text:
\section{Introduction}
\label{sec:introduction}
Score-based generative models are probabilistic generative models that estimate score functions, i.e., the gradients of the log density for some given data distributions. As described in the pioneering work~\citep{JMLR:v6:hyvarinen05a}, the process of training score-based generative models is called \textit{Score Matching} (SM), in which a score-based generative model is iteratively updated to approximate the true score function. Such a process often incurs heavy computational burdens, since the calculation of the score-matching objective involves the explicit computation of the partial derivatives of the score model during training. Therefore, a branch of study in this research domain~\citep{vincent2011connection, song2019sliced} resorts to reformulating the score-matching objective to reduce the training cost. Among these works, the author in~\citep{vincent2011connection} introduced the \textit{Denoising Score-Matching} (DSM) method. This method facilitates the training process of score-based generative models, and thus lays the foundation for a number of subsequent researches. Recently, the authors in~\citep{song2019generative} proposed an unified framework based on DSM, and achieved remarkable performance on several real-world datasets. Their success inspired several succeeding works~\citep{song2020improved, Ho2020DenoisingDP, Song2021DenoisingDI, song2021scorebased, dhariwal2021diffusion}, which together contribute to making score-based generative models an attractive choice for contemporary image generation tasks.

A favorable aspect of score-based generative models is their flexibility to be easily extended to conditional variants. This characteristic comes from a research direction that utilizes Bayes' theorem to decompose a conditional score into a mixture of scores~\citep{nguyen2017plug}. Recent endeavors followed this approach and further extended the concept of conditional score-based models to a number of application domains, including colorization~\citep{song2021scorebased}, inpainting~\citep{song2021scorebased}, and source separation~\citep{Jayaram2020SourceSW}. In particular, some recent researchers~\citep{song2021scorebased, dhariwal2021diffusion} applied this method to the field of class-conditional image generation tasks, and proposed the \textit{classifier-guidance} method. Different from the \textit{classifier-guidance-free} method adopted by~\citep{Ho2021CascadedDM}, they utilized a score model and a classifier to generate the posterior scores (i.e., the gradients of the log posterior density), with which the data samples of certain classes can be generated through the diffusion process. The authors in~\citep{dhariwal2021diffusion} showed that the classifier guidance method is able to achieve improved performance on large image generation benchmarks. In spite of their success, our analysis indicates that the conditional generation methods utilizing a score model and a classifier may suffer from a \textit{score mismatch issue}, which is the situation that the estimated posterior scores deviate from the true ones. This issue causes the samples to be guided by inaccurate scores during the diffusion process, and may result in a degraded sampling quality consequently.

To resolve this problem, we first analyze the potential causes for the score mismatch issue through a motivational low-dimensional example. Then, we formulate a new loss function called \textit{Denoising Likelihood Score-Matching} (DLSM) loss, and explain how it can be integrated into the current training method. Finally, we evaluate the proposed method under various configurations, and demonstrate its advantages in improving the sampling quality over the previous methods in terms of several evaluation metrics.
\section{Background}
\label{sec:background}
In this section, we introduce the essential background material for understanding the contents of this paper. We first introduce Langevin diffusion~\citep{bj/1178291835, roberts1998optimal} for generating data samples ${\tilde{\bm{x}}}\in{\mathbb{R}}^{d}$ of a certain dimension $d$ from an unknown probability density function (pdf) $p({\tilde{\bm{x}}})$ using the score function $\nabla_{\tilde{\bm{x}}} \log p({\tilde{\bm{x}}})$. Next, we describe Parzen density estimation~\citep{parzen1962estimation} and denoising score matching (DSM)~\citep{vincent2011connection} for approximating $\nabla_{\tilde{\bm{x}}} \log p({\tilde{\bm{x}}})$ with limited data samples. Finally, we elaborate on the conditional variant of the score function, i.e., $\nabla_{\tilde{\bm{x}}} \log p({\tilde{\bm{x}}}|{\tilde{\bm{y}}})$, and explains how it can be decomposed into $\nabla_{\tilde{\bm{x}}} \log p({\tilde{\bm{x}}})$ and $\nabla_{\tilde{\bm{x}}} \log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}})$ for some conditional variable ${\tilde{\bm{y}}}\in{\mathbb{R}}^{c}$ of dimension $c$.

\subsection{Langevin Diffusion}
\label{sec:langevin_dynamics}

Langevin diffusion~\citep{bj/1178291835, roberts1998optimal} can be used to generate data samples from an unknown data distribution $p({\tilde{\bm{x}}})$ using only the score function $\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{x}}})$, which is said to be well-defined if $p({\tilde{\bm{x}}})$ is everywhere non-zero and differentiable. Under the condition that $\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{x}}})$ is well-defined, Langevin diffusion enables $p({\tilde{\bm{x}}})$ to be approximated iteratively based on the following equation:
\begin{equation}
\label{eq:langevin_dynamics}
\begin{aligned}
{\tilde{\bm{x}}}_t = {\tilde{\bm{x}}}_{t-1} + \frac{\epsilon^2}{2} \nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{x}}}_{t-1}) + \epsilon{\bm{z}}_t,
\end{aligned}
\end{equation}
where ${\tilde{\bm{x}}}_0$ is sampled from an arbitrary distribution, $\epsilon$ is a fixed positive step size, and ${\bm{z}}_t$ is a noise vector sampled from a normal distribution $\mathcal{N}({\bm{0}},I_{{d}\times{d}})$ for simulating a ${d}$-dimensional standard Brownian motion. Under suitable regularity conditions, when $\epsilon\rightarrow 0$ and $T\rightarrow \infty$, ${\tilde{\bm{x}}}_T$ is generated as if it is directly sampled from  $p({\tilde{\bm{x}}})$~\citep{bj/1178291835,welling2011bayesian}. In practice, however, the data samples are generated with $\epsilon>0$ and $T<\infty$, which violates the convergence guarantee. Although it is possible to use Metropolized algorithms~\citep{bj/1178291835, roberts1998optimal} to recover the convergence guarantee, we follow the assumption of the prior work~\citep{song2019generative} and presume that the errors are sufficiently small to be negligible when $\epsilon$ is small and $T$ is large.

The sampling process introduced in Eq.~(\ref{eq:langevin_dynamics}) can be extended to a time-inhomogeneous variant by making $p({\tilde{\bm{x}}}_{t})$ and $\epsilon$ dependent on $t$ (i.e., $p_t({\tilde{\bm{x}}}_{t})$ and $\epsilon_t$). Such a time-inhomogeneous variant is commonly adopted by recent works on score-based generative models~\citep{song2019generative, song2021scorebased}, as it provides flexibility in controlling $p_t({\tilde{\bm{x}}}_{t})$ and $\epsilon_t$. The experimental results in~\citep{song2019generative, song2021scorebased} demonstrated that such a time-inhomogeneous sampling process can improve the sampling quality on real-world datasets. In Appendix~\ref{apx:config:noise_and_sampling}, we offer a discussion on the detailed implementation of such a time-inhomogeneous sampling process in this work.

\subsection{Parzen Density Estimation}
\label{sec:parzen_density_estimation}

Given a true data distribution $p_\mathrm{data}$, the empirical data distribution $p_0({\bm{x}})$ is constructed by sampling $m$ independent and identically distributed data points $\{{\bm{x}}^{(i)}\}_{i=1}^m$, and can be represented as a sum of Dirac functions $\frac{1}{m}\sum_{i=1}^m\delta(\lVert{\bm{x}}-{\bm{x}}^{(i)}\rVert)$. Such a discrete data distribution $p_0({\bm{x}})$
constructed from the dataset often violates the previous assumptions that everywhere is non-zero and is differentiable. Therefore, it is necessary to somehow adjust the empirical data distribution $p_0({\bm{x}})$ before applying Langevin diffusion in such cases.

To deal with the above issue, a previous literature~\citep{vincent2011connection} utilized Parzen density estimation to replace the Dirac functions with isotropic Gaussian smoothing kernels $p_{\sigma}({\tilde{\bm{x}}}|{\bm{x}})=\frac{1}{(2\pi)^{{d}/2}\sigma^{d}}e^{\frac{-1}{2\sigma^2}\lVert{\tilde{\bm{x}}}-{\bm{x}}\rVert^2}$ with variance $\sigma^2$. Specifically, Parzen density estimation enables the calculation of $p_{\sigma}({\tilde{\bm{x}}})=\frac{1}{m}\sum_{i=1}^mp_{\sigma}({\tilde{\bm{x}}}|{\bm{x}}^{(i)})$. When $\sigma>0$, the score function becomes well-defined and can thus be represented as the following:
\begin{equation}
\label{eq:oracle}
\begin{aligned}
\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}}) = \frac{\sum_{i=1}^{m}\frac{1}{\sigma^2}({\bm{x}}^{(i)}-{\tilde{\bm{x}}})p_{\sigma}({\tilde{\bm{x}}}|{\bm{x}}^{(i)})}{\sum_{i=1}^{m}p_{\sigma}({\tilde{\bm{x}}}|{\bm{x}}^{(i)})}.
\end{aligned}
\end{equation}
The proof for Eq.~(\ref{eq:oracle}) is provided in Appendix~\ref{apx:proof:parzen}. This equation can be directly applied to Eq.~(\ref{eq:langevin_dynamics}) to generate samples with Langevin diffusion. Unfortunately, this requires summation over all $m$ data points during every iteration, preventing it from scaling to large datasets due to the rapid growth in computational complexity.

\subsection{Denoising Score Matching}
\label{sec:score_matching}

Score matching (SM)~\citep{JMLR:v6:hyvarinen05a} was proposed to estimate the score function with a model ${s}({\tilde{\bm{x}}};\phi)$, parameterized by $\phi$. Given a trained score model ${s}({\tilde{\bm{x}}};\phi)$, the scores can be generated by a single forward pass, which reduces the computational complexity of Eq.~(\ref{eq:oracle}) by a factor of $m$. To train such a score model, a straightforward approach is to use the Explicit Score-Matching (ESM) loss $L_{\mathrm{ESM}}$, represented as:
\begin{equation}
\label{eq:esm}
\begin{aligned}
L_{\mathrm{ESM}}(\phi) = \mathbb{E}_{p_{\sigma}({\tilde{\bm{x}}})} \left[ \frac{1}{2}\lVert{s}({\tilde{\bm{x}}};\phi) - \nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}})\rVert^2 \right].
\end{aligned}
\end{equation}
This objective requires evaluating Eq.~(\ref{eq:oracle}) for each training step, which also fails to scale well to large datasets. Based on Parzen density estimation, an efficient alternative, called Denoising Score-Matching (DSM) loss~\citep{vincent2011connection}, is proposed to efficiently calculate the equivalent loss $L_{\mathrm{DSM}}$, expressed as:
\begin{equation}
\label{eq:dsm}
\begin{aligned}
L_{\mathrm{DSM}}(\phi) = \mathbb{E}_{p_{\sigma}({\tilde{\bm{x}}},{\bm{x}})} \left[ \frac{1}{2}\lVert{s}({\tilde{\bm{x}}};\phi) - \nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}}|{\bm{x}})\rVert^2 \right].
\end{aligned}
\end{equation}
where $\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}}|{\bm{x}})$ is simply $\frac{1}{\sigma^2}({\bm{x}}-{\tilde{\bm{x}}})$. Since the computational cost of denoising score matching is relatively lower in comparison to other reformulation techniques~\citep{JMLR:v6:hyvarinen05a, song2019sliced}, it is extensively adopted in recent score-based generative models~\citep{song2019generative, song2020improved, song2021scorebased}.

