\right)^{-d/2}\exp\left(- \frac{\alpha_{1}}{2\beta_{1}}\left\|x_{0}-\frac{x_{1}-\beta_{1}s_{1}\left(x_{1 }\right)}{\sqrt{\alpha_{1}}}\right\|_{2}^{2}\right),$ $$

and therefore

$$ $\displaystyle C_{0}(x_{0})$ $\displaystyle=\mathbb{E}_{x_{1}\sim p_{X_{1}|X_{0}}(\cdot\,|\,x_{0})}\left[- \frac{d}{2}\log\frac{2\pi\beta_{1}}{\alpha_{1}}-\frac{\alpha_{1}}{2\beta_{1}} \left\|x_{0}-\frac{x_{1}+\beta_{1}s_{1}(x_{1})}{\sqrt{\alpha_{1}}}\right\|_{2} ^{2}\right]$ $\displaystyle\overset{\text{(i)}}{=}-\frac{d}{2}\log\frac{2\pi\beta_{1}}{ \alpha_{1}}-\frac{1}{2}\mathbb{E}_{\varepsilon\sim\mathcal{N}(0,I_{d})}\left[ \|\varepsilon+\sqrt{\beta_{1}}s_{1}(\sqrt{1-\beta_{1}}x_{0}+\sqrt{\beta_{1}} \varepsilon)\|_{2}^{2}\right]$ $\displaystyle\overset{\text{(ii)}}{=}-\frac{1+\log(2\pi\beta_{1})}{2}d+\frac{d }{2}\log(1-\beta_{1})-\frac{1}{2}\beta_{1}\mathbb{E}_{\varepsilon\sim\mathcal{ N}(0,I_{d})}\big{[}\|s_{1}(\sqrt{1-\beta_{1}}x_{0}+\sqrt{\beta_{1}}\varepsilon )\|_{2}^{2}\big{]}$ $\displaystyle\quad\qquad-\sqrt{\beta_{1}}\mathbb{E}_{\varepsilon\sim\mathcal{N }(0,I_{d})}\big{[}\varepsilon^{\top}s_{1}(\sqrt{1-\beta_{1}}x_{0}+\sqrt{\beta_ {1}}\varepsilon)\big{]}.$ (D.2) $$

Here in step (i), we replace $x_{1}$ with $\sqrt{1-\beta_{1}}x_{0}+\sqrt{\beta_{1}}\varepsilon$,
which has the same distribution; step (ii) uses the fact that $\mathbb{E}[\|\varepsilon\|_{2}^{2}]=d$
for $\varepsilon\sim\mathcal{N}(0,I_{d})$. Using similar analysis
as in Proposition 2, we can show that $\sup_{x}\|\nabla^{2}\log q_{1}(x)\|\leq O(L)$
when $\beta_{1}$ is sufficiently small, as long as $\sup_{x}\|\nabla^{2}\log q_{0}(x)\|\leq L$.
Hence we have

$$ $\displaystyle\mathbb{E}_{\varepsilon\sim\mathcal{N}(0,I_{d})}\big{[}\|s_{1}( \sqrt{1-\beta_{1}}x_{0}+\sqrt{\beta_{1}}\varepsilon)\|_{2}^{2}\big{]}$ $\displaystyle\leq\mathbb{E}_{\varepsilon\sim\mathcal{N}(0,I_{d})}\big{[}\big{( }\|s_{1}(x_{0})\|_{2}+O(L)\|x_{0}-\sqrt{1-\beta_{1}}x_{0}-\sqrt{\beta_{1}} \varepsilon\|_{2}\big{)}^{2}\big{]}$ $\displaystyle\leq 2\|s_{1}(x_{0})\|_{2}^{2}+O(L^{2})\mathbb{E}_{\varepsilon \sim\mathcal{N}(0,I_{d})}\big{[}\|x_{0}-\sqrt{1-\beta_{1}}x_{0}-\sqrt{\beta_{1 }}\varepsilon\|_{2}^{2}\big{]}$ $\displaystyle=2\left\|s_{1}(x_{0})\right\|_{2}^{2}+O(L^{2}\beta_{1}).$ (D.3) $$

