Conditionally Whitened Generative Models for Probabilistic Time Series Forecasting

## Abstract

Abstract Probabilistic forecasting of multivariate time series is challenging due to non-stationarity, inter-variable dependencies, and distribution shifts. While recent diffusion and flow matching models have shown promise, they often ignore informative priors such as conditional means and covariances. In this work, we propose Conditionally Whitened Generative Models (CW-Gen), a framework that incorporates prior information through conditional whitening. Theoretically, we establish sufficient conditions under which replacing the traditional terminal distribution of diffusion models, namely the standard multivariate normal, with a multivariate normal distribution parameterized by estimators of the conditional mean and covariance improves sample quality. Guided by this analysis, we design a novel Joint Mean-Covariance Estimator (JMCE) that simultaneously learns the conditional mean and sliding-window covariance. Building on JMCE, we introduce Conditionally Whitened Diffusion Models (CW-Diff) and extend them to Conditionally Whitened Flow Matching (CW-Flow). Experiments on five real-world datasets with six state-of-the-art generative models demonstrate that CW-Gen consistently enhances predictive performance, capturing non-stationary dynamics and inter-variable correlations more effectively than prior-free approaches. Empirical results further demonstrate that CW-Gen can effectively mitigate the effects of distribution shift.

## 1 Introduction

Time series analysis has a long history, with classical approaches such as ARIMA, state-space models, and vector autoregressions (VAR) (Box & Jenkins, 1976; Durbin & Koopman, 2012; Lütkepohl, 2007). Although these methods have been widely applied, they often struggle with high-dimensionality and complex data structures that arise in modern applications. More recently, neural architectures have demonstrated superior predictive accuracy, such as recurrent neural networks (RNN), Long Short-Term Memory (LSTM), and Transformers (Sherstinsky, 2020; Hochreiter & Schmidhuber, 1997; Vaswani et al., 2017). However, these neural models primarily focus on forecasting the conditional mean of future sequences given historical observations, while providing little to uncertainty quantification.
These limitations have motivated the development of probabilistic forecasting, which seeks to model not only point predictions but also the associated uncertainty.

Multivariate time series probabilistic forecasting has recently emerged as a key methodology for quantifying predictive uncertainty,
enabling informed decision-making in numerous real-world applications in diverse domains such as finance, healthcare, environmental science, and transportation (Lim & Zohren, 2021). Formally, the task involves learning the probability distribution $P_{\textbf{X}|\textbf{C}}$ of a future time series $\textbf{X}_{0}\in\mathbb{R}^{d\times T_{f}}$ of discrete time conditioned on its corresponding historical observations $\textbf{C}\in\mathbb{R}^{d\times T_{h}}$, where the integers $T_{f}$ and $T_{h}$ denote the lengths of future and historical time series, respectively, and $d$ represents the dimensionality of each time step. However, this task still remains highly challenging, primarily due to (i) non-stationary characteristics, manifested through long-term trends, seasonal effects, and heteroscedasticity (Li et al., 2024; Ye et al., 2025); (ii) complex inter-variable dependency structures (Yuan & Qiao, 2024); (iii) inherent data uncertainty, such as short-term fluctuations (Ye et al., 2025); and (iv) potential distribution shifts between training and testing data (Kim et al., 2022).

In response to these challenges, recent advances in generative learning, especially diffusion models, focus on accurately estimating the conditional distribution $P_{\textbf{X}|\textbf{C}}$. TimeGrad employs a RNN to encode historical observations and generates forecasts autoregressively, but suffers from cumulative errors and slow computation (Rasul et al., 2021). CSDI uses a 2D-Transformer for imputation and forecasting (Tashiro et al., 2021), while SSSD employs a Structured State Space Model to reduce computational cost and emphasize temporal dependence (Alcaraz & Strodthoff, 2023). Nevertheless, CSDI, SSSD, and TimeGrad all struggle with long-term forecasting (Shen & Kwok, 2023). Diffusion-TS leverages a transformer to decompose time series into trend, seasonal, and residual components for generation, whereas FlowTS accelerates generation using rectified flow (Yuan & Qiao, 2024; Hu et al., 2025).

Although the aforementioned generative models have achieved promising performance, they ignore informative priors. Such priors, derived from historical observations or auxiliary models, can substantially improve conditional generative modeling. To the best of our knowledge, CARD is the first model to incorporate prior information into conditional diffusion models (Han et al., 2022). It pretrains a regressor to estimate the conditional mean $\mathbb{E}\left[\textbf{X}_{0}|\textbf{C}\right]$ and integrates this regressor into the diffusion process, thereby enhancing conditional generation. In time series forecasting, regressing the conditional mean and incorporating it into diffusion models as a prior has become a common practice, as it alleviates the difficulty of modeling non-stationary distributions. TimeDiff adopts a linear regressor to capture short-term patterns and employs a future mixup strategy during training to mitigate boundary disharmony (Shen & Kwok, 2023). However, its linear design limits the ability to capture complex trends and fluctuations. TMDM addresses this limitation by integrating a nonlinear regressor into the variational inference framework, enabling joint training of the regressor and the diffusion model (Li et al., 2024). The regressor for $\mathbb{E}\left[\textbf{X}_{0}|\textbf{C}\right]$ (hereafter referred to as the mean regressor) can capture trends, seasonality, and fluctuations but is vulnerable to heteroscedasticity. Building on this line, NsDiff addresses this by introducing two pretrained models: a mean regressor and a variance regressor, the latter estimating the conditional variance of each variable within a sliding window (Ye et al., 2025). By incorporating both regressors into the diffusion process, NsDiff models heteroscedasticity more effectively. Despite these innovations, the method still suffers from certain limitations, particularly the overly complex reverse process and the neglect of correlations among variables. A detailed discussion of these limitations is provided in Appendix A. Beyond diffusion models, S2DBM employs a diffusion bridge variant and incorporates the mean regressor in the same manner as CARD (Yang et al., 2024), which limits its ability to handle heteroscedasticity. TsFlow uses Gaussian Processes (GPs) as both the mean and variance regressors (Kollovieh et al., 2025), but its design is restricted to univariate forecasting with short horizons and inherits the typical drawbacks of GPs, including kernel sensitivity and cubic computational cost.

Building on the preceding literature, it is well established that carefully designed priors can substantially enhance generative models.
Yet several key questions remain unresolved: How exactly do priors contribute to these improvements, and how accurate must the mean and variance regressors be to provide tangible benefits? How can such regressors be effectively trained, and are there theoretical guarantees supporting their impact? Most existing approaches incorporate mean and variance regressors into diffusion models by following the designs of CARD and DDPM (Han et al., 2022; Ho et al., 2020). This raises a further question: is this mechanism redundant or inefficient, and could it be simplified within more flexible diffusion frameworks?