\subsection{Conditional Score Decomposition via Bayes' Theorem}
\label{sec:conditional_sampling}

Score models can be extended to conditional models when conditioned on a certain label ${\tilde{\bm{y}}}$. Similar to ${\tilde{\bm{x}}}$, the smoothing kernels with variance $\tau^2$ can be applied on ${\bm{y}}$ to meet the requirement that the pdf is everywhere non-zero. Typically, $\tau$ is assumed to be sufficiently small so that $p_{\tau}({\tilde{\bm{y}}}) \approx p({\bm{y}})$. A popular approach adopted by researchers utilizes Bayes' theorem $p_{\sigma,\tau}({\tilde{\bm{x}}}|{\tilde{\bm{y}}})=p_{\sigma,\tau}({\tilde{\bm{y}}}|{\tilde{\bm{x}}})p_{\sigma}({\tilde{\bm{x}}})/p_{\tau}({\tilde{\bm{y}}})$ to decompose the conditional score $\nabla_{\tilde{\bm{x}}}\log p_{\sigma,\tau}({\tilde{\bm{x}}}|{\tilde{\bm{y}}})$ into a mixture of scores~\citep{nguyen2017plug}, which enables conditional data generation. Following the assumptions in the previous study~\citep{song2021scorebased}, the decomposition can be achieved by taking the log-gradient on both sides of the equation, expressed as follows:
\begin{equation}
\label{eq:bayesian}
\begin{aligned}
\nabla_{\tilde{\bm{x}}}\log p_{\sigma,\tau}({\tilde{\bm{x}}}|{\tilde{\bm{y}}}) = \nabla_{\tilde{\bm{x}}}\log p_{\sigma,\tau}({\tilde{\bm{y}}}|{\tilde{\bm{x}}}) + \nabla_{\tilde{\bm{x}}}\log p_{\sigma}({\tilde{\bm{x}}}) - \underbrace{\nabla_{\tilde{\bm{x}}}\log p_{\tau}({\tilde{\bm{y}}})}_{=0},
\end{aligned}
\end{equation}
where $\nabla_{\tilde{\bm{x}}}\log p_{\sigma,\tau}({\tilde{\bm{x}}}|{\tilde{\bm{y}}})$ is the posterior score, $\nabla_{\tilde{\bm{x}}}\log p_{\sigma,\tau}({\tilde{\bm{y}}}|{\tilde{\bm{x}}})$ is the likelihood score, and $\nabla_{\tilde{\bm{x}}}\log p_{\sigma}({\tilde{\bm{x}}})$ is the prior score. Base on Eq.~(\ref{eq:bayesian}), a conditional score model $\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{x}}}|{\tilde{\bm{y}}};\phi,\theta)$ can be
represented as the combination of the log-gradient of a differentiable classifier $p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)$ and a prior score model ${s}({\tilde{\bm{x}}};\phi)$:
\begin{equation}
\label{eq:bayesian2}
\begin{aligned}
\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{x}}}|{\tilde{\bm{y}}};\theta,\phi) = \nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta) + {s}({\tilde{\bm{x}}};\phi).
\end{aligned}
\end{equation}
This formulation enables conditional data generation using a classifier $p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)$ trained with cross-entropy loss $L_\mathrm{CE}(\theta)\triangleq \mathbb{E}_{p_{\sigma,\tau}({\tilde{\bm{x}}},{\tilde{\bm{y}}})}[-\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)]$, and a score model ${s}({\tilde{\bm{x}}};\phi)$ trained with denoising score matching loss $L_\mathrm{DSM}(\phi)$. Unfortunately, a few previous studies~\citep{nguyen2017plug, dhariwal2021diffusion} have noticed that the approximation of $\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{x}}}|{\tilde{\bm{y}}};\theta,\phi)$ is empirically inaccurate (referred to as the score mismatch issue in this work), and leveraged a scaling factor ${\alpha}>0$ to adjust the likelihood score $\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)$. Such a scaling factor is a hyperparameter that empirically controls the amount of conditional information incorporated during the sampling process (see Appendix~\ref{apx:scaling_trick}). However, this usually causes the diversity of generated data samples to degrade noticeably~\citep{dhariwal2021diffusion}, which is later discussed in Section~\ref{sec:experiments:benchmarks}.

In this work, we examine and investigate the score mismatch issue from a different perspective. We first offer a motivational example to show that the issue remains even with the scaling technique. Then, we theoretically formulate a new loss term that enables a classifier $p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)$ to produce better likelihood score estimation.

\section{Analysis on the Score Mismatch Issue}
\label{sec:analysis}
\input{figures/toy_experiment.tex}

To further investigate the score mismatch issue, we first leverage a motivational experiment on the \textit{inter-twining moon} dataset to examine the extent of the discrepancy between the estimated and true posterior scores. In this experiment, we consider five different methods to calculate the posterior scores denoted as (a)$\sim$(e):
\begin{itemize}
	\item \textbf{(a) Base method.}~The posterior scores are estimated using a score model ${s}({\tilde{\bm{x}}};\phi)$ trained with $L_\mathrm{DSM}$ and a classifier $p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)$ trained with $L_\mathrm{CE}$, as described in Eq.~(\ref{eq:bayesian2}).
	\item \textbf{(b) Scaling method.}~The posterior scores are estimated in a similar fashion as method (a) except that the scaling technique~\citep{dhariwal2021diffusion} is adopted to scale the likelihood scores, i.e., $\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{x}}}|{\tilde{\bm{y}}};\theta,\phi) = {\alpha} \nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta) + {s}({\tilde{\bm{x}}};\phi)$, where ${\alpha}>0$.
	\item \textbf{(c) Posterior SM method.}~The posterior scores for different class conditions, in this case, $c_1$ and $c_2$, are separately estimated using different score models ${s}({\tilde{\bm{x}}};\phi_1)$ and ${s}({\tilde{\bm{x}}};\phi_2)$ trained with $L_\mathrm{DSM}$. Note that, since this method requires network architectures different from those used in methods (a), (b), and (d), it is only adopted for analytical purposes. For more detailed discussions about this method, please refer to Appendix~\ref{apx:conditional_frameworks}.
	\item \textbf{(d) Ours.}~The posterior scores are estimated in a similar fashion as method (a) except that the classifier is trained with the proposed loss function $L_\mathrm{Total}$, which is described in detail in Section~\ref{sec:likelihood_denoising_score_matcing}.
	\item \textbf{(e) Oracle.}~The oracle posterior scores are directly computed using the conditional variant of Eq.~(\ref{eq:oracle}), which is detailed in Appendix~\ref{apx:config:toy_experiments}.
\end{itemize}
Fig.~\ref{fig:toy_experiment} visualizes the posterior scores and the sampled points based on the five methods. It is observed that the posterior scores estimated using methods (a) and (b) are significantly different from the true posterior scores measured by method (e). This causes the sampled points in methods (a) and (b) to deviate from those sampled based on method (e). On the other hand, the estimated posterior scores and the sampled points in method (c) are relatively similar to those in method (e). The above results therefore suggest that the score mismatch issue is severe under the cases of methods (a) and (b), but is alleviated when method (c) is used.

\input{tables/toy_experiment.tex}
In order to inspect the potential causes for the differences between the results produced by methods (a), (b), and (c), we incorporate metrics for evaluating the sampling quality and the errors between the scores in an quantitative manner. The sampling quality is evaluated using the precision and recall~\citep{kynkaanniemi2019improved} metrics. On the other hand, the estimation errors of the score functions are measured by the expected values of $D_{P}$ and $D_{L}$, which are formulated according to the Euclidean distances between the estimated scores and the oracle scores. The expressions of $D_{P}$ and $D_{L}$ are represented as the following:
\begin{equation}
\label{eq:dp}
\begin{aligned}
    D_{P}({\tilde{\bm{x}}},{\tilde{\bm{y}}}) &= \lVert S_{P}({\tilde{\bm{x}}},{\tilde{\bm{y}}}) - \nabla_{\tilde{\bm{x}}}\log p_{\sigma,\tau}({\tilde{\bm{x}}}|{\tilde{\bm{y}}}) \rVert, \\
    D_{L}({\tilde{\bm{x}}},{\tilde{\bm{y}}}) &= \lVert S_{L}({\tilde{\bm{x}}},{\tilde{\bm{y}}}) - \nabla_{\tilde{\bm{x}}}\log p_{\sigma,\tau}({\tilde{\bm{y}}}|{\tilde{\bm{x}}}) \rVert,
\end{aligned}
\end{equation}
where the subscripts $P$ and $L$ denote `Posterior' and `Likelihood', while the terms $S_{P}({\tilde{\bm{x}}},{\tilde{\bm{y}}})$ and $S_{L}({\tilde{\bm{x}}},{\tilde{\bm{y}}})$ correspond to the estimated posterior score and the likelihood score for a certain pair $({\tilde{\bm{x}}},{\tilde{\bm{y}}})$, respectively.

Table~\ref{tab:toy_experiment} presents $S_{P}({\tilde{\bm{x}}},{\tilde{\bm{y}}})$ and $S_{L}({\tilde{\bm{x}}},{\tilde{\bm{y}}})$, the expectations of $D_{P}$ and $D_{L}$, and the precision and recall for different methods. It can be seen that the numerical results in Table~\ref{tab:toy_experiment} are consistent with the observations revealed in Fig.~\ref{fig:toy_experiment}, since the expectations of $D_P$ are greater in methods (a) and (b) as compared to method (c), and the evaluated recall values in methods (a) and (b) are lower than those in method (c). Furthermore, the results in Table~\ref{tab:toy_experiment} suggest that the possible reasons for the disparity of the posterior scores between methods (a)$\sim$(c) are twofold. First, adding the scaling factor $\alpha$ to the likelihood scores increases the expected distance between the estimated score and oracle score, which may exacerbate the score mismatch issue. Second, since the main difference between methods (a)$\sim$(c) lies in their score-matching objectives, i.e., the parameters $\theta$ in methods (a) and (b) are optimized using $L_\mathrm{CE}$ only while the parameters $\phi_1$ and $\phi_2$ of the score models in method (c) are optimized through $L_\mathrm{DSM}$, the adoption of the score-matching objective may potentially be the key factor to the success of method (c).

