Noise is All You Need: Solving Linear Inverse Problems by Noise Combination Sampling with Diffusion Models

## Abstract

Abstract Pretrained diffusion models have demonstrated strong capabilities in zero-shot inverse problem solving by incorporating observation information into the generation process of the diffusion models. However, this presents an inherent dilemma: excessive integration can disrupt the generative process, while insufficient integration fails to emphasize the constraints imposed by the inverse problem. To address this, we propose Noise Combination Sampling , a novel method that synthesizes an optimal noise vector from a noise subspace to approximate the measurement score, replacing the noise term in the standard Denoising Diffusion Probabilistic Models process. This enables conditional information to be naturally embedded into the generation process without reliance on step-wise hyperparameter tuning. Our method can be applied to a wide range of inverse problem solvers, including image compression, and, particularly when the number of generation steps T T is small, achieves superior performance with negligible computational overhead, significantly improving robustness and stability.

## 1 Introduction

Diffusion models have emerged as a powerful class of generative models, achieving state-of-the-art results in high-fidelity image generation, audio synthesis, video modeling, and even language modeling (Ho et al., 2020; Rombach et al., 2022; Podell et al., 2024; Ho et al., 2022; Nie et al., 2025; Gat et al., 2024). Beyond their impressive generative capabilities, researchers have also recognized their strong zero-shot potential for a variety of related tasks, including inpainting, depth estimation, segmentation, and classification (Lugmayr et al., 2022; Tian et al., 2024; Li et al., 2023).

Notably, it has been shown that diffusion models, without any additional training, can effectively solve general linear inverse problems by injecting information during the stochastic denoising steps (Wang et al., 2023; Chung et al., 2023; Cardoso et al.,; Dou & Song, 2024; He et al., 2024; Kim et al., 2025). These tasks include denoising, inpainting, and super-resolution. Such powerful plug-and-play, training-free approaches have been adopted in more complex settings, including nonlinear and deep learning-guided generation tasks such as style transfer and image editing (Yu et al., 2023; He et al., 2024; Shi et al., 2024; Ye et al., 2024). However, since the imposed guidance drives the trajectory away from the manifold of real data and disrupts the consistency of the generation process, existing methods suffer from sampling instability and rely on complex hyperparameter tuning and long sampling schedules to compensate for the limited influence of the guidance (Wang et al., 2023; Song et al., 2023; Yang et al., 2024; Zhang et al., 2025).

In this paper, we propose *Noise Combination Sampling* (NCS), a novel framework that approximates the measurement score $\nabla_{\bm{x}_{t}}\log p(\bm{y}\mid\bm{x}_{t})$ using a linear combination of Gaussian noise vectors to replace the noise term in the Denoising Diffusion Probabilistic Models (DDPM) process. By adjusting the sampling trajectory through constructed noise vectors rather than gradients, NCS mitigates the instability inherent in existing sampling-based inverse problem solvers, eliminating the need for carefully-tuned hyperparameters and preserving the denoising behavior of the generative model. Remarkably, the mathematically optimal combination weights can be derived in closed form via the Cauchy–Schwarz inequality, requiring negligible additional computation.

This approach can be seamlessly integrated into various mainstream inverse problem solvers, including Diffusion Posterior Sampling (DPS) (Chung et al., 2023) and Manifold-Preserving Gradient Descent (MPGD) (He et al., 2024), consistently yielding substantial improvements. Notably, a recently proposed impressive generative image compression method—Denoising Diffusion Codebook Models (DDCM) (Ohayon et al., 2025), can be viewed as a special case of NCS, in which the top-$m$ noise vectors ($m=1$ in DDCM) are selected from a codebook of size $K$, corresponding to an extremely low compression ratio. To improve compression quality, DDCM employs a greedy search with exponential complexity in $m$ to approximate the measurement score. In contrast, we demonstrate that NCS solution can achieve a competitive quantization result with linear complexity.

Our contributions are summarized as follows:

- •
We propose the NCS framework, which leverages the noise variables in the DDPM process to approximate the measurement score in inverse problem solving. By optimally combining multiple Gaussian noise vectors, NCS synthesizes a noise sample that closely approximates the desired conditional distribution. We show that this optimization problem admits a closed-form solution via the Cauchy–Schwarz inequality.
- •
We demonstrate that prominent diffusion-based inverse problem solvers can be naturally reformulated under the NCS framework. Using DPS and MPGD as examples, we show the effectiveness of NCS. We further conjecture that most existing gradient-based approaches admit corresponding NCS formulations, offering a unified and principled perspective.
- •
We show that the generative image compression method DDCM can be viewed as a special case of NCS. By increasing the number of combined noise vectors and reducing the number of sampling steps, we significantly accelerate both compression and decompression, with negligible quality loss and linear complexity. This is possible because the complexity of the proposed quantization of the optimal noise combination is nearly independent of $m$.
- •
We conduct experiments across multiple datasets and inverse problem tasks, showing that NCS consistently outperforms existing methods that rely on careful hyperparameter tuning. In particular, NCS demonstrates remarkable robustness and stability when the number of diffusion steps $T$ is small, achieving high-quality results at reduced sampling cost.

Figure: Figure 1: An illustration showing the difference between exact approximation methods and NCS. The intervention, i.e., the measurement score in the existing methods pushes the trajectory off the manifold $\mathcal{M}_{t-1}$ of $\bm{x}_{t-1}$. In contrast, NCS embeds the intervention into the optimal noise within an ellipsoidal subspace, defined by the span of the noise codebook. This allows NCS to naturally preserve both the position of $\bm{x}_{t-1}$ on its manifold and the consistency of the diffusion process.
Refer to caption: fig/front2.png

Figure: Figure 2: Comparison of DPS and NCS-DPS across four inverse problems under varying sampling steps. NCS-DPS yields clearer details and greater stability, especially at small step counts.
Refer to caption: fig/dps/inp.png

## 2 Background

### 2.1 Denoising Diffusion Probabilistic Models (DDPMs)

Diffusion models define the generative process as the reverse of a predefined noising process. Following the formulation of Song et al. (2021), we describe the forward (noising) process using an Itô stochastic differential equation (SDE), where $\bm{x}_{t}\in\mathbb{R}^{d}$ and $t\in[0,T]$:

$$ $d\bm{x}_{t}=f(\bm{x}_{t},t)\,dt+g(t)\,d\bm{w}_{t},$ (1) $$

where $f$ and $g$ denote the drift function and diffusion coefficient, respectively, and $\bm{w}_{t}$ is a standard Wiener process.

Assuming the initial data distribution is $\bm{x}_{0}\sim p_{\text{data}}$ and the terminal distribution $\bm{x}_{T}$ is Gaussian, i.e., $\mathcal{N}(0,\mathbf{I})$, the goal of the generative (reverse) process is to recover samples from $p_{\text{data}}$ by reversing the diffusion trajectory. According to Anderson (1982), this can be achieved by solving the reverse-time SDE:

$$ $d\bm{x}_{t}=\left(f(\bm{x}_{t},t)-g^{2}(t)\nabla_{\bm{x}_{t}}\log p_{t}(\bm{x}_{t})\right)dt+g(t)\,d\bm{w}_{t},$ (2) $$

initialized at $\bm{x}_{T}\sim\mathcal{N}(0,\mathbf{I})$.

We follow Song et al. (2021)’s definition to choose a Variance-Preserving (VP)-SDE, or DDPM schedule to show the discrete update rule. Researchers use a neural network $\bm{s}_{\theta}(\bm{x}_{t},t)$ to approximate the score function $\nabla_{\bm{x}_{t}}\log p_{t}(\bm{x}_{t})$ and makes it possible to use the reverse process to generate the data. We consider the general condition by discretizing the whole process into T bins,

$$ $\bm{x}_{t-1}=\bm{x_{t}}-f(\bm{x}_{t},t)+g^{2}(t)\bm{s}_{\theta}(\bm{x}_{t},t)+g(t)\bm{\epsilon}_{t}$ (3) $$

where $\bm{\epsilon}_{t}\sim\mathcal{N}(0,\mathbf{I})$.

Ho et al. (2020) consider the marginal distribution of $\bm{x}_{t}$ given $\bm{x}_{0}$ is Gaussian:

$$ $q(\bm{x}_{t}\mid\bm{x}_{0})=\mathcal{N}\left(\bm{x}_{t};\sqrt{\bar{\alpha}_{t}}\,\bm{x}_{0},\ (1-\bar{\alpha}_{t})\,\mathbf{I}\right),$ $$

where the noise schedule is defined via $\beta_{t}=g(t)={-2f(t)}$, $\alpha_{t}=1-\beta_{t}$, and $\bar{\alpha}_{t}=\prod_{s=1}^{t}\alpha_{s}$. This leads to the update rule of the DDPM process:

$$ $\bm{x}_{t-1}=\bm{\mu}(\bm{x}_{t},t)+\sigma_{t}\,\bm{\epsilon},\quad\bm{\epsilon}\sim\mathcal{N}(0,\mathbf{I}),$ (4) $$

where $\bm{\mu}(\bm{x}_{t},t)=\frac{1}{\sqrt{\alpha_{t}}}\left(\bm{x}_{t}-\frac{\beta_{t}}{\sqrt{1-\bar{\alpha}_{t}}}\,\bm{s}_{\theta}(\bm{x}_{t},t)\right)$,
and $\sigma_{t}=\sqrt{\beta_{t}}$ is the variance parameter governing the stochasticity of the reverse process.