Motivated by these questions, we introduce the Conditional Whitened Generative Models (CW-Gen). Our main contributions are:

- •
We develop a unified framework for conditional generation, CW-Gen, with two instantiations: the Conditional Whitened Diffusion Model (CW-Diff) and the Conditional Whitened Flow Matching (CW-Flow). Several prior methods (Han et al., 2022; Li et al., 2024; Ye et al., 2025) can be viewed as special cases of this framework. Furthermore, CW-Gen allows seamless integration
with diverse diffusion models.
- •
We provide theoretical analysis that establishes sufficient conditions under which CW-Gen improves sample quality, as stated in Theorem [1](https://arxiv.org/html/2509.20928v1#Thmtheorem1) and Theorem 2 in Appendix C.
- •
Motivated by Theorems [1](https://arxiv.org/html/2509.20928v1#Thmtheorem1) and 2, we propose a novel joint estimation procedure for the conditional mean and sliding-window covariance of time series. Empirically, it achieves high accuracy while effectively controlling covariance eigenvalues, ensuring stability and robustness in generative modeling.
- •
We integrate CW-Gen with six state-of-the-art generative models and evaluate them on five real-world datasets. Empirical results show consistent improvements in capturing non-stationarity, inter-variable dependencies, and overall sample quality, while also mitigating distribution shift.

## 2 Preliminaries

### 2.1 Denoising Diffusion Probabilistic Models (DDPM)

Most of the diffusion models discussed in Section [1](https://arxiv.org/html/2509.20928v1#S1) follow the DDPM framework (Ho et al., 2020), which we review below in a general conditional setting.
Let $(X_{0},C)$ be a random vector with the joint distribution $P_{X,C}$, where $X_{0}\in\mathbb{R}^{d_{x}}$ and $C\in\mathbb{R}^{d_{c}}$.
The (conditional) DDPM aims to learn the conditional distribution $P_{X|C}$ and generate samples that match this distribution through a forward and a reverse process. In the forward process, Gaussian noises are gradually added into $X_{0}$ by a stochastic differential equation (SDE):

$$ $dX_{\tau}=-\tfrac{1}{2}\beta_{\tau}X_{\tau}d\tau+\sqrt{\beta_{\tau}}dW_{\tau},\ \tau\in[0,1],\ X_{0}\sim P_{X|C},$ $$

where $\beta_{\tau}>0$ and $W_{\tau}$ is a Brownian motion in $\mathbb{R}^{d_{x}}$. We use $\tau$ for the time of diffusion throughout this paper, while $t$ is the index for time series. From the properties of Ornstein–Uhlenbeck-process (OU-process), we derive the marginal distribution of $X_{\tau}$:

$$ $X_{\tau}\overset{d}{=}\alpha_{\tau}X_{0}+\sigma_{\tau}\epsilon,\ \epsilon\sim N(0,I_{d_{x}}),$ $$

where $\alpha_{\tau}:=\exp\left\{-\int_{0}^{\tau}\beta_{s}ds/2\right\}$, $\sigma^{2}_{\tau}:=1-\alpha^{2}_{\tau}$, $\overset{d}{=}$ denotes equality in distribution, and $I_{d_{x}}$ is the $d_{x}$-dimensional identity matrix. By construction of $\beta_{\tau}$, the integral $\int_{0}^{1}\beta_{s}ds$ is sufficiently large, so the distribution of $X_{1}$ (the terminal distribution) is well-approximated by $N(0,I_{d_{x}})$. In the reverse process, a standard Gaussian noise $\overset{\leftarrow}{X}_{1}$ is gradually denoised by an SDE:

$$ $d\overset{\leftarrow}{X}_{\tau}=\big[-\tfrac{1}{2}\beta_{\tau}\overset{\leftarrow}{X}_{\tau}-\beta_{\tau}\nabla_{x}\log p_{\tau}(\overset{\leftarrow}{X}_{\tau}|C)\big]d\tau+\sqrt{\beta_{\tau}}d\overset{\leftarrow}{W}_{\tau},$ (1) $$

where $\tau$ starts from $\tau=1$ and ends at $\tau=\tau_{\text{min}}$, with $\tau_{\text{min}}$ being an early stopping time close to 0, and $\overset{\leftarrow}{W}_{\tau}$ is a Brownian motion. In ([1](https://arxiv.org/html/2509.20928v1#S2.E1)), $p_{\tau}(\cdot|C)$ and $\nabla_{x}\log p_{\tau}(\cdot|C)$ denote the conditional density and score function of $X_{\tau}$ given $C$, respectively. Since the conditional score function is intractable, Ho et al. (2020) and Song et al. (2021) proposed approximating it with a neural network $s_{\theta}$ parameterized by $\theta$, trained by minimizing:

$$ $\mathbb{E}_{(X_{0},C),\tau,\epsilon}\left\|s_{\theta}\left(\alpha_{\tau}X_{0}+\sigma_{\tau}\epsilon,C,\tau\right)+\epsilon/\sigma_{\tau}\right\|^{2},$ $$

where
$\tau\sim U(0,1]$ and $\epsilon\sim N(0,I_{d_{x}})$. Finally, substituting $\nabla_{x}\log p_{\tau}(\overset{\leftarrow}{X}_{\tau}|C)$ in ([1](https://arxiv.org/html/2509.20928v1#S2.E1)) with $s_{\theta}(\overset{\leftarrow}{X}_{\tau},C,\tau)$ yields the reverse process:

$$ $d\overset{\leftarrow}{X}_{\tau}=\big[-\tfrac{1}{2}\beta_{\tau}\overset{\leftarrow}{X}_{\tau}-\beta_{\tau}s_{\theta}(\overset{\leftarrow}{X}_{\tau},C,\tau)\big]d\tau+\sqrt{\beta_{\tau}}d\overset{\leftarrow}{W}_{\tau},\ \tau\in[\tau_{\text{min}},1].$ $$

### 2.2 Flow matching

Unlike diffusion models based on SDEs, Flow Matching (FM) employs an ordinary differential equation (ODE) to connect Gaussian noise $\epsilon\sim N(0,I_{d_{x}})$ with the data $X_{0}\sim P_{X|C}$ (Lipman et al., 2023):

$$ $dX_{\tau}=({\epsilon}-X_{0})d\tau,\ \tau\in[0,1].$ (2) $$

A neural network $v_{\psi}$, parameterized by $\psi$, learns the vector field of ([2](https://arxiv.org/html/2509.20928v1#S2.E2)) by minimizing:

$$ $\mathbb{E}_{(X_{0},C),\tau,\bm{\epsilon}}\|{\epsilon}-X_{0}-v_{\psi}(X_{0}+\tau({\epsilon}-X_{0}),C,\tau)\|^{2}.$ $$

Given the learned vector field, FM generates samples by solving the ODE:

$$ $d\overset{\leftarrow}{X}_{\tau}=-v_{\psi}(\overset{\leftarrow}{X}_{\tau},C,\tau)d\tau$ $$

from $\tau=1$ to $\tau=\tau_{\text{min}}$, where $\overset{\leftarrow}{X}_{1}$ is Gaussian noise. The final state $\overset{\leftarrow}{X}_{\tau_{\text{min}}}$ is the generated sample.