The above experimental clues therefore shed light on two essential issues to be further explored and dealt with. First, although employing a classifier trained with $L_\mathrm{CE}$ to assist estimating the oracle posterior score is theoretically feasible, this method may potentially lead to considerable discrepancies in practice. This implies that the score mismatch issue stated in Section~\ref{sec:conditional_sampling} may be the result of the inaccurate likelihood scores produced by a classifier. Second, the comparisons between methods (a), (b), and (c) suggest that score matching may potentially be the solution to the score mismatch issue. Based on these hypotheses, this paper explores an alternative approach to the previous works, called denoising likelihood score matching, which incorporates the score-matching technique when training the classifier to enhance its capability to capture the true likelihood scores. In the next section, we detail the theoretical derivation and the property of our method.

\section{Denoising Likelihood Score Matching}
\label{sec:likelihood_denoising_score_matcing}
In this section, we introduce the proposed denoising likelihood score-matching (DLSM) loss, a new training objective that encourages the classifier to capture the true likelihood score. In Section~\ref{sec:likelihood_denoising_score_matcing:derivation}, we derive this loss function and discuss the intuitions behind it. In Section~\ref{sec:likelihood_denoising_score_matcing:training}, we elaborate on the training procedure of DLSM, and highlight the key property and implications of this training objective.

\subsection{The Derivation of the Denoising Likelihood Score Matching Loss}
\label{sec:likelihood_denoising_score_matcing:derivation}
As discussed in Section~\ref{sec:analysis}, a score model trained with the score-matching objective can potentially be beneficial in producing a better posterior score estimation. In light of this, a classifier may be enhanced if the score-matching process is involved during its training procedure. An intuitive way to accomplish this aim is through minimizing the explicit likelihood score-matching loss $L_\mathrm{ELSM}$, which is defined as the following:
\begin{equation}
\begin{aligned}
    L_\mathrm{ELSM}(\theta) = \mathbb{E}_{p_{\sigma,\tau}({\tilde{\bm{x}}},{\tilde{\bm{y}}})}\left[\frac{1}{2} \lVert\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta) - \nabla_{\tilde{\bm{x}}}\logp_{\sigma,\tau}({\tilde{\bm{y}}}|{\tilde{\bm{x}}}) \rVert^2 \right].
\end{aligned}
\label{eq:L_ELSM}
\end{equation}
This loss term, however, involves the calculation of the true likelihood score $\nabla_{\tilde{\bm{x}}}\log p_{\sigma,\tau}({\tilde{\bm{y}}}|{\tilde{\bm{x}}})$, whose computational cost grows with respect to the dataset size.
In order to reduce the computational cost, we follow the derivation of DSM as well as Bayes' theorem, and formulate an alternative objective called DLSM loss ($L_\mathrm{DLSM}$):
\begin{equation}
\label{eq:dlsm}
\begin{aligned}
    L_\mathrm{DLSM}(\theta) = \mathbb{E}_{p_{\sigma,\tau}({\tilde{\bm{x}}},{\bm{x}},{\tilde{\bm{y}}},{\bm{y}})}\left[ \frac{1}{2} \lVert\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta) + \nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}}) - \nabla_{\tilde{\bm{x}}}\log p_{\sigma}({\tilde{\bm{x}}}|{\bm{x}}) \rVert^2 \right].
\end{aligned}
\end{equation}

\begin{theorem}
\label{theorem:DLSM}
$L_\mathrm{DLSM}(\theta) = L_\mathrm{ELSM}(\theta) + C$, where $C$ is a constant with respect to $\theta$.
\end{theorem}

The proof of this theorem is provided in Appendix~\ref{apx:proof:dlsm}. Theorem~\ref{theorem:DLSM} suggests that optimizing $\theta$ with $L_\mathrm{ELSM}(\theta)$ is equivalent to optimizing $\theta$ with $L_\mathrm{DLSM}(\theta)$. In contrast to $L_\mathrm{ELSM}$, $L_\mathrm{DLSM}$ can be approximated in a computationally feasible fashion. This is because $\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}})$ can be estimated using a score model ${s}({\tilde{\bm{x}}};\phi)$ trained with $L_\mathrm{DSM}$, and $\nabla_{\tilde{\bm{x}}}\log p_{\sigma}({\tilde{\bm{x}}}|{\bm{x}}) = \frac{1}{\sigma^2}({\bm{x}}-{\tilde{\bm{x}}})$ as described in Section~\ref{sec:score_matching}. As $\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}})$ and $\nabla_{\tilde{\bm{x}}}\log p_{\sigma}({\tilde{\bm{x}}}|{\bm{x}})$ can be computed in a tractable manner, the classifier can be updated by minimizing the approximated variant of $L_\mathrm{DLSM}$, defined as:
\begin{equation}
\label{eq:dlsmp}
\begin{aligned}
    L_\mathrm{DLSM'}(\theta) = \mathbb{E}_{p_{\sigma,\tau}({\tilde{\bm{x}}},{\bm{x}},{\tilde{\bm{y}}},{\bm{y}})}\left[ \frac{1}{2} \lVert\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta) + {s}({\tilde{\bm{x}}};\phi) - \nabla_{\tilde{\bm{x}}}\log p_{\sigma}({\tilde{\bm{x}}}|{\bm{x}}) \rVert^2 \right].
\end{aligned}
\end{equation}
The underlying intuition of $L_\mathrm{DLSM}$ and $L_\mathrm{DLSM'}$ is to match the likelihood score via matching the posterior score. More specifically, the sum of the first two terms $\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}}; \theta)$ and $\nabla_{\tilde{\bm{x}}}\log p_{\sigma}({\tilde{\bm{x}}})$ in $L_\mathrm{DLSM}$ should ideally construct the posterior score (i.e., Eq.~(\ref{eq:bayesian})). By following the posterior score, the perturbed data sample ${\tilde{\bm{x}}}$ should move towards the clean sample ${\bm{x}}$, since $\nabla_{\tilde{\bm{x}}}\log p_{\sigma}({\tilde{\bm{x}}}|{\bm{x}})$ is equal to the weighted difference $\frac{1}{\sigma^2}({\bm{x}}-{\tilde{\bm{x}}})$ (i.e., Eq.~(\ref{eq:dsm})). In the next section, we discuss the training procedure with $L_\mathrm{DLSM'}$ as well as the property of it.

\input{figures/training.tex}
\subsection{The Approximated DLSM Objective in the Classifier Training Process}
\label{sec:likelihood_denoising_score_matcing:training}
Following the theoretical derivation in Section~\ref{sec:likelihood_denoising_score_matcing:derivation}, we next discuss the practical aspects during training, and propose to train the classifier by jointly minimizing the approximated denoising likelihood score-matching loss $L_\mathrm{DLSM'}$ and the cross-entropy loss $L_\mathrm{CE}$. In practice, the total training objective of the classifier can be written as follows:
\begin{equation}
\label{eq:total_loss}
\begin{aligned}
L_\mathrm{Total}(\theta) = L_\mathrm{DLSM'}(\theta) + \lambda L_\mathrm{CE}(\theta),
\end{aligned}
\end{equation}
where $\lambda> 0$ is a balancing coefficient. According to Theorem~\ref{theorem:DLSM}, employing $L_\mathrm{DLSM'}$ allows the gradients of the classifier to estimate the likelihood scores. However, since the computation of the term $L_\mathrm{DLSM'}$ requires the approximation of $\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}})$ from a score model ${s}({\tilde{\bm{x}}};\phi)$, the training process that utilizes $L_\mathrm{DLSM'}$ alone is undesirable due to its instability on real-world datasets. To reinforce the stability of it, an additional cross-entropy loss $L_\mathrm{CE}$ is adopted during the training process of the classifier. The advantage of $L_\mathrm{CE}$ is that it leverages the ground truth labels to assist the classifier to learn to match the true likelihood density $p_{\sigma,\tau}({\tilde{\bm{y}}}|{\tilde{\bm{x}}})$, which in turn helps the score estimation to be improved.
To validate the above advantage, an ablation analysis are offered in Section~\ref{sec:experiments:ablation} to demonstrate the differences between the classifiers trained with $L_\mathrm{Total}$, $L_\mathrm{CE}$, and $L_\mathrm{DLSM'}$. The analysis reveals that $L_\mathrm{Total}$ does provide the best score-matching results while maintaining the stability.

Fig.~\ref{fig:training} depicts a two-stage training procedure adopted in this work. In stage 1, a score model ${s}({\tilde{\bm{x}}};\phi)$ is updated using $L_\mathrm{DSM}$ to match $\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}})$ (i.e., Eq.~(\ref{eq:dsm})). In stage 2, the weights of the trained score model are fixed, and a classifier $p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)$ is updated using $L_\mathrm{Total}$. After these two training stages, ${s}({\tilde{\bm{x}}};\phi)$ and $\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)$ can then be added together based on Eq.~(\ref{eq:bayesian2}) to perform conditional sampling.
\section{Experiments}
\label{sec:experiments}

In this section, we present experiments to validate the effectiveness of the proposed method. First, we provide the experimental configurations and describe the baseline methods. Next, we evaluate the proposed method quantitatively against the baselines, and demonstrate its improved performance on the Cifar-10 and Cifar-100 datasets. Lastly, we offer an ablation analysis to inspect the impacts of $L_\mathrm{CE}$ and $L_\mathrm{DLSM'}$ in $L_\mathrm{Total}$.

\subsection{Experimental Setups}
\label{sec:experiments:setup}
In this work, we quantitatively compare the base method, scaling method, and our method (i.e., (a), (b), and (d) in Section~\ref{sec:analysis}) on the Cifar-10 and Cifar-100 datasets, which contain real-world RGB images with 10 and 100 categories of objects, respectively. For both of the datasets, the image size is 32$\times$32$\times$3, and the pixel values in an image are first rescaled to $[-1,1]$ before training. For the sampling algorithm, we adopt the predictor-corrector (PC) sampler described in~\citep{song2021scorebased} with the sampling steps set to $T$=1,000. The score model architecture is exactly the same as the one used in~\citep{song2021scorebased}, while the architecture of the classifier is based on ResNet~\citep{he2016deep} with a conditional branch for encoding the information of the standard deviation $\sigma$~\citep{song2021scorebased}. When training the classifier, $\tau$ is assumed to be sufficiently small, thus the unperturbed labels ${\bm{y}}$ are adopted. Furthermore, the balancing coefficient $\lambda$ is set to 1 for the Cifar-10 and Cifar-100 datasets, and 0.125 for the motivational example. For more details about our implementation and additional experimental results, please refer to Appendix~\ref{apx:config} and Appendix~\ref{apx:additional_exp}.