The other important tool is Tweedie’s formula in Kadkhodaie & Simoncelli (2021), which can be used to estimate the original signal $\bm{x}_{0}$ from a noisy observation $\bm{x}_{t}$ during the denoising process. In practice, DDPMs approximate this expectation using the trained score network, yielding:

$$ $\tilde{\bm{x}}_{0\mid t}(\bm{x}_{t},t)=\mathbb{E}[\bm{x}_{0}\mid\bm{x}_{t}]\approx\frac{1}{\sqrt{\bar{\alpha}_{t}}}\left(\bm{x}_{t}-\sqrt{1-\bar{\alpha}_{t}}\,\bm{s}_{\theta}(\bm{x}_{t},t)\right).$ (5) $$

### 2.2 Linear Inverse Problems and Conditional Generation

Conditional generation addresses scenarios where only partial observations or measurements $\bm{y}\in\mathbb{R}^{n}$, derived from the original signal $\bm{x}_{0}\in\mathbb{R}^{d}$, are available. The corresponding *inverse problem* is typically formulated as:

$$ $\bm{y}=\mathcal{A}(\bm{x}_{0})+\bm{n},\quad\bm{x}_{0}\in\mathbb{R}^{d},\ \bm{y},\bm{n}\in\mathbb{R}^{n},$ (6) $$

where $\mathcal{A}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{n}$ is a known linear degradation operator, and $\bm{n}\sim\mathcal{N}(0,\sigma^{2}\mathbf{I})$ denotes additive white Gaussian noise. This formulation encompasses a wide range of tasks, including inpainting, super-resolution, and deblurring.

Solving the conditional generation problem with a pretrained diffusion model requires replacing the score function $\nabla_{\bm{x}_{t}}\log p_{t}(\bm{x}_{t})$ in equation [2](https://arxiv.org/html/2510.23633v1#S2.E2) with the conditional score function $\nabla_{\bm{x}_{t}}\log p_{t}(\bm{x}_{t}\mid\bm{y})$.

Under the Bayesian framework, the conditional distribution at $t$ is given by $p(\bm{x}_{t}\mid\bm{y})={p(\bm{y}\mid\bm{x}_{t})p(\bm{x}_{t})}/{p(\bm{y})}$, which indicates that $\nabla_{\bm{x}_{t}}\log p(\bm{x}_{t}\mid\bm{y})=\nabla_{\bm{x}_{t}}\log p(\bm{y}\mid\bm{x}_{t})+\nabla_{\bm{x}_{t}}\log p(\bm{x}_{t})$. Substituting the $\nabla_{\bm{x}_{t}}\log p(\bm{x}_{t}\mid\bm{y})$ into the reverse SDE in equation [2](https://arxiv.org/html/2510.23633v1#S2.E2) introduces an additional term that is not learned by the score network. The modified discrete update becomes:

$$ $\displaystyle\bm{x}_{t-1}$ $\displaystyle=\bm{x}_{t}-f(\bm{x}_{t},t)+g^{2}(t)\,\nabla_{\bm{x}_{t}}\log p_{t}(\bm{x}_{t}\mid\bm{y})+g(t)\,\bm{\epsilon}_{t}$ (7) $\displaystyle=\bm{x}_{t}-f(\bm{x}_{t},t)+g^{2}(t)\,\nabla_{\bm{x}_{t}}\log p_{t}(\bm{x}_{t})+g(t)\,\bm{\epsilon}_{t}+{\color[rgb]{1,0,0}g^{2}(t)\,\nabla_{\bm{x}_{t}}\log p_{t}(\bm{y}\mid\bm{x}_{t})}.$ (8) $$

Figure: Figure 3: The comparison of the selected noise according to the original image. We choose a downsampled 64x64 image as the target image.
Refer to caption: fig/compare/1_original_image.png

The red term in equation [7](https://arxiv.org/html/2510.23633v1#S2.E7) captures the influence of the observation $\bm{y}$ on the sampling trajectory. Since the exact posterior $\nabla_{\bm{x}_{t}}\log p(\bm{y}\mid\bm{x}_{t})$ is generally intractable, it typically requires an additional training process, such as classifier guidance or classifier-free guidance (Dhariwal & Nichol, 2021; Ho & Salimans, 2022) to be learned. The central challenge in diffusion-based inverse problem solving lies in accurately approximating $\nabla_{\bm{x}_{t}}\log p(\bm{y}\mid\bm{x}_{t})$. Most methods, including explicit approximations and variational inference techniques Daras et al. (2024), directly modify the sampling trajectory during the denoising process, which can disrupt the consistency of the denoising process. In contrast, sampling-based methods (Dou & Song, 2024; Cardoso et al.,; Wu et al., 2023) avoid altering the generative path by selecting candidate samples that satisfy the observation constraint through importance sampling or rejection. However, these approaches often suffer from weak guidance influence and high computational cost.

Figure: Figure 4: Comparison of the compression efficiency of NCS-MPGD and DDCM. For $T=1000$, we choose $K=32768$, $m=12$, $C=8$. For $T=100$, we choose $K=32768$, $m=2$, $C=0$. Our proposal get equivalent performance in fewer time.
Refer to caption: fig/compress_compare/compress.png

## 3 Noise Combination Sampling (NCS)

In the natural diffusion process, for all $t\in[0,T]$, $\bm{x}_{t}$ remains on its corresponding manifold, which is also the manifold on which the neural network $\bm{s}_{\theta}(\bm{x}_{t},t)$ was trained. As illustrated in Figure [1](https://arxiv.org/html/2510.23633v1#S1.F1), existing inverse problem solvers typically incorporate the measurement score $\nabla_{\bm{x}_{t}}\log p(\bm{y}\mid\bm{x}_{t})$ as an external term, adding it directly during the denoising steps. This intervention disrupts the generative consistency of the diffusion process. In contrast, NCS does not alter the assumption or the method for approximating $\nabla_{\bm{x}_{t}}\log p(\bm{y}\mid\bm{x}_{t})$. Instead, it implicitly embeds the conditional information into the noise component of the DDPM update, preserving the trajectory on the learned manifold. The modified update rule is given by:

$$ $\displaystyle\bm{x}_{t-1}$ $\displaystyle\approx\bm{x}_{t}-f(\bm{x}_{t},t)+g^{2}(t)\,\nabla_{\bm{x}_{t}}\log p_{t}(\bm{x}_{t})+{\color[rgb]{1,.5,0}g(t)\,\bm{\epsilon}_{t}^{*}},$ (9) $$

where $\bm{\epsilon}_{t}^{*}$ is a constructed noise vector that approximates the effect of the conditional term. NCS seeks to find an optimal noise vector $\bm{\epsilon}_{t}^{*}$ within the span of a finite set of noise vectors, referred to as the noise codebook, and synthesizes it as a linear combination of these basis vectors to align with the target conditional direction. We formalize this process below.

###### Theorem 1 (Noise Combination Sampling) .

For linear inverse problems, the optimal noise vector $\bm{\epsilon}_{t}^{*}$ that best aligns with the conditional score direction is given by:

$$ $\bm{\epsilon}_{t}^{*}=\sum_{i=1}^{K}\gamma_{i}\bm{\epsilon}_{i},$ (10) $$

where $\{\bm{\epsilon}_{i}\}_{i=1}^{K}$ are standard Gaussian vectors from a fixed noise codebook, and $\bm{\gamma}=(\gamma_{1},\ldots,\gamma_{K})$ denotes the combination weights. The optimal weights are obtained by solving the following constrained optimization problem:

$$ $\bm{\gamma}^{*}=\mathop{\operatorname{argmax}}_{\bm{\gamma}\in\mathbb{R}^{K},\ \|\bm{\gamma}\|_{2}=1}\left\langle\nabla_{\bm{x}_{t}}\log p(\bm{y}\mid\bm{x}_{t}),\ \sum_{i=1}^{K}\gamma_{i}\bm{\epsilon}_{i}\right\rangle.$ (11) $$

The synthesized noise vector $\bm{\epsilon}_{t}^{*}$ remains standard normal, i.e., $\bm{\epsilon}_{t}^{*}\sim\mathcal{N}(0,\mathbf{I})$, as a consequence of the unit-norm constraint $\|\bm{\gamma}\|_{2}=1$. A complete proof is provided in Appendix [A](https://arxiv.org/html/2510.23633v1#A1).

Figure: Algorithm 1 Noise Combination Sampling for Linear Inverse Problems

Under the NCS framework, any approximated conditional score, or intervention, can be embedded into the noise term via an optimal linear combination. In the following, we demonstrate how several representative inverse problem solvers can be reformulated as special instances of NCS.