## 3 Theory and Joint Mean–Covariance Estimator (JMCE)

### 3.1 Theoretical Foundation

A key question addressed in this subsection is how modifying the terminal distribution $N(0,I_{d_{x}})$ can enhance generation quality.
The total variation distance between the generated distribution of a diffusion model and the true distribution grows as the convergence error of the forward process increases, where the latter involves
the Kullback–Leibler divergence (KLD) between $P_{X|C}$ and the terminal distribution $D_{\mathrm{KL}}\!\left(P_{X|C}\,\|\,N(0,I_{d_{x}})\right)$ as a factor in the error
(Oko et al., 2023; Chen et al., 2023; Fu et al., 2024).
Hence, a smaller value of this KLD
leads to samples that better match $P_{X|C}$. This insight motivates replacing the standard terminal distribution $N(0,I_{d_{x}})$ with $N(\mu_{X|C},\Sigma_{X|C})$, where $\mu_{X|C}$ and $\Sigma_{X|C}$ are the true conditional mean and covariance of $X$ given $C$. Since these quantities are unknown in practice, they must be estimated by $\widehat{\mu}_{X|C}$ and $\widehat{\Sigma}_{X|C}$. The advantage of this replacement can then be measured by the reduction in

$$ $D_{\mathrm{KL}}\!\left(P_{X|C}\,\|\,N(\widehat{\mu}_{X|C},\widehat{\Sigma}_{X|C})\right)\quad\text{relative to}\quad D_{\mathrm{KL}}\!\left(P_{X|C}\,\|\,N(0,I_{d_{x}})\right).$ $$

This raises the fundamental question of when replacing the terminal distribution $N(0,I_{d_{x}})$ with $N(\widehat{\mu}_{X|C},\widehat{\Sigma}_{X|C})$ improves generation quality, which the following theorem addresses.

###### Theorem 1

Let $P_{X|C}$ denote the true conditional distribution of $X\in\mathbb{R}^{d_{x}}$ given $C$, with conditional mean $\mu_{X|C}$ and positive-definite conditional covariance $\Sigma_{X|C}$. Define $Q_{0}:=N(0,I_{d_{x}})$ and $\widehat{Q}:=N(\widehat{\mu}_{X|C},\widehat{\Sigma}_{X|C})$,
where $\widehat{\mu}_{X|C}$ and $\widehat{\Sigma}_{X|C}$ are estimators of $\mu_{X|C}$ and $\Sigma_{X|C}$, respectively. Let $\widehat{\lambda}_{X|C,i}$ denote the $i$-th eigenvalues of $\widehat{\Sigma}_{X|C}$, for $i=1,2,\ldots,d_{x}$. A sufficient condition ensuring that $D_{\mathrm{KL}}(P_{X|C}\,\|\,\widehat{Q})\leq D_{\mathrm{KL}}(P_{X|C}\,\|\,Q_{0})$ is:

$$ $\begin{split}&\left(\min_{i\in\{1,\ldots,d_{x}\}}\widehat{\lambda}_{X|C,i}\right)^{-1}\left(\|\mu_{X|C}-\widehat{\mu}_{X|C}\|_{2}^{2}+\|\Sigma_{X|C}-\widehat{\Sigma}_{X|C}\|_{N}\right)\\ &\quad+\sqrt{d_{x}}\,\|\Sigma_{X|C}-\widehat{\Sigma}_{X|C}\|_{F}\leq\|\mu_{X|C}\|_{2}^{2}.\end{split}$ (3) $$

where $\left\|\Sigma_{X|C}-\widehat{\Sigma}_{X|C}\right\|_{N}=\sum_{i=1}^{d_{x}}\widetilde{s}_{i}$ and $\widetilde{s}_{i}$ is the $i$-th singular value of $\Sigma_{X|C}-\widehat{\Sigma}_{X|C}$.

Theorem [1](https://arxiv.org/html/2509.20928v1#Thmtheorem1) states that when ([3](https://arxiv.org/html/2509.20928v1#S3.E3)) holds, replacing $Q_{0}$ with $\widehat{Q}$ reduces the KLD between $P_{X|C}$ and the terminal distribution, thereby improving generation quality. Importantly, it provides a foundation for designing loss functions to estimate $\mu_{\textbf{X}|\textbf{C}}$ and $\Sigma_{\textbf{X}|\textbf{C}}$, as detailed in Equation ([4](https://arxiv.org/html/2509.20928v1#S3.E4)) below. We emphasize that the estimators of $\mu_{X|C}$ and $\Sigma_{X|C}$ are obtained by minimizing the sample counterpart of the left-hand side of ([3](https://arxiv.org/html/2509.20928v1#S3.E3)), as detailed in the next subsection.

In order for ([3](https://arxiv.org/html/2509.20928v1#S3.E3)) to hold, it is necessary to obtain accurate estimators of both $\mu_{X|C}$ and $\Sigma_{X|C}$. The estimation accuracy of $\Sigma_{X|C}$ is measured in terms of both the Frobenius norm and the nuclear norm, with the latter characterized by $\sum_{i=1}^{d_{x}}\widetilde{s}_{i}$. We employ a Cholesky decomposition and introduce a penalty term into the loss function ([4](https://arxiv.org/html/2509.20928v1#S3.E4))
to enforce that the smallest eigenvalue,
$\min_{i\in\{1,\ldots,d_{x}\}}\{\widehat{\lambda}_{X|C,i}\}$,
remains strictly positive and bounded away from zero, as detailed in the next subsection.
Furthermore, in non-stationary time series, $\mu_{X|C}$ often exhibits sharp variations and thus deviates from zero. Consequently, ([3](https://arxiv.org/html/2509.20928v1#S3.E3)) is more likely to hold when accurate estimators of both $\mu_{X|C}$ and $\Sigma_{X|C}$ are available.

We further identify the scenarios in which our proposed replacement outperforms TMDM and NsDiff (Li et al., 2024; Ye et al., 2025), as formally established in Theorem 2 in Appendix C.