By Stein’s lemma, we can show that

$$ $\displaystyle\mathbb{E}_{\varepsilon\sim\mathcal{N}(0,I_{d})}\left[\varepsilon ^{\top}s_{1}(\sqrt{1-\beta_{1}}x_{0}+\sqrt{\beta_{1}}\varepsilon)\right]$ $\displaystyle=\sqrt{\beta_{1}}\mathbb{E}\left[\mathsf{tr}\left(\nabla^{2}\log q _{1}(\sqrt{1-\beta_{1}}x_{0}+\sqrt{\beta_{1}}\varepsilon)\right)\right]$ $\displaystyle\leq O(\sqrt{\beta_{1}}Ld).$ (D.4) $$

Substituting the bounds (D.3) and (D.4) back
into (D.2), we have

$$ $C_{0}(x_{0})=-\frac{1+\log(2\pi\beta_{1})}{2}d+O(\beta_{1})$ $$

as claimed.

#### Negligibility of L T ⁢ ( x ) subscript 𝐿 𝑇 𝑥 L_{T}(x) italic_L start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_x ) .

Since

$$ $Y_{T}\sim\mathcal{N}(0,I_{d}),\qquad\text{and}\qquad X_{T}\,|\,X_{0}=x_{0}\sim \mathcal{N}\left(\sqrt{\overline{\alpha}_{T}}x_{0},(1-\overline{\alpha}_{T})I_ {d}\right),$ $$

we can compute

$$ $\mathsf{KL}\left(p_{Y_{T}}(\cdot)\,\|\,p_{X_{T}|X_{0}}(\cdot\,|\,x_{0})\right) =\frac{1}{2}\frac{\overline{\alpha}_{T}}{1-\overline{\alpha}_{T}}\left(d+\|x_{ 0}\|_{2}^{2}\right)+\frac{d}{2}\log(1-\overline{\alpha}_{T})\leq\frac{1}{2} \frac{\overline{\alpha}_{T}}{1-\overline{\alpha}_{T}}\left(d+\|x_{0}\|_{2}^{2} \right).$ $$

Using the learning rate schedule in (D.1), we can check
that $\overline{\alpha}_{T}\leq T^{-c_{2}}$ for some large universal
constant $c_{2}>0$; see e.g., for the proof.
Therefore when $T\geq 2$, we have

$$ $\mathsf{KL}\left(p_{Y_{T}}(\cdot)\,\|\,p_{X_{T}|X_{0}}(\cdot\,|\,x_{0})\right) \leq\frac{d+\|x_{0}\|_{2}^{2}}{4T^{c_{2}}},$ $$

which is negligible when $T$ is sufficiently large.

#### Optimal solution for ( 4.5 ).

It is known that for each $1\leq t\leq T$, the score function $s_{t}^{\star}(\cdot)$
associated with $q_{t}$ satisfies

$$ $s_{t}^{\star}(\cdot)=\mathop{\arg\min}_{s(\cdot):\mathbb{R}^{d}\to\mathbb{R}^{ d}}\mathbb{E}_{x\sim q_{0},\varepsilon\sim\mathcal{N}(0,I_{d})}\left[\left\|s \left(\sqrt{\overline{\alpha}_{t}}x+\sqrt{1-\overline{\alpha}_{t}}\varepsilon \right)+\frac{1}{\sqrt{1-\overline{\alpha}_{t}}}\varepsilon\right\|_{2}^{2} \right].$ $$