###### Definition 1 (NCS-DPS) .

Chung et al. (2023) approximates the conditional score using the gradient of a likelihood loss defined between the observation $\bm{y}$ and the estimated signal $\bm{\tilde{x}}_{0|t}$. Specifically,

$$ $\displaystyle\nabla_{\bm{x}_{t}}\log p(\bm{y}\mid\bm{x}_{t})$ $\displaystyle=\nabla_{\bm{x}_{t}}\log\mathcal{N}(\bm{y};\mathbf{A}\bm{\tilde{x}}_{0|t},\sigma_{t}^{2}\mathbf{I})$ $\displaystyle=\frac{1}{\sigma_{t}^{2}}\left(\frac{\partial\bm{\tilde{x}}_{0|t}}{\partial\bm{x}_{t}}\right)^{\top}\mathbf{A}^{\top}\left(\bm{y}-\mathbf{A}\bm{\tilde{x}}_{0|t}\right).$ (12) $$

Its NCS counterpart is obtained by aligning the synthesized noise vector with this gradient direction:

$$ $\bm{\gamma}^{*}=\mathop{\operatorname{argmax}}\limits_{\bm{\gamma}\in\mathbb{R}^{K},\ \|\bm{\gamma}\|_{2}=1}\left\langle\left(\frac{\partial\bm{\tilde{x}}_{0|t}}{\partial\bm{x}_{t}}\right)^{\top}\mathbf{A}^{\top}\left(\bm{y}-\mathbf{A}\bm{\tilde{x}}_{0|t}\right),\ \sum_{i=1}^{K}\gamma_{i}\bm{\epsilon}_{i}\right\rangle.$ (13) $$

###### Definition 2 (NCS-MPGD) .

He et al. (2024) proposes performing updates directly on the estimated signal $\bm{\tilde{x}}_{0|t}$ rather than the latent variable $\bm{x}_{t}$. By isolating the additional term introduced in its update rule, we obtain the following approximation of the measurement score:

$$ $\displaystyle\nabla_{\bm{x}_{t}}\log p(\bm{y}\mid\bm{x}_{t})$ $\displaystyle=-\lambda_{t}\sqrt{\bar{\alpha}_{t}}\nabla_{\bm{\tilde{x}}_{0|t}}\|\bm{y}-\mathbf{A}\bm{\tilde{x}}_{0|t}\|_{2}^{2}$ $\displaystyle=2\lambda_{t}\sqrt{\bar{\alpha}_{t}}\mathbf{A}^{\top}(\bm{y}-\mathbf{A}\bm{\tilde{x}}_{0|t}),$ (14) $$

where $\lambda_{t}$ is a time-dependent step size. The corresponding NCS formulation is given by:

$$ $\bm{\gamma}^{*}=\mathop{\operatorname{argmax}}\limits_{\bm{\gamma}\in\mathbb{R}^{K},\ \|\bm{\gamma}\|_{2}=1}\left\langle\mathbf{A}^{\top}(\bm{y}-\mathbf{A}\bm{\tilde{x}}_{0|t}),\ \sum_{i=1}^{K}\gamma_{i}\bm{\epsilon}_{i}\right\rangle.$ (15) $$

Notably, if $\bm{\tilde{x}}_{0|t}$ is replaced with $\bm{x}_{t}$, the formulation reduces to Score-Based Annealed Langevin Dynamics (ALD) Jalal et al. (2021). For a comprehensive analysis of the connections between these methods, refer to the survey in (Daras et al., 2024).

###### Definition 3 (DDCM under the NCS Framework) .

DDCM formulates compression as a special case of inverse problems, where $\mathbf{A}=\mathbf{I}$, $\bm{n}=0$, and $\bm{y}=\bm{x}_{0}$. The conditional score is approximated based on the difference between the ground-truth signal and the estimated reconstruction:

$$ $\displaystyle\nabla_{\bm{x}_{t}}\log p(\bm{y}\mid\bm{x}_{t})\approx\frac{\sqrt{\bar{\alpha}_{t}}}{1-\bar{\alpha}_{t}}(\bm{x}_{0}-\bm{\tilde{x}}_{0|t}).$ (16) $$

DDCM selects one single noise vector from a predefined codebook via one-hot maximization:

$$ $\bm{\epsilon}_{t}^{*}=\mathop{\operatorname{argmax}}\limits_{i\in\{1,\dots,K\}}\left\langle\bm{x}_{0}-\bm{\tilde{x}}_{0|t},\ \bm{\epsilon}_{t}^{i}\right\rangle,$ (17) $$

After the reconstruction, DDCM stores these index and can reconstruct the image with index and the generative model. Ohayon et al. (2025) extend this to general linear inverse problems by:

$$ $\bm{\epsilon}_{t}^{*}=\mathop{\operatorname{argmax}}\limits_{i\in\{1,\dots,K\}}\left\langle\bm{y}-\mathbf{A}\bm{x}_{t},\ \mathbf{A}\bm{\epsilon}_{t}^{i}\right\rangle.$ (18) $$

Within the NCS framework, we replace $\bm{x}_{t}$ with the estimated $\bm{\tilde{x}}_{0|t}$ and extend the one-hot selection to a full linear combination. This yields the following optimization problem:

$$ $\bm{\gamma}^{*}=\mathop{\operatorname{argmax}}\limits_{\bm{\gamma}\in\mathbb{R}^{K},\ \|\bm{\gamma}\|_{2}=1}\left\langle\bm{y}-\mathbf{A}\bm{\tilde{x}}_{0|t},\ \mathbf{A}\sum_{i=1}^{K}\gamma_{i}\bm{\epsilon}_{i}\right\rangle.$ (19) $$

This is equivalent to NCS-MPGD, as both achieve their optimum when $\bm{\gamma}^{*}$ aligns with the same direction. A formal proof is provided in Theorem [2](https://arxiv.org/html/2510.23633v1#Thmtheorem2) and Appendix [C](https://arxiv.org/html/2510.23633v1#A3). While Ohayon et al. (2025) propose selecting the top-$m$ noise vectors from a codebook of size $K$, this can be viewed as a special top-$m$ case of NCS-MPGD with restricted support in $\bm{\gamma}$.

In the following, we describe how to compute the optimal combination weights $\bm{\gamma}^{*}$ that define the synthesized noise $\bm{\epsilon}_{t}^{*}$ under the NCS formulation.

###### Theorem 2 (Optimal Noise Combination) .

Let $\bm{c}=\nabla_{\bm{x}_{t}}\log p(\bm{y}\mid\bm{x}_{t})$ denote the approximated measurement score, and let $\mathbf{E}_{t}=[\bm{\epsilon}_{t}^{1},\ldots,\bm{\epsilon}_{t}^{K}]\in\mathbb{R}^{n\times K}$ be the matrix formed by stacking $K$ standard Gaussian noise vectors. The optimal weight vector $\bm{\gamma}^{*}\in\mathbb{R}^{K}$ that maximizes the inner product $\langle\bm{c},\mathbf{E}_{t}\bm{\gamma}\rangle$ subject to $\|\bm{\gamma}\|_{2}=1$ is given by:

$$ $\bm{\gamma}^{*}=\frac{\bm{c}^{\top}\mathbf{E}_{t}}{\|\bm{c}^{\top}\mathbf{E}_{t}\|_{2}}.$ (20) $$

By using Cauchy-Schwarz inequality, we can easily obtain the optimal weight vector, the detailed proof is provided in Appendix [B](https://arxiv.org/html/2510.23633v1#A2). The optimal noise combination can be directly computed based on inner products. This result also extends naturally to the top-$m$ case in DDCM-style problems. It significantly improves the efficiency of noise utilization: As illustrated in Fig. [3](https://arxiv.org/html/2510.23633v1#S2.F3): combining a few noise vectors can achieve an inner product magnitude comparable to that obtained by DDCM’s selection from a large codebook noise vectors.

Figure: Figure 5: Influence of iteration and codebook size. We conduct the experiment on the inpainting task (${\sigma}=0$). The heatmap shows the improvement of the NCS-DPS compared to the DPS methods.
Refer to caption: fig/heat/heatmap_diff_lpips_inp.png

Figure: Figure 6: Comparison of compression methods. NCS achieves comparable reconstruction quality while reducing the number of compression steps from 1000 to 100, demonstrating significant efficiency with minimal quality loss.
Refer to caption: fig/comp.png

As shown in Fig. [3](https://arxiv.org/html/2510.23633v1#S2.F3), the NCS method approximates the measurement score by projecting it onto an ellipsoidal subspace spanned by $K$ noise vectors. As $K$ increases, the expressiveness of the noise combination improves, while the independence among the noise vectors diminishes. The norm of the synthesized noise grows on the order of $\sqrt{d}$, where $d$ denotes the data dimensionality. This behavior suggests that the synthesized noise may increasingly deviate from the standard noise distribution. In Fig. [5](https://arxiv.org/html/2510.23633v1#S3.F5), the influence of $K$ on NCS performance appears quite robust. The NCS method performs well across a broad range of $K$ values, rendering it both practical and easy to deploy.

###### Theorem 1 (Noise Combination Sampling) .

###### Definition 1 (NCS-DPS) .

###### Definition 2 (NCS-MPGD) .

###### Definition 3 (DDCM under the NCS Framework) .

###### Theorem 2 (Optimal Noise Combination) .

### 3.1 Compression Tasks

DDCM noticed that employing multiple noise vectors can enhance compression quality. However, it requires an efficient strategy for searching and storing the indices and weights per noise. Given the definition of bits per pixel (BPP) as $\text{BPP}={(T-1)\left(\log_{2}(K)m+C(m-1)\right)}/{n^{2}}$, where $C$ represents $2^{C}$ quantization bins of vectors weights. A greedy search strategy is used to iteratively select the optimal combination of noise vectors across all quantization bins (see Ohayon et al. (2025), Appendix B.5). However, it incurs an overall complexity that grows exponentially with $C$ and $m$, making it computationally prohibitive for large-scale scenarios.

To mitigate this limitation, we propose a method that directly approximates the quantized values using the closed-form solution from Theorem [2](https://arxiv.org/html/2510.23633v1#Thmtheorem2), significantly reducing the computational burden relative to the cost of noise inner product evaluations. Our formulation enables increasing both $m$ and $C$ for higher compression fidelity while simultaneously reducing $T$. As shown in Figures [4](https://arxiv.org/html/2510.23633v1#S2.F4) and [6](https://arxiv.org/html/2510.23633v1#S3.F6), decreasing $T$ by a factor of 10, while increasing both $m$ and the number of quantization bins, preserves the compression ratio, minimizes reconstruction quality degradation, and achieves a 10$\times$ speedup in compression time. Additional implementation details are provided in Appendix [D](https://arxiv.org/html/2510.23633v1#A4).