###### Theorem 1

### 3.2 Joint Mean–Covariance Estimator (JMCE)

Theorem [1](https://arxiv.org/html/2509.20928v1#Thmtheorem1) establishes that accurate estimators of both the conditional mean and covariance can improve the quality of samples generated by diffusion models. Guided by the sufficient conditions ([3](https://arxiv.org/html/2509.20928v1#S3.E3)), we design a novel Joint Mean–Covariance Estimator (JMCE).

In terms of time series, directly estimating the true conditional covariance is extremely challenging, as it is often highly complex and non-smooth, which makes consistent estimation difficult. Instead, the sliding-window covariance is preferable, as it not only offers more accurate approximations but also improves computational efficiency (Iwakura et al., 2008; Chen et al., 2024). Motivated by this, we estimate the sliding-window conditional covariance, rather than the true conditional covariance. Let $\widetilde{\Sigma}_{\textbf{X}_{0},t}\in\mathbb{R}^{d\times d}$ denote the sliding-window covariance at time $t$, and let $\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}}\in\mathbb{R}^{d\times d}$ be an estimator of $\widetilde{\Sigma}_{\textbf{X}_{0},t}$ for $t=1,\ldots,T_{f}$. We design a non-autoregressive model to simultaneously output:

$$ $\widehat{\mu}_{\textbf{X}|\textbf{C}},\widehat{L}_{1|\textbf{C}},\ldots,\widehat{L}_{T_{f}|\textbf{C}}=\texttt{JMCE}(\textbf{C})$ $$

with $\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}}:=\widehat{L}_{t|\textbf{C}}\widehat{L}_{t|\textbf{C}}^{\top}$, for $t=1,\ldots,T_{f}$. This design, inspired by Cholesky decomposition, guarantees that all $\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}}$ are positive semi-definite (PSD). The detailed algorithm of JMCE(C) can be found in Appendix B. In our implementation, we use a Non-stationary Transformer (Liu et al., 2022) as the backbone of JMCE. Based on ([3](https://arxiv.org/html/2509.20928v1#S3.E3)) in Theorem [1](https://arxiv.org/html/2509.20928v1#Thmtheorem1), we construct the trainning loss in JMCE by combining three components: $\mathcal{L}_{2}:=\mathbb{E}_{(\textbf{X}_{0},\textbf{C})}\left\|\textbf{X}_{0}-\widehat{\mu}_{\textbf{X}|\textbf{C}}\right\|_{2}^{2},\mathcal{L}_{F}:=\mathbb{E}_{(\textbf{X}_{0},\textbf{C})}\sum_{t=1}^{T_{f}}\left\|\widetilde{\Sigma}_{\textbf{X}_{0},t}-\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}}\right\|_{F},$ and $\mathcal{L}_{\text{SVD}}:=\mathbb{E}_{(\textbf{X}_{0},\textbf{C})}\sum_{t=1}^{T_{f}}\left\|\widetilde{\Sigma}_{\textbf{X}_{0},t}-\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}}\right\|_{N}$. The smallest eigenvalues of $\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}}$ have a crucial impact on the magnitude of the left-hand side of inequality ([3](https://arxiv.org/html/2509.20928v1#S3.E3)). We thus introduce a regularization term that enforces the smallest eigenvalues of $\widehat{\Sigma}_{\mathbf{X}_{0},t|\mathbf{C}}$ to remain strictly positive and bounded away from zero, thereby avoiding numerical instability and rank deficiency.
Let $\lambda_{\min}$ be a tunable hyperparameter. The penalty term is defined as:

$$ $\mathcal{R}_{\lambda_{\text{min}}}\big(\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}}\big):=\sum_{i=1}^{d}\text{ReLU}\big(\lambda_{\text{min}}-\widehat{\lambda}_{\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}},i}\big),$ $$

where $\widehat{\lambda}_{\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}},i}(i=1,\ldots,d)$ denote the eigenvalues of $\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}}$, and $\text{ReLU}(x)=\max\{x,0\}$. It is indicated that any eigenvalue smaller than $\lambda_{\min}$ will be penalized. The overall training loss in JMCE for the conditional mean and covariance is defined as:

$$ $\mathcal{L}_{\text{JMCE}}=\mathcal{L}_{2}+\mathcal{L}_{\text{SVD}}+\lambda_{\text{min}}\sqrt{d\cdot T_{f}}\mathcal{L}_{F}+w_{\text{Eigen}}\cdot\sum_{t=1}^{T_{f}}\mathcal{R}_{\lambda_{\text{min}}}\left(\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}}\right),$ (4) $$

where $w_{\text{Eigen}}$ is a hyperparameter that controls the strength of the penalty. Empirically, we choose $w_{\text{Eigen}}\sim\mathcal{O}(\lambda_{\min}^{-1})$. It is important to note that ([4](https://arxiv.org/html/2509.20928v1#S3.E4)) is specifically designed to ensure that ([3](https://arxiv.org/html/2509.20928v1#S3.E3)) holds.

The algorithm of the joint estimator can be found in Appendix B. JMCE excels at estimating the conditional mean and covariance while controlling the minimal eigenvalue. We conduct a substantial ablation study to show the advantages, and discuss them in Appendix D.

## 4 Conditional Whitened Generative Models (CW-Gen)

In this section, we propose Conditionally whitened diffusion models (CW-Diff) and Conditionally whitened flow matching (CW-Flow). Together, we call them Conditionally Whitened Generative Models (CW-Gen).

### 4.1 Conditionally whitened diffusion models (CW-Diff)

Our JMCE outputs $\widehat{\mu}_{\textbf{X}|\textbf{C}}\in\mathbb{R}^{d\times T_{f}}$ and
$\widehat{\Sigma}_{\textbf{X}_{0}|\textbf{C}}:=[\widehat{\Sigma}_{\textbf{X}_{0},1|\textbf{C}},\ldots,\widehat{\Sigma}_{\textbf{X}_{0},T_{f}|\textbf{C}}]\in\mathbb{R}^{d\times d\times T_{f}}$. Since all $\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}}$ are positive definite, we can compute
$\widehat{\Sigma}^{k}_{\textbf{X}_{0}|\textbf{C}}:=[\widehat{\Sigma}^{k}_{\textbf{X}_{0},1|\textbf{C}},\ldots,\widehat{\Sigma}^{k}_{\textbf{X}_{0},T_{f}|\textbf{C}}]\in\mathbb{R}^{d\times d\times T_{f}}$
for $k\in\{-0.5,0.5\}$ via eigen-decomposition. Let $\bm{\epsilon}:=[\epsilon_{1},\ldots,\epsilon_{T_{f}}]\in\mathbb{R}^{d\times T_{f}}$, where each column
$\epsilon_{t}\sim N(0,I_{d})$ and the columns $\epsilon_{1},\ldots,\epsilon_{T_{f}}$ are mutually independent.
We define the tensor operation

$$ $\widehat{\Sigma}^{0.5}_{\textbf{X}_{0}|\textbf{C}}\circ\bm{\epsilon}:=[\widehat{\Sigma}^{0.5}_{\textbf{X}_{0},1|\textbf{C}}\cdot\epsilon_{1},\ldots,\widehat{\Sigma}^{0.5}_{\textbf{X}_{0},T_{f}|\textbf{C}}\cdot\epsilon_{T_{f}}]\in\mathbb{R}^{d\times T_{f}}.$ (5) $$