\input{figures/combined_benchmark.tex}

\subsection{Results on Cifar-10 and Cifar-100}
\label{sec:experiments:benchmarks}

In this section, we examine the effectiveness of the base method, the scaling method, and our proposed method on the Cifar-10 and Cifar-100 benchmarks with several key evaluation metrics. We adopt the Inception Score (IS)~\citep{barratt2018note} and the Fr\'echet Inception Distance (FID)~\citep{heusel2017gans} as the metrics for evaluating the overall sampling quality by comparing the similarity between the distributions of the generated images and the real images. We also evaluate the methods using the Precision (P), Recall (R)~\citep{kynkaanniemi2019improved}, Density (D), and Coverage (C)~\citep{ferjad2020icml} metrics to further examine the fidelity and diversity of the generated images. In addition, we report the Classification Accuracy Score (CAS)~\citep{ravuri2019classification} to measure if the generated samples bear representative class information. Given a dataset containing $m_i$ images for each class $i$, we first conditionally generate the same number of images (i.e., $m_i$) for each class. The FID, IS, P / R / D / C, and CAS metrics are then evaluated based on all the generated $m=\sum_{i=1}^c m_i$ images, where $c$ is the total number of classes. In other words, the above metrics are evaluated in an unconditional manner.

Table~\ref{tab:benchmark} reports the quantitative results of the above methods. It is observed that the proposed method outperforms the other two methods with substantial margins in terms of FID and IS, indicating that the generated samples bear closer resemblance to the real data. Meanwhile, for the P / R / D / C metrics, the scaling method is superior to the other two methods in terms of the fidelity metrics (i.e., precision and density). However, this method may cause the diversity of the generated images to degrade, as depicted in Fig.~\ref{fig:mode_collapse}, resulting in significant performance drops for the diversity metrics (i.e., the recall and the coverage metrics).

Another insight is that the base method achieves relatively better performance on the precision and density metrics in comparison to our method. However, it fails to deliver analogous tendency on the CAS metric. This behavior indicates that the base method may be susceptible to generating false positive samples, since the evaluation of the P / R / D / C metrics does not involve the class information, and thus may fail to consider samples with wrong classes. Such a phenomenon motivates us to further introduce a set of class-wise (CW) metrics, which takes the class information into account by evaluating the P / R / D / C metrics on a per-class basis. Specifically, the class-wise metrics are evaluated separately for each class $i$ with $m_i$ images. The evaluation results of these metrics shown in Fig.~\ref{fig:prdc} reveal that our method outperforms the base method for a majority of classes. Moreover, the results of the average class-wise metrics presented in the last four rows of Table~\ref{tab:benchmark} also show that our method yields a better performance as compared to the base method.

Base on these evidences, it can be concluded that the proposed method outperforms both baseline methods in terms of FID and IS, implying that our method does possess a better ability to capture the true data distribution. Additionally, the evaluation results on the CAS and the class-wise metrics suggest that our method does offer a superior ability for a classifier to learn accurate class information as compared to the base method.

\input{figures/prcd.tex}

\subsection{Ablation Analysis}
\label{sec:experiments:ablation}
\input{figures/curve.tex}
To further investigate the characteristic of $L_\mathrm{Total}$, we perform an ablation analysis on the two components $L_\mathrm{CE}$ and $L_\mathrm{DLSM'}$ in $L_\mathrm{Total}$ using the inter-twinning moon dataset, and observe the trends of (a) the score errors $\frac{1}{2}\mathbb{E}_{U({\tilde{\bm{x}}})}[D_{L}({\tilde{\bm{x}}}, c_1)] + \frac{1}{2}\mathbb{E}_{U({\tilde{\bm{x}}})}[D_{L}({\tilde{\bm{x}}}, c_2)]$, and (b) the cross entropy $\mathbb{E}_{p_{\sigma,\tau}({\tilde{\bm{x}}},{\tilde{\bm{y}}})}[-\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)]$ during the training iterations. Metrics (a) and (b) measure the accuracies of
the estimated likelihood scores $\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)$ and the estimated likelihood density $p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)$, respectively. As depicted in Fig~\ref{fig:curve}, an increasing trend is observed for metric (a) when the classifier is trained with $L_\mathrm{CE}$ alone. On the contrary, the opposite tendency is observed for those trained with $L_\mathrm{Total}$ and $L_\mathrm{DLSM'}$. The results thus suggest that matching the likelihood scores implicitly through training a classifier with $L_\mathrm{CE}$ alone leads to larger approximation errors. In contrast, $L_\mathrm{DLSM'}$ and $L_\mathrm{Total}$ explicitly encourage the classifier to capture accurate likelihood scores, since they involve the score-matching objective. On the other hand, for metric (b), the classifiers trained with $L_\mathrm{Total}$ and $L_\mathrm{CE}$ yield stable decreasing trend in comparison to that trained with $L_\mathrm{DLSM'}$ alone. These results suggest that in addition to minimizing $L_\mathrm{DLSM'}$ between $\nabla_{\tilde{\bm{x}}}\log p_{\sigma,\tau}({\tilde{\bm{y}}}|{\tilde{\bm{x}}})$ and $\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)$, the utilization of $L_\mathrm{CE}$ as an auxiliary objective enhances the stability during training. Based on the above observations, the interplay of $L_\mathrm{CE}$ and $L_\mathrm{DLSM'}$ synergistically achieves the best results in terms of metric (a) while ensuring the stability of the training process. These clues thus validate the adoption of $L_\mathrm{Total}$ during the training of the classifier.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we highlighted the score mismatch issue in the existing conditional score-based data generation methods, and theoretically derived a new denoising likelihood score-matching (DLSM) loss, which is a training objective for the classifier to match the true likelihood score. We have demonstrated that, by adopting the proposed DLSM loss, the likelihood scores can be better estimated, and the negative impact of the score mismatch issue can be alleviated. Our experimental results validated that the proposed method does offer benefits in producing higher-quality conditional sampling results on both Cifar-10 and Cifar-100 datasets.

\subsubsection*{Acknowledgments}
This work was supported by the Ministry of Science and Technology (MOST) in Taiwan under grant number MOST 111-2628-E-007-010. The authors acknowledge the financial support from MediaTek Inc., Taiwan. The authors would also like to acknowledge the donation of the GPUs from NVIDIA Corporation and NVIDIA AI Technology Center (NVAITC) used in this research work. The authors thank National Center for High-Performance Computing (NCHC) for providing computational and storage resources. Finally, the authors would also like to thank the time and effort of the anonymous reviewers for reviewing this paper.

\section{Appendix}
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\renewcommand{\thetable}{A\arabic{table}}
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In this Appendix, we first provide the definitions for the symbols used in the main manuscript and the Appendix in Section~\ref{apx:symbols}. Next, we derive the theoretical proof for the closed form of $\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}})$ in Section~\ref{apx:proof:parzen}. Then, we offer a number of discussions on the scaling technique in Section~\ref{apx:scaling_trick}, and different conditional score-based data generation methods in Section~\ref{apx:conditional_frameworks}. Subsequently, we offer the theoretical proof for Theorem~\ref{theorem:DLSM} in Section~\ref{apx:proof:dlsm}. Finally, in Sections~\ref{apx:config} and~\ref{apx:additional_exp}, we provide the detailed experimental setups and additional experimental results.

\subsection{The List of Symbols Used in the Paper}
\label{apx:symbols}
In this section, we provide the list of symbols used throughout the main manuscript and the Appendix.  These symbols are summarized in Tables~\ref{tab:notation} and \ref{tab:notation_2}, containing their notations as well as their detailed descriptions.
\input{tables/symbols}

\subsection{A Proof of the Closed Form for $\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}})$}
\label{apx:proof:parzen}
In Sections~\ref{sec:parzen_density_estimation} and \ref{sec:analysis}, we utilized the closed form formula to calculate the true scores $\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}})$. To show the equivalence holds for Eq.~(\ref{eq:oracle}), we provide a formal derivation as shown in the following proposition.
\input{proofs/prop0.tex}

\subsection{A Discussion on the Impacts of the Scaling Technique}
\label{apx:scaling_trick}

In this section, we first provide a proof to demonstrate the reason behind the non-equivalence between the scaling technique and the re-normalization of a likelihood density. Then, we present the derivation of the posterior scores based on a re-normalized likelihood density. Finally, we adopt a low-dimensional example to explain the effect of re-normalization on a posterior density.

The authors in~\citep{dhariwal2021diffusion} leveraged a scaling factor ${\alpha}>0$ to adjust the likelihood score $\nabla_{\bm{x}}\log p({\bm{y}}|{\bm{x}})$. To demonstrate the effect of such a scaling factor ${\alpha}$, the authors claimed that there always exists a normalizing constant $Z$ such that $\alpha \nabla_{\bm{x}}\log p({\bm{y}}|{\bm{x}})=\nabla_{\bm{x}} \log \frac{1}{Z}p({\bm{y}}|{\bm{x}})^{\alpha}$, where $\frac{1}{Z}p({\bm{y}}|{\bm{x}})^{\alpha}$ is a re-normalized likelihood density. However, such an equality only holds for specific conditions, and is not applicable to general cases, as explained in \textbf{Proposition~\ref{sup:thm:scaling}}.

\begin{prop}
\label{sup:thm:scaling}
Given a constant ${\alpha}>0$, and a probability density function (pdf) $p({\bm{y}}\vert{\bm{x}})$ that is everywhere non-zero and differentiable, there does not exist a constant $Z$ such that $\frac{1}{Z}p({\bm{y}}\vert{\bm{x}})^{\alpha}$ is a pdf for all ${\bm{x}}$ in general.
\end{prop}

\begin{proof}
The proposition can be proved by contradiction. Assume that such a constant $Z$ exists and consider the following case:
$$
\alpha=2,
$$
$$
x,y\in{\mathbb{R}},
$$
$$
f(y;\mu,\sigma^2)=(2\pi\sigma^2)^\frac{-1}{2}e^\frac{-(y-\mu)^2}{2\sigma^2},
$$
$$
p(y\vert x)=\frac{1}{2(x^2+1)}\left((x+1)^2 f(y;0,1)+(x-1)^2 f(y;0,\frac{1}{2})\right).
$$
Note that $p(y\vert x)$ is a pdf and is everywhere non-zero and differentiable.