## 4 Experiments

**Table 1: Quantitative comparison of baseline solvers and their NCS variants on FFHQ dataset (Inpainting and SR 4×). Each cell shows PSNR / FID / LPIPS.**
|  |  | PSNR(↑) / FID(↓) / LPIPS(↓) |  |  |
| --- | --- | --- | --- | --- |
| Task | Method | 20 | 100 | 1000 |
| Inpainting<br>(Box) | DPS | 12.52 / 133.9 / 0.497 | 18.67 / 95.65 / 0.286 | 22.71 / 57.11 / 0.139 |
| NCS-DPS | 19.16 / 116.0 / 0.323 | 22.31 / 71.38 / 0.170 | 23.41 / 39.82 / 0.088 |  |
| MPGD | 16.84 / 107.3 / 0.220 | 17.26 / 107.6 / 0.164 | 13.51 / 241.7 / 0.387 |  |
| NCS-MPGD | 19.00 / 99.31 / 0.277 | 20.53 / 62.50 / 0.153 | 20.96 / 44.42 / 0.101 |  |
|  | DAPS | 22.01 / 53.52 / 0.209 | 22.56 / 46.49 /  0.197 | 24.20 / 39.79 / 0.168 |
|  | NCS-DAPS | 22.33 / 49.71 / 0.205 | 22.47 / 45.01 / 0.195 | 24.24 / 39.98 / 0.168 |
| Inpainting<br>(Random) | DPS | 13.13 / 132.8 / 0.472 | 19.26 / 96.09 / 0.278 | 27.35 / 58.85 / 0.126 |
| NCS-DPS | 21.20 / 87.55 / 0.297 | 27.31 / 42.34 / 0.137 | 31.57 / 14.28 / 0.042 |  |
| MPGD | 21.80 / 76.86 / 0.172 | 25.45 / 45.73 / 0.100 | 25.05 / 69.68 / 0.214 |  |
| NCS-MPGD | 20.25 / 102.7 / 0.290 | 25.21 / 61.82 / 0.126 | 28.71 / 31.91 / 0.049 |  |
|  | DAPS | 14.08 / 259.2 / 0.625 | 16.31 / 226.4 / 0.543 | 25.33 / 67.69 / 0.238 |
|  | NCS-DAPS | 16.83 / 202.8 / 0.557 | 19.50 / 141.9 / 0.456 | 25.73 / 63.56 / 0.230 |
| SR 4× | DPS | 12.87 / 121.9 / 0.480 | 16.70 / 98.36 / 0.338 | 23.53 / 69.54 / 0.171 |
| NCS-DPS | 21.07 / 111.5 / 0.290 | 26.33 / 62.51 / 0.133 | 26.59 / 31.57 / 0.084 |  |
| MPGD | 19.35 / 90.92 / 0.246 | 22.59 / 63.68 / 0.148 | 20.46 / 84.88 / 0.490 |  |
| NCS-MPGD | 22.83 / 92.41 / 0.231 | 25.82 / 49.53 / 0.115 | 25.85 / 35.83 / 0.161 |  |
|  | DAPS | 25.52 / 89.95 / 0.329 | 26.31 / 79.90 / 0.329 | 28.22 / 52.80 / 0.200 |
|  | NCS-DAPS | 25.48 / 91.50 / 0.328 | 26.22 / 82.45 / 0.328 | 28.15 / 57.00 / 0.207 |

### 4.1 Experimental Setup

All experiments were conducted on an NVIDIA RTX 4090 GPU. In Fig. [2](https://arxiv.org/html/2510.23633v1#S1.F2), we choose pretrained models from Ho et al. (2020), and for experiments on FFHQ (Karras et al., 2019) and ImageNet (Deng et al., 2009) in Table [1](https://arxiv.org/html/2510.23633v1#S4.T1) and [2](https://arxiv.org/html/2510.23633v1#S4.T2), we choose models from Dhariwal & Nichol (2021). For compression experiments, we choose stable diffusion models 2.0 in Rombach et al. (2022).

### 4.2 Inverse Problem Solving

We evaluated our method on several challenging inverse problems: image inpainting, super-resolution, gaussian deblurring and motion deblurring. For inpainting, we tested both center box and random mask settings. For super-resolution, we examined 4× and 8×. Except for the experiments in Fig. [2](https://arxiv.org/html/2510.23633v1#S1.F2), we choose DPS, MPGD, and DAPS Zhang et al. (2025) as the baseline solvers, inverse problems setting are the same as Chung et al. (2023), all tasks choose $\sigma=0.05$. Detailed setting is shown in Appendix [E](https://arxiv.org/html/2510.23633v1#A5). We use black bold font to highlight the better result between its baseline and NCS counterpart under the same setting. We use blue bold font to highlight the better result among all methods under the same setting. For other experiments, please refer to Appendix [F](https://arxiv.org/html/2510.23633v1#A6).

### 4.3 Compression Experiments

We conducted experiments to evaluate the compression performance of our NCS-based approach. We compared our method against several state-of-the-art compression algorithms including PSC-P, BPG, and HiFiC (Albalawi et al., 2015; Elata et al., 2024; Muckley et al., 2023; Mentzer et al., 2020) on Kodak24 dataset (Franzen, 1999) and ImageNet dataset, using the same experimental setup as in DDCM Ohayon et al. (2025). The results demonstrate that our method maintains high fidelity while achieving significant storage savings compared to baseline approaches.

**Table 2: Quantitative comparison of baseline solvers and their NCS variants on four inverse problems on ImageNet. Each cell shows PSNR / FID / LPIPS.**
|  |  | PSNR(↑) / FID(↓) / LPIPS(↓) |  |  |
| --- | --- | --- | --- | --- |
| Task | Method | 20 | 100 | 1000 |
| Inpainting<br>(Box) | DPS | 11.67 / 272.9 / 0.682 | 15.34 / 234.6 / 0.550 | 18.52 / 134.8 / 0.308 |
| NCS-DPS | 15.77 / 279.2 / 0.623 | 18.19 / 159.2 / 0.362 | 19.38 / 88.65 / 0.190 |  |
| MPGD | 13.94 / 201.0 / 0.364 | 14.94 / 205.9 / 0.336 | 14.01 / 218.5 / 0.421 |  |
| NCS-MPGD | 16.12 / 226.9 / 0.466 | 16.36 / 152.3 / 0.270 | 16.57 / 126.1 / 0.226 |  |
|  | DAPS | 18.55 / 159.9 / 0.269 | 26.31 / 140.5 / 0.329 | 20.83 / 115.8 / 0.231 |
|  | NCS-DAPS | 17.78 / 152.5 / 0.268 | 26.22 / 154.8 / 0.328 | 20.49 / 114.2 / 0.233 |
| Inpainting<br>(Random) | DPS | 12.88 / 271.9 / 0.671 | 16.86 / 231.9 / 0.545 | 23.96 / 101.2 / 0.278 |
| NCS-DPS | 17.66 / 262.1 / 0.596 | 23.70 / 110.1 / 0.300 | 28.69 / 37.77 / 0.097 |  |
| MPGD | 17.15 / 121.8 / 0.285 | 19.32 / 170.9 / 0.435 | 16.38 / 263.9 / 0.770 |  |
| NCS-MPGD | 17.47 / 238.2 / 0.495 | 22.05 / 102.0 / 0.222 | 24.02 / 58.80 / 0.178 |  |
|  | DAPS | 25.59 / 60.41 / 0.252 | 26.31 / 44.64 / 0.329 | 28.79 / 23.83 / 0.146 |
|  | NCS-DAPS | 25.97 / 55.28 / 0.240 | 25.75 / 42.57 / 0.318 | 28.64 / 25.44 / 0.149 |
| SR 4× | DPS | 12.12 / 255.6 / 0.688 | 14.92 / 248.3 / 0.587 | 20.40 / 144.6 / 0.365 |
| NCS-DPS | 17.60 / 260.9 / 0.595 | 22.62 / 112.4 / 0.328 | 23.78 / 44.12 / 0.195 |  |
| MPGD | 16.41 / 191.0 / 0.420 | 18.60 / 96.65 / 0.327 | 10.43 / 292.1 / 1.109 |  |
| NCS-MPGD | 19.83 / 165.5 / 0.417 | 22.35 / 76.57 / 0.223 | 16.94 / 92.43 / 0.704 |  |
|  | DAPS | 23.25 / 135.9 / 0.368 | 23.02 / 257.4 / 0.358 | 25.21 / 105.0 / 0.301 |
|  | NCS-DAPS | 23.19 / 123.9 / 0.369 | 22.82 / 257.9 / 0.364 | 25.05 / 108.2 / 0.302 |

## 5 Conclusion

In this work, we introduced NCS, a principled framework that approximates the measurement score in diffusion models through an optimal linear combination of noise vectors. We derived a closed-form solution via the Cauchy–Schwarz inequality and showed that NCS unifies and generalizes existing inverse problem solvers, achieving strong performance with fewer diffusion steps and improved stability. With the NCS framework and its closed-form solution, we also accelerate DDCM while preserving similar reconstruction quality. As future work, we will explore applying NCS to compression, where its closed-form structure may inspire more efficient quantization schemes.

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## Appendix A Proof of Gaussianity and Optimal Weights

We first show that, under a unit-norm constraint on the combination weights, the synthesized noise remains standard normal, and then derive the closed-form optimizer for equation [11](https://arxiv.org/html/2510.23633v1#S3.E11).