Accordingly, we say that a tensor follows $\mathcal{N}(\widehat{\mu}_{\textbf{X}|\textbf{C}},\widehat{\Sigma}_{\textbf{X}_{0}|\textbf{C}})$ if it has the same distribution as
$\widehat{\Sigma}^{0.5}_{\textbf{X}_{0}|\textbf{C}}\circ\bm{\epsilon}+\widehat{\mu}_{\textbf{X}|\textbf{C}}$. With this formulation, we define the forward process:

$$ $d\big(\textbf{X}_{\tau}-\widehat{\mu}_{\textbf{X}|\textbf{C}}\big)=-\tfrac{1}{2}\beta_{\tau}\big(\textbf{X}_{\tau}-\widehat{\mu}_{\textbf{X}|\textbf{C}}\big)d\tau+\sqrt{\beta_{\tau}}\cdot\widehat{\Sigma}^{0.5}_{\textbf{X}_{0}|\textbf{C}}\circ d\textbf{W}_{\tau},\ \tau\in[0,1],\ \textbf{X}_{0}\sim P_{\textbf{X}|\textbf{C}},$ (6) $$

where $\textbf{W}_{\tau}$ is a Brownian motion in $\mathbb{R}^{d\times T_{f}}$. By the property of the OU-process, the terminal distribution of $\textbf{X}_{1}$ is close to
$\mathcal{N}(\widehat{\mu}_{\textbf{X}|\textbf{C}},\widehat{\Sigma}_{\textbf{X}_{0}|\textbf{C}})$.
A formal proof of the terminal distribution of ([6](https://arxiv.org/html/2509.20928v1#S4.E6)) is provided in Appendix C. Furthermore, the following SDE is equivalent to ([6](https://arxiv.org/html/2509.20928v1#S4.E6)):

$$ $d\ \widehat{\Sigma}^{-0.5}_{\textbf{X}_{0}|\textbf{C}}\circ\big(\textbf{X}_{\tau}-\widehat{\mu}_{\textbf{X}|\textbf{C}}\big)=-\tfrac{1}{2}\beta_{\tau}\cdot\widehat{\Sigma}^{-0.5}_{\textbf{X}_{0}|\textbf{C}}\circ\big(\textbf{X}_{\tau}-\widehat{\mu}_{\textbf{X}|\textbf{C}}\big)d\tau+\sqrt{\beta_{\tau}}d\textbf{W}_{\tau},\ \tau\in[0,1],$ $$

which implies that the diffusion processes can be directly performed on $\textbf{X}^{\text{CW}}_{0}:=\widehat{\Sigma}_{\textbf{X}_{0}|\textbf{C}}^{-0.5}\circ\big(\textbf{X}_{0}-\widehat{\mu}_{\textbf{X}|\textbf{C}}\big)$. We call this operation conditional whitening (CW). Subtracting $\widehat{\mu}_{\textbf{X}|\textbf{C}}$ removes the non-stationary trends and seasonal effects in $\textbf{X}_{0}$, while being operated by $\widehat{\Sigma}_{\textbf{X}_{0}|\textbf{C}}^{-0.5}$ addresses heteroscedasticity and mitigates linear correlations among features. The CW operation thus renders the data as stationary as possible and enables diffusion models to more effectively capture temporal and higher-order dependencies. Moreover, since it is a full-rank linear transformation, CW is entirely invertible. Building on these properties, we now formally write the forward process of the Conditional Whitened Diffusion Model (CW-Diff) as follows:

$$ $d\textbf{X}^{\text{CW}}_{\tau}=-\tfrac{1}{2}\beta_{\tau}\textbf{X}^{\text{CW}}_{\tau}d\tau+\sqrt{\beta_{\tau}}d\textbf{W}_{\tau},\ \tau\in[0,1],$ (7) $$

with the initial state $\textbf{X}^{\text{CW}}_{0}$ satisfying $\big(\widehat{\Sigma}_{\textbf{X}_{0}|\textbf{C}}^{0.5}\circ\textbf{X}^{\text{CW}}_{0}+\widehat{\mu}_{\textbf{X}|\textbf{C}}\big)\sim P_{\textbf{X}|\textbf{C}}$. Correspondingly, we use a neural network $s_{\theta}^{\text{CW}}$ to learn the score function of $\textbf{X}^{\text{CW}}_{\tau}$ given C by minimizing the following loss function:

$$ $\mathbb{E}_{(\textbf{X}^{\text{CW}}_{0},\textbf{C}),\tau,\bm{\epsilon}}\|s_{\theta}^{\text{CW}}\left(\alpha_{\tau}\textbf{X}^{\text{CW}}_{0}+\sigma_{\tau}\bm{\epsilon},\textbf{C},\tau\right)+\bm{\epsilon}/\sigma_{\tau}\|^{2}.$ $$

Let $\overset{\leftarrow}{\textbf{X}}{}^{\text{CW}}_{1}\sim\mathcal{N}(0,I_{d\times d\times T_{f}})$, where $I_{d\times d\times T_{f}}:=[I_{d},\ldots,I_{d}]\in\mathbb{R}^{d\times d\times T_{f}}$. Then, the reverse process of CW-Diff is given by:

$$ $d\overset{\leftarrow}{\textbf{X}}{}^{\text{CW}}_{\tau}=\left[-\tfrac{1}{2}\beta_{\tau}\overset{\leftarrow}{\textbf{X}}{}^{\text{CW}}_{\tau}-\beta_{\tau}s_{\theta}^{\text{CW}}(\overset{\leftarrow}{\textbf{X}}{}^{\text{CW}}_{\tau},\textbf{C},\tau)\right]d\tau+\sqrt{\beta_{\tau}}d\overset{\leftarrow}{\textbf{W}}_{\tau},$ $$

where $\tau$ decreases from $1$ to $\tau_{\min}$, with $\tau_{\min}$ being an early stopping time close to 0.
Finally, we obtain

$$ $\overset{\leftarrow}{\textbf{X}}_{\tau_{\text{min}}}=\widehat{\Sigma}_{\textbf{X}_{0}|\textbf{C}}^{0.5}\circ\overset{\leftarrow}{\textbf{X}}{}^{\text{CW}}_{\tau_{\text{min}}}+\widehat{\mu}_{\textbf{X}|\textbf{C}}$ $$

by inverting the original CW operation. $\overset{\leftarrow}{\textbf{X}}_{\tau_{\text{min}}}$ is the final sample generated by CW-Diff approximating $P_{\textbf{X}|\textbf{C}}$.