Since a pdf sums to 1 (i.e., $\int_y\frac{1}{Z}p(y\vert x)^{\alpha} d y=1,\forall x$),
by utilizing the Gaussian integral and the above assumption, the following derivation holds:
\begin{equation*}
\begin{aligned}
1&=\int_y\frac{1}{Z}p(y\vert (-1))^{\alpha} dy=\int_y\frac{1}{Z}p(y\vert 1)^{\alpha} dy \\
\Leftrightarrow Z&=\int_y p(y\vert (-1))^2 dy=\int_y p(y\vert 1)^2 dy \\
\Leftrightarrow Z&=\int_y f(y;0,\frac{1}{2})^2 dy=\int_y f(y;0,1)^2 dy \\
\Leftrightarrow Z&=\int_y \left((\pi)^\frac{-1}{2}e^{-y^2}\right)^2 dy=\int_y \left((2\pi)^\frac{-1}{2}e^\frac{-y^2}{2}\right)^2 dy \\
\Leftrightarrow Z&=\frac{1}{\pi}\int_y \left(e^{-y^2}\right)^2 dy=\frac{1}{2\pi}\int_y \left(e^\frac{-y^2}{2}\right)^2 dy \\
\Leftrightarrow Z&=\frac{1}{\pi}\int_y e^{-2y^2} dy=\frac{1}{2\pi}\int_y e^{-y^2} dy \\
\Leftrightarrow Z&=\frac{1}{\sqrt{2}\pi}\int_u e^{-u^2} du=\frac{1}{2\pi}\int_y e^{-y^2} dy \\
\Leftrightarrow Z&=\frac{1}{\sqrt{2}\pi}\sqrt{\pi}=\frac{1}{2\pi}\sqrt{\pi} \\
\Leftrightarrow Z&=\frac{1}{\sqrt{2\pi}}=\frac{1}{2\sqrt{\pi}}
\end{aligned}
\end{equation*}
The final equation leads to a contradiction for the value of $Z$, which completes the proof of \textbf{Proposition~\ref{sup:thm:scaling}}.
Therefore, we concluded that there does not exist a constant $Z$ such that $\frac{1}{Z}p({\bm{y}}\vert{\bm{x}})^{\alpha}$ is a pdf for all ${\bm{x}}$ in general.
\end{proof}

To the best of our knowledge, the scaling technique still lacks a theoretical justification. However, if $Z$ is allowed to be a function of ${\bm{x}}$ rather than a constant, re-normalization of the likelihood density becomes feasible. In other words, setting $Z({\bm{x}})=\int_{\bm{y}} p({\bm{y}}\vert{\bm{x}})^{\alpha} d{\bm{y}}$ enables $\int_{\bm{y}}\frac{1}{Z({\bm{x}})}p({\bm{y}}\vert{\bm{x}})^{\alpha} d{\bm{y}}=1$, where $\frac{1}{Z({\bm{x}})}p({\bm{y}}\vert{\bm{x}})^{\alpha} \triangleq p_{{\alpha}}({\bm{y}}\vert{\bm{x}})$ is a re-normalized likelihood density. In this case, the score of the re-normalized likelihood density can be written as,
\begin{equation}
\label{eq:scale}
\begin{aligned}
\nabla_{\bm{x}}\log p_{{\alpha}}({\bm{y}}\vert{\bm{x}}) &=\nabla_{\bm{x}}\log\left( \frac{1}{Z({\bm{x}})}p({\bm{y}}\vert{\bm{x}})^{\alpha}\right) \\
&=\nabla_{\bm{x}} \Big( \log p({\bm{y}}|{\bm{x}})^{\alpha}-\log Z({\bm{x}}) \Big)\\
&=\nabla_{\bm{x}} \log p({\bm{y}}|{\bm{x}})^{\alpha}- \nabla_{\bm{x}} \log Z({\bm{x}}) \\
&={\alpha} \nabla_{\bm{x}} \log p({\bm{y}}|{\bm{x}})- \nabla_{\bm{x}} \log Z({\bm{x}}).
\end{aligned}
\end{equation}
Given the joint density $p_{{\alpha}}({\bm{x}},{\bm{y}}) \triangleq p_{{\alpha}}({\bm{y}}\vert{\bm{x}})p({\bm{x}})$, the posterior of the re-normalized density can be derived by Bayes' Theorem, i.e., $p_{{\alpha}}({\bm{x}}|{\bm{y}})=p_{{\alpha}}({\bm{y}}\vert{\bm{x}})p({\bm{x}})/{\int_{\bm{x}} p_{{\alpha}}({\bm{x}},{\bm{y}}) d{\bm{x}}}$. Based on Eq.~(\ref{eq:bayesian}), the posterior score of the re-normalized density can be written as:
\begin{equation}
\label{eq:scale_post}
\begin{aligned}
\nabla_{\bm{x}} \log  p_{{\alpha}}({\bm{x}}|{\bm{y}}) &= \nabla_{\bm{x}} \log  p_{{\alpha}}({\bm{y}}|{\bm{x}}) + \nabla_{\bm{x}}\log p({\bm{x}})\\
&={\alpha} \nabla_{\bm{x}} \log p({\bm{y}}|{\bm{x}})- \nabla_{\bm{x}} \log Z({\bm{x}}) + \nabla_{\bm{x}}\log p({\bm{x}}). \\
\end{aligned}
\end{equation}
Through the above equation, conditional sampling based on the re-normalized likelihood density becomes feasible.

\input{figures/distribution.tex}
To further investigate the effect of re-normalization on $p_{{\alpha}}({\bm{x}}|{\bm{y}})$, we provide an illustrative example in Fig.~\ref{fig:distribution}. In this example, the condition ${\bm{y}}$ is assumed to be sampled from $\{ {\bm{y}}_1, {\bm{y}}_2\}$ with equal probability $1/2$. The original posterior densities $p({\bm{x}}|{\bm{y}}_1)$ and $p({\bm{x}}|{\bm{y}}_2)$ are normal distributions with variance $2.25$ centered at ${\bm{x}}=-1.5$ and ${\bm{x}}=1.5$, which are plotted in orange and green, respectively. On the other hand, the prior density $p({\bm{x}})\triangleq \frac{1}{2}p({\bm{x}}|{\bm{y}}_1)+\frac{1}{2}p({\bm{x}}|{\bm{y}}_2)$ is plotted in blue, and the likelihood density $p({\bm{y}}|{\bm{x}})= p({\bm{x}}|{\bm{y}})p({\bm{y}})/p({\bm{x}})$ is defined according to Bayes' theorem.

Subsequently, the re-normalized posterior densities $p_{\alpha}({\bm{x}}|{\bm{y}}_1)$ and $p_{\alpha}({\bm{x}}|{\bm{y}}_2)$ can be constructed based on the equation $\frac{1}{Z({\bm{x}})}p({\bm{y}}_1 \vert{\bm{x}})^{\alpha}$, $\frac{1}{Z({\bm{x}})}p({\bm{y}}_2 \vert{\bm{x}})^{\alpha}$, and Bayes' theorem. In Fig.~\ref{fig:distribution}, it is observed that under the case that ${\alpha}=5$, the re-normalized posterior densities are more concentrated. Conversely, when ${\alpha}=0.2$, the re-normalized posterior densities are much more dispersed. These results therefore suggest that the re-normalized posterior density can be adjusted by controlling the value of $\alpha$.

\input{figures/methods.tex}
\subsection{Conditional Score-based Data Generation Methods}
\label{apx:conditional_frameworks}
As described in Section~\ref{sec:analysis}, methods (a), (b), and (d) utilize a classifier $p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta)$ and a score model ${s}({\tilde{\bm{x}}};\phi)$ to estimate the posterior scores. In contrast, method (c) separately trains the posterior score models ${s}({\tilde{\bm{x}}};\phi_1)$ and ${s}({\tilde{\bm{x}}};\phi_2)$ for different class conditions. Although method (c) achieves better evaluation results, it requires training score models for each possible ${\tilde{\bm{y}}}$, which is usually considered impractical in real-world applications. A potential solution to this problem is to train a score model ${s}({\tilde{\bm{x}}},{\tilde{\bm{y}}};\phi)$ that can estimate the scores for the joint probability of ${\tilde{\bm{x}}}$ and ${\tilde{\bm{y}}}$ (denoted as `Method (c$^*$)' in Fig.~\ref{fig:methods}). This method, however, is inflexible in comparison to methods (a), (b), and (d), since any change on the condition ${\tilde{\bm{y}}}$ could lead to re-training the joint score model ${s}({\tilde{\bm{x}}},{\tilde{\bm{y}}};\phi)$. Therefore, in this work, we only compare the performance of methods (a), (b), and (d), and consider method (c) to be another orthogonal research direction.

\subsection{A Proof for Denoising Likelihood Score Matching}
\label{apx:proof:dlsm}
In Section~\ref{sec:likelihood_denoising_score_matcing}, we stated the equivalence between $L_\mathrm{ELSM}(\theta)$ and $L_\mathrm{DLSM}(\theta)$ when optimizing $\theta$. To demonstrate how such an equivalence stands, we provide the theoretical proof as follows.
\input{proofs/prop1}

\subsection{The Detailed Experimental Configurations}
\label{apx:config}
In this section, we elaborate on the implementation details and the experimental setups. In Section~\ref{apx:config:noise_and_sampling}, we describe the noise perturbation method and the sampling algorithm adopted in this work. In Sections~\ref{apx:config:toy_experiments} and~\ref{apx:config:benchmark}, we present the training configurations for the experiments on the inter-twinning moon, Cifar-10, and Cifar-100 datasets. The code implementation for these experiments is provided in the Github repository (\url{https://github.com/chen-hao-chao/dlsm}).

\subsubsection{Noise Perturbation and Sampling Algorithm}
\label{apx:config:noise_and_sampling}
\paragraph{Sampling Algorithm.}~In this work, we adopt the PC sampler (VE-SDE) identical to that in~\citep{song2021scorebased}, which is a time-inhomogeneous sampling algorithm. The PC sampler (VE-SDE) is described in \textbf{Algorithm 1}. In \textbf{Algorithm 1}, a noise-conditioned score model ${s}({\tilde{\bm{x}}};\phi,\sigma_t)$ is adopted, where ${\tilde{\bm{x}}}$ is perturbed by noises sampled from Gaussian distributions using different standard deviations $\sigma_t$. The standard deviations of the noises, denoted as $\sigma_t=\sigma_{\text{min}} (\frac{\sigma_{\text{max}}}{\sigma_{\text{min}}})^{\frac{t}{T}}$, are determined by the time step $t$ in \textbf{Algorithm 1}. We set $\sigma_{\text{min}}=0.01$ for all of our experiments. On the other hand, we set $\sigma_{\text{max}}=50$ for the Cifar-10 and Cifar-100 datasets, and $\sigma_{\text{max}}=10$ for the inter-twinning moom dataset. As for the total time steps, the value of $T$ is set to 1,000.