###### Lemma 1 (Gaussianity of unit-norm combinations) .

Let $\{\bm{\epsilon}_{i}\}_{i=1}^{K}$ be mutually independent with $\bm{\epsilon}_{i}\sim\mathcal{N}(\bm{0},\mathbf{I})$. For any $\bm{\gamma}=(\gamma_{1},\ldots,\gamma_{K})\in\mathbb{R}^{K}$ with $\|\bm{\gamma}\|_{2}=1$ that is deterministic (or independent of $\{\bm{\epsilon}_{i}\}_{i=1}^{K}$), the linear combination

$$ $\bm{\epsilon}_{t}^{*}\;=\;\sum_{i=1}^{K}\gamma_{i}\bm{\epsilon}_{i}$ $$

satisfies $\bm{\epsilon}_{t}^{*}\sim\mathcal{N}(\bm{0},\mathbf{I})$.

###### Proof.

By linearity of expectation,
$\mathbb{E}[\bm{\epsilon}_{t}^{*}]=\sum_{i=1}^{K}\gamma_{i}\,\mathbb{E}[\bm{\epsilon}_{i}]=\bm{0}$.
For the covariance, independence and isotropy give

$$ $\operatorname{Cov}(\bm{\epsilon}_{t}^{*})=\sum_{i=1}^{K}\sum_{j=1}^{K}\gamma_{i}\gamma_{j}\,\mathbb{E}\!\big[\bm{\epsilon}_{i}\bm{\epsilon}_{j}^{\top}\big]=\sum_{i=1}^{K}\gamma_{i}^{2}\,\mathbf{I}=\|\bm{\gamma}\|_{2}^{2}\,\mathbf{I}=\mathbf{I}.$ $$

Since $(\bm{\epsilon}_{1},\ldots,\bm{\epsilon}_{K})$ is jointly Gaussian and $\bm{\epsilon}_{t}^{*}$ is a linear transformation of it, $\bm{\epsilon}_{t}^{*}$ is Gaussian with mean $\bm{0}$ and covariance $\mathbf{I}$; hence $\bm{\epsilon}_{t}^{*}\sim\mathcal{N}(\bm{0},\mathbf{I})$.
∎

###### Lemma 1 (Gaussianity of unit-norm combinations) .

###### Proof.

#### Conclusion.

Combining Lemma [1](https://arxiv.org/html/2510.23633v1#Thmlemma1) with Lemma [2](https://arxiv.org/html/2510.23633v1#Thmtheorem2), we obtain that the synthesized noise is standard normal whenever the synthesis codebook is independent of the weight computation, i.e., $\bm{\epsilon}_{t}^{*}\sim\mathcal{N}(\bm{0},\mathbf{I})$.

In specific optimization scenarios, as the number of noise sources increases linearly, the inner product also increases nearly linearly according to log(K). This approach achieves significantly higher efficiency than selecting a single noise source within K (in DDCM). Furthermore, the magnitude of the optimal noise remains constant over a considerable range (approximately equal to the average magnitude of the noise sources). Specific variations can be observed in Fig. [3](https://arxiv.org/html/2510.23633v1#S2.F3).

## Appendix B Proof of Theorem 2

###### Proof.

Define $\bm{v}=\bm{c}^{\top}\mathbf{E}_{t}\in\mathbb{R}^{K}$. Our goal is to solve:

$$ $\bm{\gamma}^{*}=\mathop{\operatorname{argmax}}_{\|\bm{\gamma}\|_{2}=1}\langle\bm{v},\bm{\gamma}\rangle.$ $$

By the Cauchy–Schwarz inequality,

$$ $\langle\bm{v},\bm{\gamma}\rangle\leq\|\bm{v}\|_{2}\|\bm{\gamma}\|_{2}=\|\bm{v}\|_{2},$ $$

with equality if and only if $\bm{\gamma}$ is aligned with $\bm{v}$, i.e., $\bm{\gamma}=\lambda\bm{v}$ for some scalar $\lambda$. Enforcing the constraint $\|\bm{\gamma}\|_{2}=1$ gives $|\lambda|=1/\|\bm{v}\|_{2}$, and the maximum is attained at:

$$ $\bm{\gamma}^{*}=\frac{\bm{v}}{\|\bm{v}\|_{2}}=\frac{\bm{c}^{\top}\mathbf{E}_{t}}{\|\bm{c}^{\top}\mathbf{E}_{t}\|_{2}}.$ $$

∎

###### Proof.

## Appendix C Proof of Equivalence between NCS Optimization and NCS-MPGD

###### Theorem 3 (Equivalence of NCS Formulations) .

For NCS optimization problem [11](https://arxiv.org/html/2510.23633v1#S3.E11), we propose the following formulation: in which we replaced the $\bm{x}_{0}$ in Ohayon et al. (2025) with the more accurate reconstruction $\bm{\tilde{x}}_{0|t}$.

$$ $\bm{\gamma}^{*}=\mathop{\operatorname{argmax}}\limits_{\bm{\gamma}\in\mathbb{R}^{K},\|\bm{\gamma}\|_{2}=1}\left\langle\bm{y}-\mathbf{A}\tilde{\bm{x}}_{0\mid t},\mathbf{A}\sum_{i=1}^{K}\gamma_{i}\bm{\epsilon}_{i}\right\rangle,$ (21) $$

It is equivalent to the NCS-MPGD formulation:

$$ $\bm{\gamma}^{*}=\mathop{\operatorname{argmax}}\limits_{\bm{\gamma}\in\mathbb{R}^{K},\,\|\bm{\gamma}\|_{2}=1}\left\langle-\mathbf{A}^{\top}(\bm{y}-\mathbf{A}\bm{\tilde{x}}_{0|t}),\ \sum_{i=1}^{K}\gamma_{i}\bm{\epsilon}_{i}\right\rangle.$ (22) $$

That is, both optimization problems have the same optimal solution $\bm{\gamma}^{*}$.

###### Proof.

We prove this equivalence by showing that the two objective functions are identical for any feasible $\bm{\gamma}$.

Let $\bm{c}=\bm{y}-\mathbf{A}\tilde{\bm{x}}_{0\mid t}$ and $\bm{\varepsilon}=\sum_{i=1}^{K}\gamma_{i}\bm{\epsilon}_{i}$. The standard NCS objective can be written as:

$$ $\displaystyle\left\langle\bm{c},\mathbf{A}\bm{\varepsilon}\right\rangle$ $\displaystyle=\bm{c}^{\top}(\mathbf{A}\bm{\varepsilon})$ (23) $\displaystyle=(\bm{c}^{\top}\mathbf{A})\bm{\varepsilon}$ (24) $\displaystyle=\bm{\varepsilon}^{\top}(\bm{c}^{\top}\mathbf{A})^{\top}$ (25) $\displaystyle=\bm{\varepsilon}^{\top}\mathbf{A}^{\top}\bm{c}$ (26) $\displaystyle=\left\langle\mathbf{A}^{\top}\bm{c},\bm{\varepsilon}\right\rangle.$ (27) $$

The NCS-MPGD objective is:

$$ $\displaystyle\left\langle-\mathbf{A}^{\top}\bm{c},\bm{\varepsilon}\right\rangle$ $\displaystyle=\left\langle-\mathbf{A}^{\top}\bm{c},\sum_{i=1}^{K}\gamma_{i}\bm{\epsilon}_{i}\right\rangle$ (28) $\displaystyle=\sum_{i=1}^{K}\gamma_{i}\left\langle-\mathbf{A}^{\top}\bm{c},\bm{\epsilon}_{i}\right\rangle.$ (29) $$

Since the two inner products are identical for every $\bm{\varepsilon}$ (and thus for every $\bm{\gamma}$), and the feasible set $\{\bm{\gamma}\in\mathbb{R}^{K}\mid\|\bm{\gamma}\|_{2}=1\}$ is the same for both problems, their maxima and argmax sets are identical.

Furthermore, since each objective is a linear functional of $\bm{\gamma}$, the maximum over the unit sphere occurs when $\bm{\gamma}$ is aligned with the functional’s direction. For the standard NCS formulation, this gives:

$$ $\bm{\gamma}^{*}=\frac{(\mathbf{A}^{\top}\bm{c})^{\top}\mathbf{E}_{t}}{\|(\mathbf{A}^{\top}\bm{c})^{\top}\mathbf{E}_{t}\|_{2}},$ (30) $$

where $\mathbf{E}_{t}=[\bm{\epsilon}_{1},\ldots,\bm{\epsilon}_{K}]$ is the matrix of noise vectors.

For the NCS-MPGD formulation, the optimal solution is:

$$ $\bm{\gamma}^{*}=\frac{(-\mathbf{A}^{\top}\bm{c})^{\top}\mathbf{E}_{t}}{\|(-\mathbf{A}^{\top}\bm{c})^{\top}\mathbf{E}_{t}\|_{2}}=\frac{-(\mathbf{A}^{\top}\bm{c})^{\top}\mathbf{E}_{t}}{\|(\mathbf{A}^{\top}\bm{c})^{\top}\mathbf{E}_{t}\|_{2}}.$ (31) $$

The negative sign in the NCS-MPGD formulation is due to the maximization of the negative inner product, which is equivalent to minimizing the positive inner product. However, since we are maximizing the absolute value of the alignment, both formulations yield the same optimal direction (up to a sign, which is normalized out by the unit norm constraint).