The forward process in Equation ([7](https://arxiv.org/html/2509.20928v1#S4.E7)) is consistent with that of DDPM. Furthermore, CW-Diff is readily extendable to TMDM, NsDiff, and other diffusion models. This extension is accomplished by replacing the initial variable $\textbf{X}_{0}$ with its CW-transformed form $\textbf{X}^{\text{CW}}_{0}$. Within this framework, the task of learning the mean and sliding-window covariance in $\textbf{X}^{\text{CW}}_{0}$ may be interpreted as a form of residual learning, analogous to the mechanisms used in GBDT and XGBoost (Chen & Guestrin, 2016).

Figure: Figure 1: The flow chat of JMCE, CW-Diff and CW-Flow.
Refer to caption: x1.png

### 4.2 Conditionally whitened flow matching (CW-Flow)

In CW-Diff, the inverse matrices of $\widehat{\Sigma}_{\textbf{X}_{0},t|\textbf{C}}$ are computed via eigen-decomposition, which requires a computational complexity of $\mathcal{O}(d^{3}T_{f})$. To reduce this cost and improve efficiency, we transition to the FM framework introduced in Section [2.2](https://arxiv.org/html/2509.20928v1#S2.SS2), where the estimated mean and covariance can be incorporated in a more efficient way.

The Conditional Whitened Flow Matching (CW-Flow) model employs an ODE to connect $\textbf{X}_{0}\sim P_{\textbf{X}|\textbf{C}}$ with a noise $\bm{\epsilon}^{\text{CW}}\sim\mathcal{N}(\widehat{\mu}_{\textbf{X}|\textbf{C}},\widehat{\Sigma}_{\textbf{X}_{0}|\textbf{C}})$:

$$ $d\textbf{X}^{\text{CW}}_{\tau}=\big(\bm{\epsilon}^{\text{CW}}-\textbf{X}_{0}\big)d\tau,\ \tau\in[0,1].$ $$

Accordingly, the CW-Flow network $v_{\psi}^{\text{CW}}$ is trained by minimizing:

$$ $\mathbb{E}_{(\textbf{X}_{0},\textbf{C}),\tau,\bm{\epsilon}^{\text{CW}}}\left\|\bm{\epsilon}^{\text{CW}}-\textbf{X}_{0}-v_{\psi}^{\text{CW}}(\textbf{X}_{0}+\tau(\bm{\epsilon}^{\text{CW}}-\textbf{X}_{0}),\textbf{C},\tau)\right\|^{2}.$ $$

CW-Flow then generates samples by solving the following ODE:

$$ $d\overset{\leftarrow}{\textbf{X}}{}^{\text{CW}}_{\tau}=-v_{\psi}^{\text{CW}}(\overset{\leftarrow}{\textbf{X}}{}^{\text{CW}}_{\tau},\textbf{C},\tau)d\tau,\ \overset{\leftarrow}{\textbf{X}}{}^{\text{CW}}_{1}\sim\mathcal{N}(\widehat{\mu}_{\textbf{X}|\textbf{C}},\widehat{\Sigma}_{\textbf{X}_{0}|\textbf{C}}),$ $$

where $\tau$ starts from $\tau=1$ and ends at $\tau=\tau_{\min}$. $\overset{\leftarrow}{\textbf{X}}{}^{\text{CW}}_{\tau_{\min}}$ is the final sample generated by CW-Flow approximating $P_{\textbf{X}|\textbf{C}}$. Compared with CW-Diff, CW-Flow does not require computing inverse matrices or reversing the CW operation of the final sample $\overset{\leftarrow}{\textbf{X}}{}^{\text{CW}}_{\tau_{\min}}$. The algorithms of CW-Diff and CW-Flow are provided in Appendix B. The flow chart of CW-Diff and CW-Flow can be found in Figure [1](https://arxiv.org/html/2509.20928v1#S4.F1).

## 5 Experiments

**Table 1: Dataset descriptions, including dimensions $d$, frequencies, total length of time series, length of historical observations $T_{h}$, length of future time series $T_{f}$, and win rates of our CW methods. Win rate refers to the proportion that our CW method outperforms original method.**
| Dataset | Dimension | Frequency | Total length | $T_{h}$ | $T_{f}$ | Win rate of CW-Gen |
| --- | --- | --- | --- | --- | --- | --- |
| ETTh1 | 7 | 1 Hour | 14,400 | 168 | 192 | $22/24\approx 91.67\%$ |
| ETTh2 | 7 | 1 Hour | 14,400 | 168 | 192 | $22/24\approx 91.67\%$ |
| ILI | 7 | 1 Week | 966 | 52 | 36 | $21/24\approx 87.50\%$ |
| Weather | 21 | 10 Minutes | 52,696 | 168 | 192 | $22/24\approx 91.67\%$ |
| Solar Energy | 137 | 10 Minutes | 52,560 | 168 | 192 | $19/24\approx 79.17\%$ |

**Table 2: Metrics for models trained on original ETTh1 (Raw) and conditionally whitened ETTh1 (CW). Each experiment is repeated by 10 times, and standard deviations are provided in brackets. The better results between Raw and CW are underlined. The win rates of every metric of Raw and CW-Gen models are also provided.**
| Model | CRPS ($\downarrow$) | QICE ($\downarrow$) | ProbCorr ($\downarrow$) | Conditional FID ($\downarrow$) |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| (ETTh1) | Raw | CW | Raw | CW | Raw | CW | Raw | CW |
| TimeDiff | 0.736 | 0.530 | 9.513 | 12.610 | 0.299 | 0.238 | 14.081 | 3.998 |
| (2023) | (0.031) | (0.019) | (0.589) | (0.904) | (0.017) | (0.020) | (6.778) | (0.386) |
| SSSD | 0.836 | 0.541 | 11.624 | 4.710 | 0.326 | 0.254 | 40.887 | 35.645 |
| (2023) | (0.153) | (0.064) | (1.312) | (1.555) | (0.032) | (0.030) | (17.601) | (19.248) |
| Diffusion | 0.641 | 0.453 | 6.742 | 2.856 | 0.337 | 0.261 | 21.098 | 18.432 |
| -TS (2024) | (0.027) | (0.023) | (1.610) | (1.281) | (0.029) | (0.028) | (6.391) | (7.516) |
| TMDM | 0.472 | 0.469 | 3.360 | 3.205 | 0.230 | 0.205 | 9.931 | 9.846 |
| (2024) | (0.031) | (0.027) | (1.055) | (0.731) | (0.014) | (0.009) | (4.439) | (4.037) |
| NSDiff | 0.407 | 0.416 | 1.792 | 1.534 | 0.214 | 0.213 | 35.261 | 20.278 |
| (2025) | (0.032) | (0.015) | (0.682) | (0.314) | (0.014) | (0.008) | (7.785) | (5.912) |
| FlowTS | 0.578 | 0.467 | 6.300 | 3.456 | 0.284 | 0.227 | 19.442 | 14.553 |
| (2025) | (0.065) | (0.014) | (1.329) | (0.607) | (0.024) | (0.011) | (13.874) | (9.103) |
| Win rate | 16.7$\%$ | 83.3$\%$ | 16.7$\%$ | 83.3$\%$ | 0.0$\%$ | 100.0$\%$ | 0.0$\%$ | 100.0$\%$ |