See e.g., Chen et al., (, Appendix A) for the proof. Recall that $\varepsilon_{t}^{\star}(\cdot)=\sqrt{1-\overline{\alpha}_{t}}s_{t}^{\star}(\cdot)$,
then we have

$$ $\varepsilon_{t}^{\star}(\cdot)=\mathop{\arg\min}_{\varepsilon(\cdot):\mathbb{R }^{d}\to\mathbb{R}^{d}}\mathbb{E}_{x\sim q_{0},\varepsilon\sim\mathcal{N}(0,I_ {d})}\Big{[}\big{\|}\varepsilon-\varepsilon(\sqrt{\overline{\alpha}_{t}}x+ \sqrt{1-\overline{\alpha}_{t}}\varepsilon)\big{\|}_{2}^{2}\Big{]}.$ $$

Therefore the global minimizer for (4.5) is
$\widehat{\varepsilon}_{t}(\cdot)\equiv\varepsilon_{t}^{\star}(\cdot)$
for each $1\leq t\leq T$.

### D.2 Technical details in Section 4.2

By checking the optimality condition, we know that $(D_{\lambda},G_{\lambda})$
is a Nash equilibrium if and only if

$$ $\displaystyle D_{\lambda}(x)$ $\displaystyle=\frac{p_{\mathsf{data}}(x)}{p_{\mathsf{data}}(x)+p_{G_{\lambda}} (x)},\qquad\text{(optimality condition for }D_{\lambda})$ (D.5) $$

where $p_{G_{\lambda}}=(G_{\lambda})_{\#}p_{\mathsf{noise}}$, and
there exists some constant $c$ such that

$$ $\begin{cases}-\log D_{\lambda}(x)+\lambda L(x)=c,&\text{when}\quad x\in\mathsf {supp}(p_{G_{\lambda}}),\\ -\log D_{\lambda}(x)+\lambda L(x)\geq c,&\text{otherwise}.\end{cases}\qquad( \text{optimality condition for }G_{\lambda})$ (D.6) $$

Taking the approximation $L(x)\approx-\log p_{\mathsf{data}}(x)+C_{0}^{\star}$
as exact, we have

$$ $D_{\lambda}(x)=\begin{cases}e^{\lambda C_{0}^{\star}-c}p_{\mathsf{data}}^{- \lambda}(x),&\text{for }x\in\mathsf{supp}(p_{G_{\lambda}}),\\ 1,&\text{for }x\notin\mathsf{supp}(p_{G_{\lambda}}).\end{cases}$ (D.7) $$

where the first and second cases follow from (D.6) and
(D.5) respectively. Then we derive a closed-form expression
for $p_{G_{\lambda}}$.

- •
For any $x\in\mathsf{supp}(p_{G_{\lambda}})$, by putting (D.5)
and (D.7) together, we have
$e^{\lambda C_{0}^{\star}-c}p_{\mathsf{data}}^{-\lambda}(x)=\frac{p_{\mathsf{
data}}(x)}{p_{\mathsf{data}}(x)+p_{G_{\lambda}}(x)},$
which further gives
$p_{G_{\lambda}}(x)=p_{\mathsf{data}}(x)\big{(}e^{-\lambda C_{0}^{\star}+c}p_{
\mathsf{data}}^{\lambda}(x)-1\big{)}.$
(D.8)
- •
For any $x\notin\mathsf{supp}(p_{G_{\lambda}})$, we have
$-\log D_{\lambda}(x)+\lambda L(x)\overset{\text{(i)}}{=}\lambda L(x)\overset{
\text{(ii)}}{=}-\lambda\log p_{\mathsf{data}}(x)+\lambda C_{0}^{\star}\overset
{\text{(iii)}}{\geq}c,$
where step (i) follows from $D_{\lambda}(x)=1$, which follows from
(D.7); step (ii) holds when we take the approximation
$L(x)\approx-\log p_{\mathsf{data}}(x)+C_{0}^{\star}$ as exact; and
step (iii) follows from (D.6). This immediately gives
$e^{-\lambda C_{0}^{\star}+c}p_{\mathsf{data}}^{\lambda}(x)-1=\log\left(-
\lambda C_{0}^{\star}+c+\lambda\log p_{\mathsf{data}}(x)\right)-1\leq 0.$
(D.9)