\paragraph{Training with Different Standard Deviations.}~Since the PC sampler (VE-SDE) described in \textbf{Algorithm 1} requires the score model ${s}({\tilde{\bm{x}}};\phi,\sigma_t)$ conditioned on different values of standard deviation $\sigma_t$, the training objectives described in Eqs.~(\ref{eq:dsm}) and (\ref{eq:total_loss}) are modified accordingly. For the DSM loss described in Eq.~(\ref{eq:dsm}), the modified loss $L_{\mathrm{DSM}}(\phi) $ is represented as:
\begin{equation}
\label{eq:dsm_mod}
\begin{aligned}
\mathbb{E}_{\mathcal{U}_{[0,T]}(t)} \left[  \lambda_{\mathrm{DSM}}(t) \mathbb{E}_{p_{\sigma_t}({\tilde{\bm{x}}},{\bm{x}})} \left[ \lVert{s}({\tilde{\bm{x}}};\phi,\sigma_t) - \nabla_{\tilde{\bm{x}}}\log p_{\sigma_t}({\tilde{\bm{x}}}|{\bm{x}})\rVert^2 \right]\right],
\end{aligned}
\end{equation}
where $\mathcal{U}_{[0,T]}(t)$ is a uniform distribution on the interval $[0,T]$. In Eq.~(\ref{eq:dsm_mod}), The additional coefficient $\lambda_{\mathrm{DSM}}(t)$ is multiplied with the original DSM loss for better training results~\citep{song2019generative}. For the approximated DLSM loss described in Eq.~(\ref{eq:dlsm}), the modified loss $L_\mathrm{DLSM'}(\theta)$ is represented as:
\begin{equation}
\begin{aligned}
     \mathbb{E}_{\mathcal{U}_{[0,T]}(t)} \left[ \lambda_{\mathrm{DLSM}} (t) \mathbb{E}_{p_{\sigma_t,\tau}({\tilde{\bm{x}}},{\bm{x}},{\tilde{\bm{y}}},{\bm{y}})}\left[\lVert\nabla_{\tilde{\bm{x}}}\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta,\sigma_t) + {s}({\tilde{\bm{x}}};\phi,\sigma_t) - \nabla_{\tilde{\bm{x}}}\log p_{\sigma_t}({\tilde{\bm{x}}}|{\bm{x}}) \rVert^2\right] \right],
\end{aligned}
\end{equation}
where $ p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta,\sigma_t)$ is a noise-conditioned classifier. $\lambda_{\mathrm{DLSM}}(t)$ is the balancing coefficient for different $\sigma_t$. For the cross-entropy loss, the modified loss $L_\mathrm{CE}(\theta)$ is represented as:
\begin{equation}
\begin{aligned}
    \mathbb{E}_{\mathcal{U}_{[0,T]}(t)} \left[ \lambda_{\mathrm{CE}}(t) \mathbb{E}_{p_{\sigma_t,\tau}({\tilde{\bm{x}}},{\tilde{\bm{y}}})}[-\log p({\tilde{\bm{y}}}|{\tilde{\bm{x}}};\theta,\sigma_t)] \right],
\end{aligned}
\end{equation}
where $\lambda_{\mathrm{CE}}(t)$ is the balancing coefficient for different $\sigma_t$. In all of our experiments, $\lambda_{\mathrm{CE}}(t)$ is set to $1$. The values of $\lambda_{\mathrm{DSM}}(t)$ and $\lambda_{\mathrm{DLSM}}(t)$ are customized according to different settings, and specified in Sections~\ref{apx:config:toy_experiments} and \ref{apx:config:benchmark}. Please note that $\lambda_{\mathrm{DSM}}(t)$, $\lambda_{\mathrm{DLSM}}(t)$, and $\lambda_{\mathrm{CE}}(t)$ are adopted to balance the loss functions for different $\sigma_t$, and are different from the balancing coefficient $\lambda$ in Eq.~(\ref{eq:total_loss}).

\input{algorithm/pc_algorithm.tex}

\subsubsection{The Experiments on the Inter-twinning Moon Dataset}
\label{apx:config:toy_experiments}
\paragraph{Dataset.}~We perform our motivational experiments on the inter-twinning moon dataset provided by the \texttt{sklearn} library in \texttt{python}. In this dataset, the data points are sampled from two interleaving crescents. The data points of the upper crescent are generated based on $(\cos(x),\sin(x)),\forall x \in [0,\pi]$ and are labeled as $c_1$. On the other hand, the data points of the lower crescent are generated based on $(1-\cos(x),0.5-\sin(x)),\forall x \in [0,\pi]$ and are labeled as $c_2$. Before training, the data points are first normalized such that their mean become 0 and then their coordinates are scaled by a factor of 20.

\paragraph{Setups and Evaluation Method.}~In this experiment, the scores $\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}})$ and $\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}}|{\tilde{\bm{y}}})$ are obtained by calculating Eq.~(\ref{eq:oracle}). The likelihood scores are computed according to $\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{y}}}|{\tilde{\bm{x}}})=\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}}|{\tilde{\bm{y}}})-\nabla_{\tilde{\bm{x}}}\logp_{\sigma}({\tilde{\bm{x}}})$ in Eq.~(\ref{eq:bayesian}). As for the $D_P$ and $D_L$ metrics in Table~\ref{tab:toy_experiment}, the errors between two score functions are measured using 1,225 data points uniformly sampled from the two-dimensional subspace $[-25,25]\times[-40,40]$. The scaling factor $\alpha$ adopted in method (b) is set to 10. The values of $\lambda_{\mathrm{DSM}}(t)$ and $\lambda_{\mathrm{DLSM}}(t)$ are both set to $\frac{1}{\sigma_t^4}$, and the value of $\lambda$ is set to 0.125.

\paragraph{Training and Implementation Details.}~The network architectures for the score model and the classifier are simple three-layer multilayer perceptrons (MLPs) with ReLu as the activation function. These networks are implemented using the \texttt{pytorch} library with \texttt{nn.Linear} layers that contain (128, 64, 32) neurons per layer for both of the score model and the classifier. The score model is trained using the Adam optimizer with a learning rate of $6.5\times 10^{-4}$ and a batch size of 4,000 for 40,000 iterations. The classifier is trained using the Adam optimizer with a learning rate of $2.0\times 10^{-5}$ and a batch size of 4,000 for 40,000 iterations.

\subsubsection{The Experiments on the Cifar-10 and Cifar-100 Datasets}
\label{apx:config:benchmark}
\paragraph{Dataset.}~We adopt the Cifar-10 and Cifar-100 datasets~\citep{krizhevsky2009learning} for evaluating our design on image generation tasks. Cifar-10 consists of 60,000 $32\times32$ RGB images with 10 classes, where each class contains 6,000 images. Cifar-100 is similar to Cifar-10, except it has 100 classes, where each class contains 600 images. The images are first normalized to $[-1,1]^d$ before training.

\paragraph{Setups and Evaluation Method.}~The results on the CAS metric are evaluated using a classifier (ResNet-50~\citep{he2016deep}) trained with the generated samples. This classifier is trained to minimize the cross-entropy loss with the Adam optimizer using a learning rate of $2.0\times 10^{-4}$ with a batch size of 150. The results on the P / R / D / C metrics are calculated using the officially released \href{https://github.com/clovaai/generative-evaluation-prdc}{\underline{code}}. The hyperparameter $k$ used in the P / R / D / C metrics is fixed to 3. The results on the FID and IS metrics are computed on 50,000 samples using the \texttt{tensorflow$\_$gan} library in \texttt{python}. The value of $\lambda_{\mathrm{DSM}}(t)$ is set to $\frac{1}{\sigma_t^2}$, while the value of $\lambda_{\mathrm{DLSM}}(t)$ is set to $1$. The value of $\lambda$ is set to 1.

\paragraph{Training and Implementation Details.}~The network architecture for the score model is the same as~\citep{song2021scorebased}. The network architecture for the classifier is a modified ResNet-18~\citep{he2016deep} for Cifar-10 and a modified ResNet-34 for Cifar-100. The modification only involves an additional conditional branch for encoding the standard deviation $\sigma_t$. The score model is trained with the Adam optimizer using a learning rate of $2.0\times 10^{-4}$ and a batch size of 128. The classifier is trained with the Adam optimizer using a learning rate of $2.0\times 10^{-4}$ and a batch size of 100.

\subsection{Additional Experiments}
\label{apx:additional_exp}
In this section, we present a number of additional experimental results and provide discussions on them. We first report the results of our method using different values of $\lambda$ on Cifar-10 to demonstrate that $L_\mathrm{Total}$ is less sensitive to the choices of $\lambda$. Next, we apply the scaling technique on both the base method and our method with different choices of $\alpha$ to demonstrate the influence of the hyperparameter $\alpha$ on the evaluation results of several metrics. Finally, we provide uncurated examples on the Cifar-10 and Cifar-100 datasets.

\input{tables/toy_experiment_ablation.tex}
\subsubsection{Ablation Analysis on the Inter-twinning Moon Dataset}
\label{apx:additional_exp:abation}
Table~\ref{tab:toy_experiment_ablation} compares the evaluation results on the $\mathbb{E}_{U({\tilde{\bm{x}}})}[D_{P}({\tilde{\bm{x}}},{\tilde{\bm{y}}})]$ and $\mathbb{E}_{U({\tilde{\bm{x}}})}[D_{L}({\tilde{\bm{x}}},{\tilde{\bm{y}}})]$ metrics when the classifiers are trained with $L_\mathrm{CE}$, $L_\mathrm{DLSM'}$, and $L_\mathrm{Total}$, respectively. It is observed that the score errors $\mathbb{E}_{U({\tilde{\bm{x}}})}[D_{P}({\tilde{\bm{x}}},{\tilde{\bm{y}}})]$ and $\mathbb{E}_{U({\tilde{\bm{x}}})}[D_{L}({\tilde{\bm{x}}},{\tilde{\bm{y}}})]$ are the lowest when $L_\mathrm{Total}$ is adopted. The experimental results therefore validate the effectiveness of the proposed loss $L_\mathrm{Total}$.

\subsubsection{The Experiments on the Balancing Coefficient}
\label{apx:additional_exp:balancing_coefficient}
Table~\ref{tab:lambda} demonstrates the evaluation results on FID, IS, and P / R / D / C metrics using $L_\mathrm{Total}$ with different values of $\lambda$. It is observed that the evaluation results only slightly fluctuate on the six metrics with different choices of $\lambda$. The results thus experimentally validate that $L_\mathrm{Total}$ is less sensitive to the choice of $\lambda$.
\input{tables/lambda.tex}

\input{tables/alpha.tex}
\subsubsection{The Experiments on the Scaling Technique}
\label{apx:additional_exp:scaling_factor}
Table~\ref{tab:alpha} demonstrates the evaluation results of the base method, the scaling method using different values of $\alpha$, our method, and our method with the scaling technique using different values of $\alpha$ (i.e., `ours + scaling') on the Cifar-10 dataset. It is observed that the performance of the `scaling method' and `ours + scaling' method on the fidelity metrics, i.e., precision and density, improves as the value of $\alpha$ grows. Conversely, the performance of the `scaling method' and `ours + scaling' method on the FID, recall, and coverage metrics degrades as the value of $\alpha$ grows. This results thus experimentally demonstrate the effects of the scaling technique on the performance of the base method and our method.