Datasets: We evaluate CW-Gen on five representative time series datasets—ETTh1, ETTh2, ILI, Weather, and Solar Energy—spanning various domains and temporal resolutions. Further details of the datasets can be found in Appendix D. For the ETT datasets, the training/validation/test split follows a 3:1:1 ratio, while for the other datasets we adopt a 7:1:2 ratio. Table [1](https://arxiv.org/html/2509.20928v1#S5.T1) presents the dataset properties and the win rate of CW-Gen, computed as the proportion of cases where CW-Gen outperforms competing methods, based on the results in Tables [2](https://arxiv.org/html/2509.20928v1#S5.T2)-6.

Baselines: We evaluate five diffusion models and one flow matching model for time series forecasting (denoted as Raw), and further integrate all six generative models with our CW-Diff and CW-Flow approaches (denoted as CW). Specifically, the baselines include TimeDiff (Shen & Kwok, 2023), SSSD (Alcaraz & Strodthoff, 2023), Diffusion-TS (Yuan & Qiao, 2024), TMDM (Li et al., 2024), NsDiff (Li et al., 2024), and FlowTS (Hu et al., 2025). Among them, TimeDiff, TMDM, and NsDiff are prior-informed methods.

Metrics: We evaluate the predictive performance with six metrics: Continuous Ranked Probability Score (CRPS) (Matheson & Winkler, 1976), Quantile Interval Coverage Error (QICE) (Han et al., 2022), Probabilistic Correlation score (ProbCorr), Conditional Context Fréchet Inception Distance (Conditional FID) (Yue et al., 2022), Probabilistic mean square error (ProbMSE), and Probabilistic mean average error (ProbMAE). Formal definitions can be found in Appendix D. We also provide the results for ProbMSE and ProbMAE in Tables 7 and 8 in Appendix D.

Settings: During evaluation, $\textbf{X}_{0}$ and C refers to non-overlapping subsequences drawn from the test set, where C denotes the historical observations and $\textbf{X}_{0}$ the corresponding future series. We adopt the widely used long-term forecasting setting with a historical length of 168 and a future horizon of 192 (Shen & Kwok, 2023; Ye et al., 2025). The sliding-window covariance is computed with a window size of 95, except for ILI, where it is set to 15. In the JMCE loss ([4](https://arxiv.org/html/2509.20928v1#S3.E4)), $\lambda_{\min}$ is fixed at 0.1, and the penalty weight $w_{\text{Eigen}}$ is set to 50. All diffusion models follow their default diffusion schedules, and the number of sampling steps is set to 50 (20 for NsDiff). We train JMCE and CW-Gen on the training set, select the model checkpoint with the lowest loss on validation set, and then perform evaluation on the test set. Each model generates 100 samples for evaluation. On each dataset, we train every model 10 times with different random seeds and report the mean and one standard deviation of the four metrics. We also conduct extensive ablation studies on JMCE, which can be found in Appendix D. The other parameters are provided in Appendix E.

Results: As shown in Tables [2](https://arxiv.org/html/2509.20928v1#S5.T2)-6, CW-Gen reduces CRPS and QICE in a substantial number of cases, indicating improvements in predictive accuracy. Moreover, it consistently lowers ProbCorr and Conditional FID, with only minor exceptions, showing that CW-Gen enables models to better capture feature correlations in time series and to enhance overall sample quality. Moreover, as shown in Tables 7 and 8, our CW-Gen method improves the ProbMSE metric in 80.00$\%$ and the ProbMAE metric in 73.33$\%$ of the evaluated model–dataset combinations. This demonstrates that, in addition to enhancing probabilistic forecasting ability, CW-Gen also strengthens the point forecasting performance of the models.

Illustrations: In Figure [2](https://arxiv.org/html/2509.20928v1#S5.F2), we illustrate representative results of representative generative models combined with CW-Gen. Among them, Diffusion-TS serves as a typical diffusion model, NsDiff is a diffusion based model augmented by priors, and FlowTS is based on flow matching. Comparing NsDiff and CW-Gen with the other models, we observe that generative models without priors tend to generate sample with shifted means, which we attribute to distribution shifts between the training and test sets. This observation highlights the benefit of incorporating priors in probabilistic time series forecasting, as they can effectively mitigate such distribution shifts. In contrast, CW-Diffusion-TS and CW-FlowTS, which leverage JMCE as priors, exhibit no noticeable mean shift compared to Diffusion-TS and FlowTS. Moreover, the samples generated by CW-Diffusion-TS and CW-FlowTS achieve finer resolution and better capture the peaks in Dimension 1 than their non-CW counterparts. Compared with NsDiff, CW-NsDiff produces more accurate sample means and smaller standard deviations in Dimension 1, which contributes to more reliable uncertainty quantification. More illustrations can be found in Figure 3 in Appendix D.

Figure: Figure 2: Comparison of Diffusion-TS, NsDiff, FlowTS, and their CW variants on ETTh1 across Dimensions 1 and 2. True ETTh1 means the real time series from ETTh1 dataset. Sample mean and standrad deviation refer to the mean and standrad deviation of 100 samples generated by generative models. One sample refers to a randomly chosen instance among the 100 generated samples.
Refer to caption: x2.png

## 6 Conclusion

In this work, we establish for the first time a sufficient condition that reduces the KL divergence between a conditional distribution and the terminal distribution of a diffusion model. By tightening this KL divergence, we obtain a sharper bound on the total variation distance between the generated distribution of the diffusion model and the true distribution. Building on this result, we design the Joint Mean–Covariance Estimator (JMCE), which jointly estimates the conditional mean and the conditional sliding-window covariance while controlling the behavior of the minimal eigenvalue. We then use JMCE as a data-driven prior to conditionally whiten the original data, and train diffusion models on the whitened space, yielding the Conditionally Whitened Diffusion Model (CW-Diff). Similarly, by modifying the terminal distribution of flow matching, we introduce the Conditionally Whitened Flow Model (CW-Flow). Together, we refer to these as CW-Gen. We evaluate CW-Gen on five real-world time series datasets using six generative models and four evaluation metrics. Experimental results demonstrate that CW-Gen consistently improves model performance in most cases.