Taking (D.8) and (D.9) collectively,
we can write

$$ $p_{G_{\lambda}}(x)=p_{\mathsf{data}}(x)\big{(}e^{-\lambda C_{0}^{\star}+c}p_{ \mathsf{data}}^{\lambda}(x)-1\big{)}_{+}.$ (D.10) $$

On the other hand, we can check that (D.7) and (D.10)
satisfies the optimality conditions (D.5) and (D.6),
which establishes the desired result.

## Peer Reviews

confidence: 4
contribution: 1
initial_rating: 3
presentation: 2
questions: Please refer to weaknesses  section. Also, why do the authors consider the particular SDE in lines 76-77 and not a more generic one?
soundness: 2
strengths: The paper focuses on the theoretical understanding of score-based generative models (SGMs) by applying a density formula to explain why optimizing the evidence lower bound (ELBO) effectively supports training for diffusion models like DDPMs. By investigating the theoretical aspects  behind ELBO optimization, the authors promote a more rigorous basis for diffusion models.

Additionally, the paper extends the implications of this analysis to areas such as GAN regularization, diffusion classifiers, and autoregressive models, illustrating the potential for these findings to enhance model training practices across various generative frameworks
summary: This paper considers a density formula based on score estimation to analyze Denoising Diffusion Probabilistic Models (DDPMs). Using this formula, the authors provide a theoretical basis for why optimizing the evidence lower bound (ELBO) serves as an effective approach for training these models. The paper addresses the problem of the understanding of ELBO optimization for diffusion models, adding theoretical context to a widely used empirical technique.

The analysis extends to practical implications across different generative modeling contexts, including applications to GAN regularization, diffusion classifiers, and autoregressive models. By investigating these areas, the authors demonstrate how insights from the density formula can support training and optimization practices in various generative frameworks. This broad applicability suggests that the theoretical findings may be interesting to both foundational research and practical applications in generative modeling.
weaknesses: The biggest weakness of the work lies in the lack of novelty and the positioning with respect to the literature. 
In particular, it is not evident how the work differs from known results such as [1], where Thm 4+ eq (25) and the comment in eq(29) seems to provide a result which is even more general than the one discussed by the authors.  It is worth mentioning that related results which the authors do not cite in their work are also presented in [2], in particular Thm 3.


One other big limitation is that the authors derive their connection between continuous and discrete time (with an **approximated score**) by a sequence of approximations, without discussing properly the impact of these. A clear quantification analysis would greatly strengthen the paper. 



[1] Huang et al., A Variational Perspective on Diffusion-Based Generative Models and Score Matching, (NeurIPS 2021)

[2] Song et al., Maximum Likelihood Training of Score-Based Diffusion Models (NeurIPS 2021)

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confidence: 3
contribution: 2
initial_rating: 3
presentation: 2
questions: your SDE $d X_t = - \frac{1}{2(1-t)} X_t d t + \frac{1}{\sqrt{1-t}} d B_t $  is a special case in Song's SDE in [1], which follows that

$$d X_t = -\frac{1}{2} \beta_t X_t dt + \sqrt{\beta_t} d B_t$$

It appears that there are countless choices of $\beta_t$. Why do you claim $\beta_t = \frac{1}{1-t}$ is the continuous-time limit of the aforementioned forward process in section 2.1 and is more preferred than Song's linear version $\beta_t = \beta_{\min} + t (\beta_{\max} - \beta_{\min})$?  