\subsubsection{Uncurated Examples}
\label{apx:additional_exp:uncurrated}
Figs.~\ref{fig:uncurrated_cifar10_standard}, \ref{fig:uncurrated_cifar10_scaled}, \ref{fig:uncurrated_cifar10_ours}, and \ref{fig:uncurrated_cifar100} depict a few uncurated examples that qualitatively compare the effectiveness of different methods.

\input{figures/uncurrated_cifar10_standard.tex}
\input{figures/uncurrated_cifar10_scaled.tex}
\input{figures/uncurrated_cifar10_ours.tex}
\input{figures/uncurrated_cifar100.tex}

redpajama_set_name: RedPajamaArXiv


text:
You are at:Home»News»Inside Dallas ISD»Ellie the Elephant spreads message of kindness, empathy to students
Dan D. Rogers Elementary students catch a first glimpse of Ellie the Elephant.
Ellie, a realistic-looking animatronic robot, shares a story of capture, abuse and rescue.
Students saw posters showing a few things elephants enjoy doing in the wild.
Principal Lisa Lovato, left, helps pre-K students get a good look at Ellie.
Ellie's message is one of empathy – students thought about what it would be like to live the elephant's life.
Ellie the Elephant spreads message of kindness, empathy to students
By The Hub on	 March 30, 2017 Inside Dallas ISD
It's not every day that an elephant shows up at your school.
But that's exactly what happened on Thursday, March 30, at Dallas ISD's Dan D. Rogers Elementary School. Well, sort of.
Ellie the Elephant is a realistic-looking, animatronic robot elephant that talks. Ellie tells a story about being captured in the wild, abused and forced to work in the circus. The program aims to teach empathy to schoolchildren about how animals are sometimes treated in captivity. Her story ends happily, as she is rescued and sent to live in an animal sanctuary.
This year, Ellie has spoken to 25,000 students across the country about the importance of kindness.
At Rogers, the program also touched on how abusive treatment of animals is a form of bullying.
The program is coordinated by Teach Kind, the educational arm of the People for the Ethical Treatment of Animals (PETA) organization.
Dan D. Rogers Elementary School Trustee District 2
Thomas Jefferson teacher receives grant from the American Geographical Society

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Beano and the Isle of Man Post Office are today celebrating 80 years of the world's longest running and most loved comic with a special collection of stamps, featuring Beano's iconic characters Dennis & Gnasher on the Isle of Man.
Representatives for the Isle of Man Post Office say they feel extremely proud to be working with Beano on such a special, vibrant and colourful stamp collection, celebrating eighty years of fun filled mischief.
Over the festive period Dennis & Gnasher are stranded, without their mobile phones, on the Isle of Man to visit a mysterious uncle named Paul, who has a dark secret: he's a postman!!! After this nightmarish revelation (especially for Gnasher) it seems that Christmas is officially cancelled! But just when Dennis & Gnasher give up on the holidays Uncle Paul gives them hope by letting them into a secret: Writing a letter is like sending an epic text message that is then delivered by your own personal Postie Butler!
Dennis & Gnasher are immediately hooked and with Uncle Paul's help, they even create their own stamps that they then use to send letters and cards to their friends in Beanotown. Not only do their friends write back, but they promise to meet together on what's quickly become the Christmas Isle of Mischief, to share the festive fun.
The artwork for the stamp collection and the accompanying comic strip have been drawn exclusively by Nigel Parkinson, who, since 2012, is the official Dennis & Gnasher artist and is one of the most prolific Beano artists since 1999.
We are overjoyed that Dennis & Gnasher, along with all his pals in the Beano Studio of DC Thomson in Dundee, worked so hard to capture well known Manx locations and traditions in an exclusive Beano comic strip titled A Write Christmas on the Isle of Man'.
We hope that we can encourage people of all ages to discover and engage with sending Christmas greetings. The moral of the tale is that by writing to friends and family by post is still the best way to catch up at Christmas or share good wishes at any time of the year.
We're delighted to work with the Isle of Man Post Office to create a unique festive collection of stamps. We look forward to seeing Dennis & Gnasher on Christmas cards, letters and parcels across the world.


Herbes de Provence are a special blend of seasonings, including summer savory, basil, marjoram, fennel, bay, thyme, orange zest, and our very sweetest English lavender. Add to olive oil for a wonderful salad dressing or a zesty bruschetta. May also be used to season fish, poultry, red meats, pasta, and more. Comes in an 8-ounce bottle.


Timber flooring has an exquisite appeal, a reason why many people prefer going for it. The aesthetic appearance makes the interior of the house look unique and more natural than other flooring types. Timber floors designed and fitted by a professional can make the house value increase as well. Timber flooring is also very easy to clean because dust and stains rarely stick to these materials.
The primary advantage of using our timber flooring services in Hurstville, compared to other types of flooring, is that it is very durable. When designing a home to have a timber floor installed there are many factors to consider. In the Hurstville area there are still a large number of older style homes that are renovating and considering having a new solid timber floor installed. Our expertise can help with selecting the best-suited species to your homes design.
Timber flooring has many different types, but the best is created from hardwoods. These are very durable and can survive for more than fifty years without developing cracks. They are maintained by regular polishing which can make them look new.
Call our friendly team on 02 9756 4242 today for free advice on how we can help you make the best timber flooring selection for your house in Hurstville!


text:
Sue Evans
Drum, Musical Artist
Who is Sue Evans?
Sue Evans is an American jazz, pop, classical, and studio percussionist/drummer.
Evans played piano, violin and clarinet as a young child before switching to drums. She studied under Warren Smith and Sonny Igoe, and graduated in 1969 from the High School of Music and Art. She soon became one of the top recording percussionists in New York,recording jingles, movie scores, and numerous albums with many jazz, folk and pop artists. She was Judy Collins's touring drummer from 1969 to 1973, and worked with Gil Evans from 1969 to 1982. In the 1970s she worked with Steve Kuhn, Art Farmer, Bobby Jones, George Benson, Urbie Green and Roswell Rudd's Jazz Composers Orchestra, in addition to playing with The New York Pops, the New York Philharmonic, the Brooklyn Philharmonic and the New Jersey Symphony Orchestra. In the 1980s, she worked with Michael Franks, Suzanne Vega, Tony Bennett, and Morgana King. Other associations include touring or recording with Aretha Franklin, Sting, Spike Lee, James Brown, Billy Cobham, Blood, Sweat and Tears, Philip Glass, Peter, Paul, and Mary, Don Sebesky, Sadao Watanabe, Hubert Laws, Randy Brecker, David Sanborn and Terence Blanchard. She also played the Tony Awards for several years, as well as the Grammy Awards.
"Sue Evans." Biographies.net. STANDS4 LLC, 2020. Web. 21 Jan. 2020. <https://www.biographies.net/people/en/sue_evans>.
Discuss this Sue Evans biography with the community:
Evans, Sue
The High School of Music & Art


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1.What's the relationship between Xiamen C&D Inc. and C&D Group?
怎样看懂快三彩票走势图Xiamen C&D Inc. is a listed subsidiary of C&D Group, floated on the Shanghai Stock Exchange in 1998 with its turnover of 128 billion yuan in 2015. As the parent company of Xiamen C&D Inc., C&D Group is a large-scale investment-oriented enterprise ( 46% holding shares in Xiamen C&D Inc.) and ranked first in the top 100 enterprises of Fujian Province for many years.
2.What is the main business of Xiamen C&D Inc.?
怎样看懂快三彩票走势图Xiamen C&D Inc. is a modern service-oriented enterprise engaged in supply chain operation, which started its business in 1980. The traditional core business of C&D Inc. lies in supply chain operation. With rich experience in that field, the company owns a sound business network and well-deserved reputation. It has been ranked among the top 100 Chinese listed companies and Forbes Global 2000. The company has formed 11 core business areas such as pulp and paper, steel, mineral products, agricultural products, textile, chemical engineering, electromechanical products, automobile, alcohol, logistics and financial services. Its 20-odd branches are located in Shanghai, Beijing, Tianjin, Guangzhou, Qingdao, Chengdu, Wuhan, Hong Kong, the United States, Canada, Singapore and so on.
3.Where do Xiamen C&D Inc. offer employment?
With the rapid development of supply chain operation, Xiamen C&D Inc. has established branches in major cities such as Shanghai, Beijing, Guangzhou, Tianjin, Hong Kong, Wuhan, Chengdu, Fuzhou, Qingdao and Ningbo as well as other countries such as the United States, Canada and Singapore. Moreover, the company has built partnerships with more than 70 countries and regions. Besides more opportunities of work and self-improvement, the company provides a broad stage of development and hopes employees can make a figure and achieve their dreams.
4.How do I apply for a job in Xiamen C&D Inc. ?
(1) Social recruitment is geared to social persons with work experience. Please pay attention to the job responsibility and application requirement of each post. As a general rule, there will be 1 to 3 rounds of interviews. If necessary, a professional written examination and an online evaluation will be needed.
(2) Campus recruitment caters for fresh graduates without work experience. Please view the website for campus recruitment of Xiamen C&D Inc.. The general application process: resume delivery online or in recruitment seminar/job fair — 1 to 3 rounds of interviews — online evaluation — written examination — a tripartite agreement.
Honors of The Enterprise
Names, trademarks and services of other companies mentioned in the website are their registered trade marks and owned by them respectively. 建发, 厦门建发, CND, CHINACND, C & D are trade marks of Xiamen C & D Inc. Co,. Ltd or its companies of controlling interest, or joint stock companies.
怎样看懂快三彩票走势图 Please observe stipulations of applicable law when visiting our website.
怎样看懂快三彩票走势图 The information on the website of Xiamen C&D Co. Ltd. is released by the company as the principle of "according to the facts". The company doesn't guarantee for any information from website, doesn't guarantee for anything including marketing & channel、business use or non-loss, whatever indication or implication.
怎样看懂快三彩票走势图 Xiamen C&D Co. Ltd. will take necessary measures to ensure the veracity of the website information, but doesn't guarantee any accuracy or integrity. We won't offer further notice if there is any change or update to the information. Xiamen C&D reserves the right to modify this declaration at anytime. We strongly suggest you to visit this webpage regularly to realize the latest declaration.
怎样看懂快三彩票走势图 Xiamen C&D Co. Ltd. and its sub-companies will not assume any responsibility for the direct or indirect loss caused by using the information on our website. Any responsibilities and results shall be assumed by users.
怎样看懂快三彩票走势图 For users' convenience and information, the company will allow the user to access other websites via our website regularly. We don't take any responsibility for the websites, which the user access through our links. No comment toward the quality, property or reliability of these websites. This shall not represent that Xiamen C&D offers the contents and information to the websites which not belong to Xiamen C&D. Xiamen C&D Co. Ltd. reserves the right to delete the links at anytime without any notice to the third party.
We don't keep the confidentiality for the information which you reply to the website of Xiamen C&D Co. Ltd. Once you reply to our website, you have authorized us the irrepealably and unlimited right to use, copy, distribute, modify your information, as well as the agreement on that Xiamen C&D Co. Ltd. is allowed to use any plan or concept which included in your response for any use for free. However, we will not disclose your name unless the law requires it or we acquire your authorization in written form.
Never sell or lease the personal information of the client, potential client and website visitor to anyone. Xiamen C&D guarantee to protect the personal privacy.
We will take necessary preventive measures to reinforce the monitoring and control of Spam email.