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## Peer Reviews

confidence: 2
contribution: 3
initial_rating: 6
presentation: 3
questions: See above
soundness: 3
strengths: 1. The paper establishes conditions that justify why replacing the traditional terminal Gaussian distribution with one parameterized by estimated conditional mean and covariance improves sample quality.
2. The JMCE simultaneously learns accurate conditional means and sliding-window covariances with eigenvalue control to ensure stability and robustness—this nuanced approach effectively addresses non-stationarity and heteroscedasticity.
3. Experiments show the outperformed performance over baselines.
4. The algorithms and theoretical proofs are clearly detailed, and code for experiment reproduction is available.
summary: The paper introduces CW-Gen, including CW-Diff and CW-Flow. It proposes a novel JMCE to simultaneously estimate the conditional mean and sliding-window covariance of future time series, which guides the whitening process. The authors provide theoretical guarantees showing conditions under which their approach improves the generative model's sample quality by reducing the KL divergence between the conditional distribution and the model's terminal distribution. Experimental evaluations demonstrate improvements in probabilistic forecasting accuracy, capturing non-stationary dynamics and inter-variable correlations better than baselines.
weaknesses: 1. The approach involves computationally expensive operations, particularly eigen-decomposition for whitening covariance matrices, which scales cubically with dimensionality. For very high-dimensional datasets, CW-Gen can become quite slow, limiting real-time deployment scenarios.
2. The framework's reliance on complex joint estimators and whitening transformations may impose a higher barrier to adoption compared to simpler baseline models.
3. The paper ablates hyperparameters but does not isolate the individual contributions of conditional mean estimation versus covariance estimation. Which component drives most improvements?

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 3
questions: - Can you be very explicit on how your proposed method compare to other (univariate) flow-based and diffusion-based methods that introduced flexible priors, e.g., [1,2]
- Can you add a short experiment showing numeric values of both sides of Equation (3) on training/validation examples so readers can see how far the estimators are from satisfying the bound in practice.
- Choice of sliding window length (95 for most datasets, 15 for ILI) needs justification. Why 95? How sensitive are results to this hyperparameter?


1. Modeling temporal data as continuous functions with stochastic process diffusion, ICLR 2023
2. Flow Matching with Gaussian Process Priors for Probabilistic Time Series Forecasting, ICLR 2025
soundness: 3
strengths: - Tackle a challenging, important and underrepresented aspect for time-series modelling: multivariate time-series.
- Clear motivation: Proposes a method that allows for a new prior, that is closer to the data-distribution.
- Well written and theoretically founded.
- Extensive evaluation, which show that the proposed prior actually improves state-of-the art models.
summary: The paper proposes CW-Gen, which whitens data using learned conditional means and covariances from a Joint Mean-Covariance Estimator before training generative models. The idea to align the model priors with data statistics is sound and yields consistent empirical gains across datasets.
weaknesses: - Unclear connection of DKL reduction in Theorem 1 to actual practical guarantees: Can you (a) clarify the tightness of the bound; (b) give intuition / toy examples showing when the condition is achievable (or not); (c) explicitly discuss regimes where the condition can fail
- Unclear estimator quality: Covariance targets are noisy; stability and regularization not analyzed.
- The paper notes that CW requires eigen-decomposition per sample / per time step but then glosses over practical limitations. More precise complexity analysis and discussion of remedies (approximate eigen/svd, diagonal + low-rank approximation, block-diagonalization, randomized SVD or factor models) could benefit the paper.
- The paper argues joint mean+full covariance is important. But it’s not fully convincing which component drives gains. There are some ablations (backbone/wEigen) but I don’t see a simple controlled ablation that compares: (a) subtract mean only, (b) subtract mean and scale by diagonal variance (NsDiff style), (c) full covariance JMCE. Can you add a direct ablation and show where full covariance helps.

---

confidence: 4
contribution: 2
initial_rating: 6
presentation: 3
questions: - Can the model be trained in an end-to-end fashion or does it require a two-stage process?
- Did you try diagonal covariance parametrizations to reduce the runtime complexity?
- Can you elaborate more on the performance of the JMCE model itself? How does it compare in forecasting?
soundness: 3
strengths: - The methodology is theoretically justified, and the authors show under which conditions the distance between the prior and conditional distribution is minimized.
- A simple mean and covariance estimator is introduced and well-motivated to parametrize the generative process.
- The proposed conditional whitening leads to empirical performance improvements.
- The framework includes diffusion and flow matching.
summary: The paper introduces conditionally whitened generative models that incorporate information in the noising process. The authors tailor diffusion and flow matching processes by including conditional information obtained via a mean and covariance estimator (termed JMCE). Their approach is theoretically motivated and empirically demonstrated to improve generative performance.
weaknesses: - The model requires a two-stage process now. First fit JMCE, then train the generative model.
- The derived diffusion process requires the inversion of the covariance matrix, resulting in a higher runtime complexity compared to standard diffusion models. A runtime comparison would aid the comparison.
- The limitations of the method should be discussed more thoroughly. Furthermore, I recommend separating the related work section from the introduction.

Minors:

- L260: Sentence incomplete

---

confidence: 3
contribution: 4
initial_rating: 6
presentation: 3
questions: 1. Do you train the JMCE model jointly with the generative model?
1. Line 313: How can this be done more efficiently exactly?
1. Are you modifying the generative models themselves? Or do you train the models on the conditionally whitened data as they are?
1. Have you ablated the different components of your JMCE loss?
soundness: 3
strengths: 1. Consistent improvements in experimental results
1. Clear graphical illustration of method
summary: The authors propose to use a separate model to predict sliding-window means and covariances and subsequently use those to whiten to prediction targets of generative forecasting models.
Their method improves forecasting results for a broad selection of generative forecasting models on several datasets.
weaknesses: 1. As far as I understand from the algorithms in the appendix, conditional whitening with a JMCE is a wrapper or pre/postprocessing around a generative forecasting model. While this can be interwoven with the diffusion/flow matching dynamics in Section 4, this does not seem essential. If I am correct, I think it would be an advantage to highlight the simplicity of the final method.


Decision: Accept (Poster)