Therefore, the two optimization problems are not merely equivalent—they are literally the same problem written in two different notations, and they achieve their maximum when the optimal $\bm{\gamma}^{*}$ is the same (up to normalization).
∎

###### Theorem 3 (Equivalence of NCS Formulations) .

###### Proof.

## Appendix D Quantization by NCS closed-form solution

To utilize extra noise vectors to approximate the measurement score, Ohayon et al. (2025) proposes to use greedy search to find the noise vector that maximize the inner product between the measurement score and the noise vector, and to search the optimal quantization parameters on the selected noise vector and the next one. This process is computationally expensive. It will cost hours to search the optimal quantization parameters. If $C=10$ and $m=10$ $K=1024$, it will cost around 10 hours to search the optimal quantization parameters.

#### Problem.

Let $\{\bm{\epsilon}_{i}\}_{i=1}^{N}$ be given noise vectors and let $\bm{c}$ be the target.
Define $b_{i}:=\langle\bm{\epsilon}_{i},\bm{c}\rangle$ and (optionally) align signs so $b_{i}\geq 0$ by replacing $\bm{\epsilon}_{i}\leftarrow\operatorname{sgn}(b_{i})\bm{\epsilon}_{i}$.
We form a mixture

$$ $\bm{m}(\bm{\gamma})\;=\;\sum_{i=1}^{m}\gamma_{i}\,\bm{\epsilon}_{(i)},\qquad\bm{\gamma}=(\gamma_{1},\ldots,\gamma_{m})^{\top},\quad\sum_{i=1}^{m}\gamma_{i}^{2}=1,$ $$

where $(i)$ indexes an ordered subset of $m$ atoms (e.g., the Top-$m$ by $|b_{i}|$).
The objective is to maximize alignment, i.e.

$$ $\max_{\bm{\gamma}}\;\langle\bm{m}(\bm{\gamma}),\bm{c}\rangle\;=\;\max_{\bm{\gamma}}\;\sum_{i=1}^{m}\gamma_{i}\,b_{(i)}\quad\text{s.t.}\quad\sum_{i=1}^{m}\gamma_{i}^{2}=1.$ $$

#### Closed-form (continuous) solution.

Without quantization, the optimal coefficients on a fixed support are

$$ $\bm{\gamma}^{\,*}\;=\;\frac{\bm{b}_{S}}{\|\bm{b}_{S}\|_{2}},\qquad\bm{b}_{S}=\big(b_{(1)},\ldots,b_{(m)}\big)^{\top},$ $$

by Cauchy–Schwarz. If the support $S$ is free, it is optimal to take the $m$ indices with largest $|b_{i}|$.

#### Definition (quantization via ℓ 2 \ell_{2} stick-breaking).

We quantize by parameterizing $\bm{\gamma}$ through a stick-breaking map using

$$ $u_{i}\in\mathcal{Q}\subset(0,1],\qquad i=1,\ldots,2^{C}-1,$ $$

where $\mathcal{Q}$ is a finite grid with $2^{C-1}-1$ elements (e.g., if m = 3, then if selected (not 0), $\mathcal{Q}=\{0.33,0.66,1\}$).
The coefficients are then

$$ $\displaystyle\gamma_{1}$ $\displaystyle=\sqrt{u_{1}},$ (32) $\displaystyle\gamma_{i}$ $\displaystyle=\Big(\prod_{j=1}^{i-1}\sqrt{1-u_{j}}\Big)\sqrt{u_{i}},\quad i=2,\ldots,m-1,$ (33) $\displaystyle\gamma_{m}$ $\displaystyle=\prod_{j=1}^{m-1}\sqrt{1-u_{j}}.$ (34) $$

By construction $\sum_{i=1}^{m}\gamma_{i}^{2}=1$ for any choices of $\{u_{i}\}$, so no final normalization is required.
The inverse map (from any feasible $\bm{\gamma}$ with $\sum\gamma_{i}^{2}=1$) is

$$ $\displaystyle u_{1}$ $\displaystyle=\gamma_{1}^{2},$ (35) $\displaystyle u_{i}$ $\displaystyle=\frac{\gamma_{i}^{2}}{1-\sum_{t=1}^{i-1}\gamma_{t}^{2}},\quad i=2,\ldots,m-1.$ (36) $$

#### Using the closed-form to obtain a quantized solution.

We obtain a quantized solution directly from $\bm{\gamma}^{\,*}$ in two simple steps:

- 1.
Project $\bm{\gamma}^{\,*}$ into stick-breaking space.
Compute $\{u_{i}^{*}\}_{i=1}^{m-1}$ from $\bm{\gamma}^{\,*}$ using the inverse map above.
(When $b_{(1)}\geq b_{(2)}\geq\cdots$, $\gamma_{i}^{*}\propto b_{(i)}$ is non-increasing, which matches the stick-breaking order.)
- 2.
Quantize and reconstruct.
Independently quantize each stage by nearest-neighbor projection onto the grid,
$\hat{u}_{i}\;=\;\arg\min_{u\in\mathcal{Q}}\big|u-u_{i}^{*}\big|,\qquad i=1,\ldots,m-1,$
then form $\hat{\bm{\gamma}}$ from $\{\hat{u}_{i}\}$ via the forward stick-breaking map.

This yields $\hat{\bm{\gamma}}$ in $O(m)$ time and preserves $\sum_{i}\hat{\gamma}_{i}^{2}=1$ by construction.

#### Remark (stage-wise closed form and exact discrete refinement).

If one optimizes stage-wise in the *continuous* domain, the optimal fraction at stage $i$ has the closed form

$$ $u_{i}^{\star}\;=\;\frac{b_{(i)}^{\,2}}{b_{(i)}^{\,2}+v_{i+1}^{\,2}},\qquad v_{m}=b_{(m)},\quad v_{i}=b_{(i)}\sqrt{u_{i}^{\star}}+v_{i+1}\sqrt{1-u_{i}^{\star}}.$ $$

A discretized variant replaces $u_{i}^{\star}$ by the nearest grid point in $\mathcal{Q}$ at each stage (still $O(m)$).
For the *exact* discrete optimum on $\mathcal{Q}$ one can use a 1D dynamic program:

$$ $v_{m}=b_{(m)},\qquad v_{i}=\max_{u\in\mathcal{Q}}\Big\{\,b_{(i)}\sqrt{u}+v_{i+1}\sqrt{1-u}\,\Big\},$ $$

which selects $\hat{u}_{i}\in\mathcal{Q}$ per stage and remains $O\big(m\,|\mathcal{Q}|\big)$.

## Appendix E Inverse Problems Setting

We choose the same setting of inverse problems with Chung et al. (2023). Detailed setting is shown in Table [3](https://arxiv.org/html/2510.23633v1#A6.T3) and [4](https://arxiv.org/html/2510.23633v1#A6.T4).

As for the NCS method, because the strength of the measurement would influence the dimension of the noise space that is available for us to approximate, we set the $K$ for different tasks as follows:

- •
Super-resolution $\times$4 (ImageNet/FFHQ): $K=512$
- •
Super-resolution $\times$8 (ImageNet/FFHQ): $K=64$
- •
Inpainting box (ImageNet/FFHQ): $K=64$
- •
Inpainting random (ImageNet/FFHQ): $K=512$
- •
Gaussian deblurring (ImageNet/FFHQ): $K=256$
- •
Motion deblurring (ImageNet/FFHQ): $K=512$
- •
Phase retrieval (ImageNet/FFHQ): $K=512$

## Appendix F Additional Experiments Outcomes

We record experiments on Super-resolution $\times$8, deblurring, gaussian blurring, in Table. [3](https://arxiv.org/html/2510.23633v1#A6.T3) and [4](https://arxiv.org/html/2510.23633v1#A6.T4)

Figure: Figure 7: Visual comparison on Gaussian Blur task. Each row shows results for one image across different methods. The three sections correspond to timesteps 20, 100, and 1000 respectively.
Refer to caption: fig/results_selected/gaussian_blur/sampler_dps/timestep_20/model_imagenet256/input/00015.png

Figure: Figure 8: Visual comparison on Inpainting Random task. Each row shows results for one image across different methods. The three sections correspond to timesteps 20, 100, and 1000 respectively.
Refer to caption: fig/results_selected/inpainting_random/sampler_dps/timestep_20/model_imagenet256/input/00000.png

Figure: Figure 9: Visual comparison on Motion Blur task. Each row shows results for one image across different methods. The three sections correspond to timesteps 20, 100, and 1000 respectively.
Refer to caption: fig/results_selected/motion_blur/sampler_dps/timestep_20/model_ffhq_10m/input/00010.png

Figure: Figure 10: Visual comparison on Super Resolution x4 task. Each row shows results for one image across different methods. The three sections correspond to timesteps 20, 100, and 1000 respectively.
Refer to caption: fig/results_selected/super_resolution_x4/sampler_dps/timestep_20/model_ffhq_10m/input/00014.png