The second annual WIndiana conference opens today, July 21, at the Indianapolis Convention Center (full agenda). The 2-day conference focuses on wind resources in Indiana for both utility level wind development, urban or smaller wind power, helping Indiana manufacturers enter or expand their presence in the wind industry supply chain, and what the future holds for the wind industry in the state. Both Governor Mitch Daniels and Lt. Governor Becky Skillman are addressing the conference.
Along with general sessions, WIndiana includes two separate tracks, Wind Resources and Indiana's Wind Supply Chain. The three supply chain sessions will include topics such as Indiana Site Selection for Manufacturing Opportunities, Supply Chain Basics, and Supply Chain Nuts and Bolts.
The American Wind Energy Association has ranked Indiana as the fastest-growing state for wind energy in 2008 and by the end of 2009, Indiana is on track for growth this year of more than 600%, which would again lead the nation. Indiana is on track to have 5,000 MW of wind capacity by 2013 and Indiana has thus far made good use of its endowments, and its current installed wind power capacity ranks it 14th among states.
Indiana is ranked sixth in the nation for manufacturing job creation in the wind industry. There are 45,794 Hoosiers working in jobs directly related to the renewable energy industry, generating $11.17 Billion in annual output, which is over 2 percent of Indiana's total output.
Additional information is available at the OED Web site, www.energy.IN.gov.


1) A temperature is quoted as 60.16 oF. What is it in oC (see sheet 3)?
2) A temperature difference is quoted as 22.27 oC. What is the temperature difference quoted in oK (see sheet 3) ? Indicate with a positive (negative) sign whether the relation between DT in oC and oK is (not) the same for DT in oC and oF.
Change in temperature will be the same in degrees Kelvin.
3) A substance with a density of 0.5031 g/cm3 occupies a volume of 2.905 cm3. What is the number of Atomic Mass Units (see sheet 6) ? Indicate with a negative (positive) sign whether the chemical composition of the substance has (does not have) to be known.
4) What is the mass of 2.828 moles of H2O in kg (see sheet 6,8') ? Indicate with a negative (positive) sign whether the mass of the same number of hydrogen moles and oxygen moles is (not) the same.
So for three, I just used the numbers from the given question above and I get an answer pretty close to the one provided, but I don't even know if this is the way the notes say how to do it because I didn't look at them, i just used chemistry knowledge.
The third decimal place is off so I'm not sure exactly what to do but maybe someone else can work off of this to find the right way?
And my guess is that it would be positive since I didn't even know the composition while attempting this. Any input?
for this question u=m/(1.66e-27 kg).
i dont understand what you mean..
I don't know why it is 1.66e-27, maybe guest145 could explain where they got that?
well, of course i can explain. The answer is indeed in slide 6, in which 1 u=1.66 e-27kg, and we all know that u is the atomic mass unit!
Ok i finally got it now...thanks for ur help!!
as an old chem lab Prof once said,"Think about your units."
you are multiplying g/cm^3 and cm^3. cm^3 cancel out, you are only left with grams. Grams must be converted then to kg by multiplying by 0.001 or (as posted above) divide by 1000. Entering your answer by moving the decimals and having all of those zero's is accepted by MapleTA, so count carefully before your enter your answer!
Use capital " E "


text:
Charles Lapointe,  (born July 17, 1944) is a Canadian businessman and former politician and public servant.

Lapointe was first elected to the House of Commons of Canada in the 1974 federal election as a Liberal Member of Parliament for Charlevoix. He served as Canadian delegate to the United Nations General Assembly in 1976, and parliamentary secretary to the Transport minister from 1977 to 1979.

He was re-elected in the 1979 federal election that defeated the Liberal government. When the Liberals returned to power in the 1980 election, Lapointe was appointed by Prime Minister Trudeau to the cabinet as Minister of State for Small Businesses and Tourism.

In 1982, he became Minister of State for External Relations and, in 1983, he was promoted to Minister of Supply and Services and Receiver-General.

When John Turner succeeded Trudeau as Liberal leader and prime minister in June 1984, he retained Lapointe as Minister of Supply and Services while giving him the additional portfolio of Minister of Public Works. The added responsibility was short-lived, however, as both the Liberal government and Lapointe were defeated in the subsequent fall federal election.

Returning to private life, Lapointe became president of International Aeroplane Company, and then vice-president of business development for Lavalin. In 1989, he became president and chief executive officer of the Greater Montreal Convention and Tourism Bureau (Tourisme Montréal), retaining the position until his retirement in 2013. In 2002, he also became chairman of the Canadian Tourism Commission.

In January 2007, as head of Tourism Montreal, he said the city was filthy and more should be done to make it tidy for visitors, leading Mayor Gérald Tremblay to ask for his resignation. Supported by many, Lapointe did not resign and continued in his current position.

Lapointe was openly gay among his caucus colleagues, but never publicly spoke about his sexuality to the media, during his time in Parliament. With Tourisme Montréal, however, beginning in 1994 he created a then-innovative campaign to increase the city's visibility as a gay tourism destination, building civic partnerships with LGBT community events such as Divers/Cité and the Black and Blue Festival, and culminating in the city's hosting of the World Outgames in 2006. In 2013 he received the Hanns Ebensten Hall of Fame Award from the International Gay and Lesbian Travel Association.

References

External links

1944 births
Living people
Liberal Party of Canada MPs
Members of the House of Commons of Canada from Quebec
Members of the King's Privy Council for Canada
Canadian LGBT Members of Parliament
Gay politicians
People from Côte-Nord
Members of the 23rd Canadian Ministry
21st-century Canadian LGBT people
Canadian gay men

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Eggplant, the provider of user-centric, intelligent testing and performance solutions, today announced the latest enhancements to its Digital Automation Intelligence (DAI) Suite. The advancements support Eggplant's mission to use AI, machine learning, and analytics to optimize the digital experience, allowing teams to keep up with pace of DevOps, and improve time to market and customer satisfaction.
The latest product releases will enable Eggplant to continue to transform testing by increasing automation and visibility, helping teams to understand and test the user journey and gain valuable insight into the entire user experience.
Eggplant AI, the AI-powered test modeling and automated test creation product, is enhanced to incorporate the concept of prioritized user journeys. This allows QA professionals to ensure that all common user journeys are tested before switching to AI-powered test mode — which is analogous to switching to test autopilot. This functionality blends the best of human expertise and AI insight. Eggplant AI can now also automate the capture and analysis of runtime performance data and enable the testing process to "learn" where potential bottlenecks are and eradicate them.
Eggplant Functional, the market-leading automation engine, is enhanced with AI-powered application discovery capabilities. The solution can now automatically build a GUI interaction model by routinely learning about the elements that compose an application screen. Companies can then use this information to automatically populate the automation logic in an Eggplant AI application model.
Eggplant Performance, a powerful load and performance testing product, is enhanced with support for JMeter, a popular and simple-to-use open source performance testing framework. By including support for JMeter, Eggplant now enables developers and testers to utilize the power of Eggplant Performance with existing JMeter assets, as well as use the test controller and analyzer technology for better runtime test visibility and improved analysis and reporting capabilities. Creating load test scripts is simple, and the new, real-time KPIs dramatically shorten the feedback loop between testing and results analysis.
Eggplant Manager and Eggplant Automation Cloud, which offer testers the ability to create a scalable device lab that virtualizes and manages thousands of devices, test cases and users, is also enhanced. Both solutions feature a 10x performance and scalability improvement as well as automatic database backups during upgrades. Updates to Eggplant Manager also include better support for command-line repositories, plus the ability to export filtered HTML and PDF versions of results.


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Home / Sports / Gary Lineker Took To Social Media To Thank Gabriel Martinelli For Opening The Scoring For Arsenal In Their 4-1 Win Over Leeds
Gary Lineker Took To Social Media To Thank Gabriel Martinelli For Opening The Scoring For Arsenal In Their 4-1 Win Over Leeds
By Stephen on December 19, 2021
Leeds pulled a goal back in the 75th minute when Raphinha converted his penalty after Ben White's foul on Joe Gelhardt in the box.
Gary Lineker took to social media to thank Gabriel Martinelli for opening the scoring for Arsenal in their 4-1 win over Leeds United after the rest of the day's Premier League games were postponed due to Covid-19.
The 20-year-old has been in excellent form for the Gunners in recent games and he netted the opener in the 16th minute with a strong finish from close range after he pounced on a loose ball in the box.
Martinelli then made it 2-0 to the visitors when he fired home a dinked finish following Granit Xhaka's through-ball in the 28th minute.
Bukayo Saka then made it three to Mikel Arteta's men with a deflected close-range effort two minutes before the break.
However, substitute Emile Smith Rowe came off the bench to fire home a fourth for the Gunners in the 84th minute to secure Arsenal their third win in a row in the English top flight.
With all of the other day's games postponed due to Covid-19, England legend and Match of the Day host Lineker was understandably pleased to see Martinelli open the scoring and provide a talking point ahead of the programme.
Posting on Twitter after the opener, Lineker wrote: "We've got a goal. Match of the Day has a goal. Thank you Gabriel Martinelli."
Gary Lineker Arsenal Martinelli
After the second goal, Lineker added: "We've doubled our goals tally on the telly thanks to Martinelli."
And, when jokingly asked by Piers Morgan to "clear" the Match of the Day schedule for Arsenal's game, Lineker replied: "We've got bugger all else, so that would be the case regardless of this excellent Arsenal display."
Martinelli has been in great form for the Gunners in recent games – and he has now netted three goals in his last two outings for the north London side, who are fourth and now four points clear of fifth-placed West United.
"It was a very good result," Martinelli told Sky Sports after the game. "It's so important for the team. All the team played well and we're so happy with the work and the result.
"That's what Mikel [Arteta] asks from us. Every time we go into the final third we need to be dangerous.
"As I always say, we have a young team but a very strong team with good players. Emile [Smith Rowe] came from the bench and scored another goal. I'm so happy for the team."
Next up for Arsenal is their League Cup quarter-final clash with Sunderland at The Emirates on Tuesday night. They will then travel to Norwich City in the Premier League on Boxing Day.
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