[1] Score-Based Generative Modeling through Stochastic Differential Equations. ICLR'21.
soundness: 4
strengths: The analysis of the variational gap for the continuous-time approximation of the discrete alternative of diffusion models, such as DDPM, is missing. The authors conducted clear and solid derivations to show why the variational gap is negligible, which provides proof for the empirical usage that the ELBO matches the true objectives.
summary: Despite empirical advances, the theoretical foundation for why optimizing the evidence lower bound (ELBO) on the log-likelihood is effective for training diffusion generative models, such as DDPMs, remains largely unexplored.  The authors proposed to address this question by establishing a density formula for a continuous-time diffusion process, which can be viewed as the continuous-time
limit of the forward process in an SGM. The formula shows that the variational gap is negligible.
weaknesses: 1. despite the soundness of the derivations, I found that the goal of this paper is not very interesting.

2. insights on GANs are not clear to me (a layperson in GAN).

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confidence: 3
contribution: 2
initial_rating: 3
presentation: 2
questions: 1. Several existing works also explore the relationship between the density and the optimization objectives of diffusion models, e.g. [1] and [2]. What's the relation between the current results and those in existing works. 

    [1] Kong et al. "Information Theoretic Diffusion."

    [2] Song et al. "Maximum likelihood training of score-based diffusion models." 

2. In general variational inference, for fixed observations, maximizing the ELBO is equivalent to minimizing the KL divergence. What's new in the current results compared to the general observation? 

3. Are there error bounds for the various approximations? Without such bounds, why do we expect the interpretation using approximations to be better and more useful than the interpretation using lower bounds? 

4. Existing theoretical results have provided error bounds for KL divergence. What's the connection between those results and the current results? What additional insights can the current results bring?

5. What's the advantage of the SDE in (2.4) over the more commonly used O-U process?
soundness: 3
strengths: 1. The paper provides a formula relating the target density and the scores at different time steps. 

2. The paper shows that maximizing the ELBO in DDPM is approximately equivalent to minimizing the KL divergence of the target distribution and the learned distribution.
summary: This paper derives a density formula for a continuous-time diffusion process, which can be viewed as the continuous-time limit of the forward process of an SGM. The formula relates the target density and the score functions at different time steps. The authors use the formula to show that maximizing the ELBO in DDPM is approximately equivalent to minimizing the KL divergence of the target distribution and the learned distribution.  The authors also apply the approximation to explain the use of score-matching regularization in GAN training, ELBO in diffusion classifier, and diffusion loss is autoregressive models.
weaknesses: 1. The contribution is unclear. The discussion about some important existing results are missing. 

2. The applications of the density formula presented here involves approximations, but there is no characterization of the approximation errors. 

3. The presentation needs improvement.

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confidence: 4
contribution: 2
initial_rating: 3
presentation: 1
questions: * In light of the comments about weaknesses above, can you clearly spell out what are the novel contributions of the submitted article? I am not against revisiting known results to set the stage for the main contributions, but I have the impression that most of the conclusions drawn in Sec. 4, which is where the authors use their revisited formulation of the log data density, have been known to the community, also from the theoretical point of view, and not only from an heuristic perspective. Can you also answer to the questions raised in the "weaknesses" section of this review?

* Despite the intelligible intent of reuniting continuous-time and discrete-time models, I find the exposition of results in Sec. 2 and Sec. 3, according to slightly different formulations than those existing in the literature, is confusing. Is there a way to organize this work such that contributions are more clear, and the implications of the presented theory spelled out well?

* Would you feel comfortable by stating that your exact formulation of the log density of the data distribution as a function of the drift and diffusion terms of the SDEs, or equivalently the log density of the data distribution as a function of the transition kernels and noise of the discrete-time diffusion, as a novel result that has not been discussed in the literature?
soundness: 2
strengths: * It is important to revisit known results that might have been obtained through carefully engineered heuristics, through the lenses of a sound theoretical formalism, such that the community can validate existing choices. The endeavor of this work is in line with this objective, which I think is valuable.
* This work shows that the theory developed to derive an expression for the density of the data distribution can be applied to numerous modeling approaches to generative modeling.
* The mathematical derivations in Appendix A (which are the most important to me), seem correct.
summary: This work is presented as a theoretical contribution to the field of diffusion models. 