**Table 3: Quantitative comparison of baseline solvers and their NCS variants on the FFHQ dataset (Inpainting and SR 4×). Each cell shows PSNR / FID / LPIPS. Bold indicates the better result between a baseline and its NCS counterpart under the same setting. Values are formatted for visual alignment.**
|  |  | PSNR(↑) / FID(↓) / LPIPS(↓) |  |  |
| --- | --- | --- | --- | --- |
| Task | Method | 20 | 100 | 1000 |
| SR 8× | DPS | 11.84 / 118.3 / 0.506 | 15.05 / 102.1 / 0.387 | 20.87 / 76.90 / 0.220 |
| NCS-DPS | 20.83 / 109.4 / 0.290 | 24.09 / 67.34 / 0.154 | 21.97 / 49.35 / 0.266 |  |
| MPGD | 17.60 / 93.26 / 0.303 | 20.08 / 77.28 / 0.213 | 18.08 / 87.93 / 0.560 |  |
| NCS-MPGD | 22.10 / 92.13 / 0.246 | 23.29 / 65.00 / 0.170 | 21.67 / 60.28 / 0.352 |  |
|  | DAPS | 23.66 / 202.8 / 0.387 | 24.02 / 270.5 / 0.337 | 25.20 / 188.4 / 0.278 |
|  | NCS-DAPS | 23.74 / 199.9 / 0.382 | 24.14 / 273.6 / 0.334 | 25.23 / 188.6 / 0.279 |
| Gaussian Deblur | DPS | 12.12 / 139.4 / 0.495 | 20.02 / 88.92 / 0.253 | 24.81 / 60.57 / 0.130 |
| NCS-DPS | 21.59 / 115.1 / 0.311 | 26.68 / 52.14 / 0.115 | 27.02 / 49.90 / 0.083 |  |
| MPGD | 21.36 / 78.46 / 0.206 | 23.85 / 64.22 / 0.133 | 24.80 / 74.67 / 0.130 |  |
| NCS-MPGD | 25.42 / 74.20 / 0.184 | 26.58 / 48.04 / 0.110 | 26.36 / 86.73 / 0.166 |  |
|  | DAPS | 25.80 / 85.06 / 0.319 | 26.63 / 71.26 / 0.267 | 28.38 / 50.10 / 0.187 |
|  | NCS-DAPS | 25.80 / 83.67 / 0.318 | 26.57 / 72.99 / 0.268 | 28.34 / 51.49 / 0.189 |
| Motion Deblur | DPS | 12.21 / 140.3 / 0.493 | 20.19 / 88.61 / 0.251 | 25.87 / 56.65 / 0.118 |
| NCS-DPS | 22.05 / 114.9 / 0.303 | 28.28 / 47.25 / 0.097 | 29.50 / 28.41 / 0.044 |  |
| MPGD | 21.86 / 76.79 / 0.191 | 24.88 / 53.05 / 0.107 | 25.54 / 54.78 / 0.142 |  |
| NCS-MPGD | 26.22 / 64.51 / 0.153 | 27.76 / 39.31 / 0.078 | 28.17 / 41.91 / 0.083 |  |
|  | DAPS | 26.25 / 66.91 / 0.289 | 27.77 / 57.02 / 0.232 | 30.52 / 34.34 / 0.139 |
|  | NCS-DAPS | 26.19 / 65.73 / 0.290 | 27.68 / 54.91 / 0.233 | 30.51 / 33.59 / 0.139 |

**Table 4: Quantitative comparison of baseline solvers and their NCS variants on four inverse problems on ImageNet. Each cell shows PSNR / FID / LPIPS. Bold indicates the better result between a baseline and its NCS counterpart under the same setting. Values are formatted for visual alignment.**
|  |  | PSNR(↑) / FID(↓) / LPIPS(↓) |  |  |
| --- | --- | --- | --- | --- |
| Task | Method | 20 | 100 | 1000 |
| SR 8× | DPS | 11.16 / 257.5 / 0.702 | 13.78 / 246.6 / 0.616 | 18.45 / 179.9 / 0.415 |
| NCS-DPS | 17.54 / 254.3 / 0.583 | 21.02 / 118.0 / 0.328 | 18.79 / 84.54 / 0.453 |  |
| MPGD | 15.46 / 226.4 / 0.500 | 16.98 / 145.9 / 0.383 | 10.86 / 315.1 / 1.037 |  |
| NCS-MPGD | 19.26 / 176.1 / 0.461 | 19.74 / 118.3 / 0.371 | 14.64 / 172.8 / 0.798 |  |
|  | DAPS | 21.43 / 202.8 / 0.368 | 21.32 / 270.5 / 0.472 | 25.21 / 188.4 / 0.301 |
|  | NCS-DAPS | 21.50 / 199.9 / 0.369 | 21.24 / 273.6 / 0.472 | 22.31 / 188.6 / 0.424 |
| Gaussian Deblur | DPS | 12.02 / 265.5 / 0.675 | 17.32 / 213.8 / 0.493 | 21.01 / 105.2 / 0.298 |
| NCS-DPS | 13.18 / 247.3 / 0.656 | 22.29 / 87.41 / 0.298 | 23.87 / 72.03 / 0.215 |  |
| MPGD | 16.53 / 196.2 / 0.413 | 14.72 / 261.9 / 0.253 | 10.69 / 317.4 / 1.079 |  |
| NCS-MPGD | 22.06 / 114.6 / 0.376 | 22.73 / 93.45 / 0.258 | 16.97 / 206.9 / 0.720 |  |
|  | DAPS | 23.42 / 135.2 / 0.370 | 26.62 / 120.6 / 0.267 | 24.89 / 102.6 / 0.275 |
|  | NCS-DAPS | 23.38 / 127.5 / 0.371 | 26.57 / 123.3 / 0.268 | 24.85 / 106.5 / 0.281 |
| Motion Deblur | DPS | 12.05 / 265.5 / 0.676 | 17.39 / 218.4 / 0.502 | 22.75 / 87.47 / 0.268 |
| NCS-DPS | 13.39 / 250.8 / 0.648 | 24.13 / 77.85 / 0.255 | 27.25 / 34.19 / 0.103 |  |
| MPGD | 16.99 / 165.3 / 0.375 | 14.81 / 242.0 / 0.253 | 13.99 / 227.4 / 0.886 |  |
| NCS-MPGD | 22.83 / 90.49 / 0.283 | 23.90 / 62.78 / 0.169 | 17.46 / 157.6 / 0.648 |  |
|  | DAPS | 24.54 / 80.48 / 0.297 | 25.21 / 68.35 / 0.272 | 27.93 / 43.42 / 0.181 |
|  | NCS-DAPS | 24.42 / 80.41 / 0.298 | 24.83 / 69.94 / 0.278 | 27.85 / 44.81 / 0.185 |

## Appendix G Compression by NCS-DPS

Figure: Figure 11: Compression results on Kodak24 images. Each row corresponds to a different image. Columns: Original, DDCM (100 steps), DDCM (1000 steps), NCS-DPS (100 steps), NCS-DPS (1000 steps).
Refer to caption: fig/compress_new/5/origin.png

Since we have unified various methods under the NCS framework, this implies that—aside from DDCM, which is a special case equivalent to NCS-MPGD, NCS-DPS can also be employed for compression tasks. To evaluate this, we conducted experiments on several images from the Kodak24 dataset, using the Stable Diffusion 2 model as the pre-trained backbone (Rombach et al., 2022). We adopted a codebook of size 1024 but restricted the selection to a single optimal noise vector to maintain consistency with the experimental settings of the DDCM paper.

As shown in Fig. [11](https://arxiv.org/html/2510.23633v1#A7.F11), the compression efficiency of NCS-DPS is inferior to that of DDCM. The compressed images appear overly smooth, exhibiting a significant loss of fine details. This degradation may stem from inaccuracies in the gradient used during the reverse computation. Nonetheless, when using fewer denoising steps, the images compressed by NCS-DPS appear to retain richer semantic structures.

## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 4
presentation: 3
questions: 1. Could the authors provide a theoretical or empirical explanation for why replacing the noise term with an optimally combined version improves stability or reconstruction quality? Is there any evidence (e.g., manifold distance, variance analysis, or effective step size) showing that the NCS trajectory stays closer to the learned data manifold?

2. In practice, NCS appears mathematically similar to taking a guided noise step proportional to $
\nabla_x \log p(y\mid x)$. Could the authors clarify how NCS differs in effect from existing gradient-based corrections? Are there cases where the two produce substantially different trajectories?

3. How sensitive is performance to the codebook size K and the number of combined noise vectors m? Does increasing K always help, or does it introduce variance and instability due to correlation among noise samples? The method is restricted to linear inverse problems. Are there conceptual or mathematical obstacles to extending NCS to nonlinear or learned degradation operators (e.g., differentiable renderers, neural forward models)?
soundness: 3
strengths: 1. The paper is clearly written, with logical organization and careful presentation of derivations, figures, and tables. The notation is consistent, and the main idea is easy to follow even for readers without deep expertise in diffusion models.
2. The derivation of the optimal noise combination through the Cauchy–Schwarz inequality is elegant. It provides a compact, closed-form solution that is computationally lightweight and easy to implement.
3. The authors successfully demonstrate that several well-known inverse problem solvers (DPS, MPGD, DDCM) can be interpreted as instances of the same general principle. This unifying viewpoint could be valuable for researchers seeking a more cohesive understanding of guidance mechanisms in diffusion models. Also, across multiple datasets and problem types, the NCS variants perform as well as or slightly better than their corresponding baselines, especially when the number of diffusion steps is small. This consistency suggests that the approach is robust and stable in practice. More importantly, the method requires no extra training and introduces negligible additional computational cost. The linear complexity with respect to the codebook size makes it practical and accessible for a wide range of applications. By relating the approach to DDCM, the paper opens the possibility of extending diffusion-based generative compression with simpler noise quantization schemes.
summary: This paper proposes Noise Combination Sampling (NCS), a new approach to solving linear inverse problems using pretrained diffusion models without retraining. The main idea is to replace the standard Gaussian noise term in the denoising process with a synthesized noise vector, formed as an optimal linear combination of Gaussian samples drawn from a codebook. The combination weights are derived via a simple closed-form expression based on the Cauchy–Schwarz inequality, which aligns the noise with the conditional measurement score.