The overall structure of the paper mimics the intended objective of this work: to revisit both continuous-time and discrete-time diffusion models to arrive at the (exact and approximate) definition of a density expression for the log data density (in continuous and discrete time), that is used to: i) discuss the validity of an ELBO formulation for the optimization of the parameters of the denoising network of discrete diffusion models, ii) understand the optimization objective in generative adversarial networks, iii) provide a justification for classifier-based guidance in diffusion models, and iv) show that the diffusion loss used in autoregressive models corresponds to an approximate maximum likelihood solution.
weaknesses: * Sec. 2: this section repurposes known results from the literature, including [1,2,3], in which it has been shown the equivalence between discrete-time and continuous-time variants of diffusion. Note also that [3], which is not cited by the authors, shows that "*the log-likelihood of score-based models can be tractably computed through a connection to continuous normalizing flows, even though the log-likelihood is not directly optimized by the weighted combination of score matching losses. In particular, it is shown that under a specific weighting scheme the score-matching objective upper bounds the negative log-likelihood, thus enabling approximate maximum likelihood training of score-based models.*"
* In sec 2.1, DDPM are revisited, but mixed with score functions, yielding Eq. 2.3. Why and how does the score function appears in discrete-time diffusion?
* In sec 2.2, I am curious to learn why Eq. 2.4 has been chosen to be so specific, instead of using a more general form with a functional drift term. Here you specify a linear drift whose coefficients explode, compared to the typical variance preserving formulation from [1], as time $t \to 1$.

[1] Song et al. “Score-Based Generative Modeling through Stochastic Differential Equations”, https://arxiv.org/abs/2011.13456

[2] Ho et al. “Denoising Diffusion Probabilistic Models”, https://arxiv.org/abs/2006.11239

[3] Song et al. “Maximum Likelihood Training of Score-Based Diffusion Models”, https://arxiv.org/abs/2101.09258

* Sec. 3: This section displays some calculations that rely on the continuous-time formulation of diffusion processes. Sec. 3.1 begins by focusing on Eq. 2.4, which is the linear variance preserving SDE discussed above. Sec. 3.2 continues the derivations, to relate continuous-time and discrete-time known results, and Sec. 3.3 discusses known results on the equivalence to a probability flow ODE and more recent results on density estimation. What are the main take home messages here? What is the original contribution the authors would like to put forward in this section?
To the best of my understanding, the result in Eq. 3.1.a is an exact formulation for the log likelihood of the data distribution that did not require, as done in [1,3], probability flow ODE equivalence. I followed the proof in Appendix A, and to my eyes it seems correct.
Sec 3.2 should also deserve more insights provided by the authors, as it gives an approximate log density for the discrete case, bypassing the need to work directly in discrete-time. Can we quantify the discretization errors that are introduced by relying on Eq 3.1.c?

* Sec. 4: This is an “application” of the exact log density expression for the data distribution form Sec. 3.
Sec. 4.1 aims at discussing the validity of the ELBO formulation as a good proxy for the log likelihood, to demonstrate that in DDPM optimizing the ELBO is a valid replacement for optimizing log likelihood. This can also be understood from [2] and [1] above, and, for continuous time, is readily discussed in [3], which also shows the similarity (modulo discretization errors and constants) between continuous-time and discrete time formulations. So, what do we learn from the derivations presented in this section that were not directly discussed in these earlier work?
Sec. 4.2 begs for the same question, and should be reviewed in light of an overloaded notation: please check that $z$ is used both as a random variable sampled from a noise distribution, and as a normalizing factor.
Sec 4.3 revisits classifier guidance mechanisms for conditional generation using diffusion models, and offers a critic to some practical heuristics used in recent work, based on the density defined in this paper.
Similarly, Sec. 4.4 revisits autoregressive models in light of the proposed density definition, and suggest that the training objective used in the literature can be viewed as approximate maximum likelihood training.