The authors claim that this procedure naturally embeds conditional information into the generation process while avoiding the instability and heavy hyperparameter tuning required by prior guidance-based inverse solvers such as Diffusion Posterior Sampling (DPS) and Manifold-Preserving Gradient Descent (MPGD). They further argue that existing diffusion-based inverse problem solvers can be interpreted as special cases of NCS, including the recently proposed Denoising Diffusion Codebook Models (DDCM). Empirical results on FFHQ and ImageNet show moderate improvements in PSNR and LPIPS, particularly for low sampling step counts (T = 20–100).
weaknesses: While the paper is neat and clearly executed, the conceptual advance feels incremental and its underlying mechanism insufficiently explored. The proposed idea of aligning a noise combination to a measurement gradient is mathematically simple and closely related to existing guidance schemes. The derivation relies on a direct application of Cauchy–Schwarz, and it is unclear why this reformulation should lead to qualitatively better sampling.

Moreover, the claimed advantages (manifold preservation, stability, robustness to step size) remain intuitive hypotheses rather than demonstrated phenomena. No quantitative measure or theoretical justification is provided for why embedding the conditional information into the noise term, as opposed to the mean term, should improve results. The experimental section reports moderate numerical gains but does not investigate what aspects of NCS drive these improvements—whether due to the noise combination itself, implicit regularization effects, or differences in implementation.

Overall, the work reads as a well-presented reformulation of known methods rather than a fundamentally new contribution. It could be strengthened by deeper theoretical analysis and more targeted experiments probing why the approach works. If the authors are able to answer the following questions in the next section, it would be a big help for us.

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 2
questions: 1. Can NCS be adapted to nonlinear inverse problems? If not, what theoretical modifications would be required?

2. Is there any theoretical guideline for K? For example, how should K scale with data dimensionality (d) or task complexity to balance noise expressiveness and independence?
soundness: 3
strengths: 1. NCS embeds conditional constraints into the DDPM noise term instead of directly modifying the sampling trajectory to avoid pushing generated samples off the data manifold and breaking diffusion generation consistency. This design aligns with the intrinsic stochasticity of diffusion models and has solid theoretical motivation.

2. NCS does not redesign the approximation of conditional score and is expected to expand the performance of many existing methods that apply data consistency through conditional gradient score approximation.
summary: This paper proposes the Noise Combination Sampling (NCS) framework to address the stability dilemma in solving linear inverse problems with diffusion models. Instead of directly modifying the sampling trajectory, which risks disrupting data manifolds and generative consistency, NCS embeds conditional information into the noise term of Denoising Diffusion Probabilistic Models (DDPM). It synthesizes an optimal noise vector from a predefined noise codebook to approximate the measurement score, with closed-form optimal weights derived via the Cauchy–Schwarz inequality to ensure that the synthesized noise remains standard normal. NCS addresses the core dilemma of diffusion-based inverse problems that excessive integration disrupts generation and insufficient integration ignores constraints by synthesizing optimal noise vectors to embed conditional information, rather than directly modifying the sampling trajectory.
weaknesses: 1. The paper acknowledges unstable results on nonlinear tasks (e.g., phase retrieval) but provides no theoretical analysis on the reason why NCS fails here. For example, whether the noise subspace can approximate non-linear conditional gradient score or if the closed-form weight solution breaks down under non-linear constraints.

2. The choice of noise codebook size K (e.g., 512 for 4× super-resolution and 64 for 8× super-resolution) is purely empirical. In the paper, although K is noted to work "across a broad range", there does not exist a quantifiable relationship between K, data dimensionality, or task complexity, nor does it define the boundary where increasing K reduces noise independence (mentioned in Section 3.2) and yields degraded performance. Ablation studies are also missing for key parameters related to noise combination such as the number of combined noise vectors K on performance. This makes it hard to validate the robustness of NCS to parameter changes.

---

confidence: 4
contribution: 2
initial_rating: 2
presentation: 3
questions: 1. *Regarding weakness 3:* Can the authors clarify how the Gaussianity of $ε^*_t$ is preserved in practice given that $𝛾$ depends on {$ε_i$}? If the independence assumption is violated, what is the actual distribution of the constructed noise, and how does this affect the validity of the diffusion process?
2. *Regarding weakness 4:* Can the authors explain why stronger and more recent zero-shot inverse solvers such as were omitted from comparison? Additionally, can they clarify whether NCS actually improves DAPS, given the quantitative results reported in Tables 2,3 and 4?
3. *Regarding weakness 5:* How does NCS behave when the forward operator is ill-conditioned or nonlinear? Does the Gaussianity assumption still hold in such cases, and how exactly was phase retrieval implemented or adapted within the proposed framework?
4. Can the authors report the additional runtime and GPU memory consumption introduced by NCS for typical values of K?
soundness: 2
strengths: 1. The paper is generally well-written and clearly structured, making the proposed method easy to follow.
2. The idea of modifying the diffusion noise term instead of using explicit gradient guidance is interesting and practically implementable.
3. The authors conduct some ablations such as varying codebook size K and report quantitative metrics across multiple datasets, showing some empirical rigor.
summary: The paper proposes Noise Combination Sampling (NCS), a method for solving diffusion-based linear inverse problems by replacing the stochastic noise term in the diffusion update rule with a constructed linear combination of Gaussian noise vectors. The combination weights are chosen to align with the estimated measurement score direction, derived via a closed-form expression using the Cauchy–Schwarz inequality. Experiments on standard inverse problems such as inpainting, deblurring, super-resolution, and compression are presented, showing moderate performance gains and faster convergence in some settings.
weaknesses: 1. The review template seems to be missing some elements, such as line numbers and the header stating “Under review as a conference paper at ICLR 2026.” The former, in particular, would have made it easier to reference specific parts of the text when providing comments.
2. NCS primarily reparameterizes the noise term in existing diffusion posterior sampling methods, offering little conceptual or theoretical advancement. The “closed-form” derivation is a straightforward application of the Cauchy–Schwarz inequality, and the claimed improvements over prior approaches are minimal. Overall, the contribution feels incremental rather than a substantive methodological innovation.
3. The paper claims that the constructed noise combination remains Gaussian if the combination weights are independent of the noise codebook. However, in the proposed method, the weights $𝛾_i$ are computed from the inner products between the codebook vectors and the measurement score, creating explicit statistical dependence between $𝛾_i$ and $ε_i$. This invalidates the independence assumption underlying the “Gaussianity lemma” and breaks the formal justification that the composed noise $ε^*_t$ follows $\mathcal{N\mathrm{(\mathbf{0},\mathbf{I})}}$.
4. Methods like DPS and MPGD are reasonable baselines but outdated, as many faster samplers now exist (e.g., ΠGDM, DDNM, DiffPIR, MGPS) that achieve strong performance within 50-100 NFEs. It remains unclear whether NCS-DPS and NCS-MPGD can outperform these state-of-the-art methods, and Table 2 does not clearly demonstrate whether NCS provides any real improvement over DAPS.
5. In Section 3, the authors derive all formulas under the assumption of linear inverse problems, yet later report results for the nonlinear phase retrieval task without providing its explicit formulation or explaining how NCS applies in this setting.
6. The authors claim that NCS unifies most existing gradient-based approaches, but this appears largely superficial, consisting mainly of algebraic reformulations where the guidance term is replaced by a projected noise. There is no shared probabilistic interpretation or derivation showing these methods as genuine special cases of a single framework.
7. The authors report that they used $σ_y$ = 0.05, but the qualitative figures suggest $σ_y$ ~= 0.0 (e.g., Figure 2, box inpainting).

---

confidence: 4
contribution: 2
initial_rating: 4
presentation: 3
questions: Please refer to the weaknesses.
soundness: 3
strengths: 1. The closed-form solution for optimal weights (Theorem 2) is strictly derived using inner products and the Cauchy–Schwarz inequality, eliminating the need for heuristic iterative optimization. This ensures reproducibility—researchers can directly compute \gamma^* without tuning empirical parameters.
2. Unlike baselines (e.g., DPS) that suffer severe quality degradation when diffusion steps T are reduced (to cut costs), NCS maintains high robustness. Experiments show it achieves high-quality results even with small T (e.g., 100 steps vs. 1000 steps), as the optimal noise combination preserves manifold consistency without relying on excessive iterations.
3. This method replaces DDCM’s exponential-complexity noise selection, where C is quantization bins) with NCS’s linear-complexity combination. For example, combining 3 noise vectors achieves inner-product magnitudes comparable to DDCM’s search over 1024 vectors, slashing storage (smaller codebooks) and computation (fewer inner-product calculations) costs.
summary: This paper proposes the Noise Combination Sampling (NCS) framework to address the key challenge of balancing observation constraint integration and diffusion process consistency in diffusion-based inverse problem solving. NCS constructs an optimal noise vector (via linear combination of base Gaussian noise from a codebook) to implicitly embed observation information, ensuring the noise remains standard normal and the sampling trajectory stays on the data manifold. It provides a closed-form solution for optimal weights (via Cauchy–Schwarz inequality), unifies existing methods (DPS, MPGD, DDCM) as special cases, and achieves both better performance (higher PSNR, lower LPIPS) and efficiency across tasks like inpainting, super-resolution, and deblurring.
weaknesses: 1. This method lacks clear guidelines for selecting codebook size K and base noise distribution. A small K restricts the noise combination space (failing to match rare observation directions), while a large K increases memory usage and inner-product computation time. No adaptive mechanism (e.g., dynamic K based on task complexity) is proposed.
2. The authors state that the NCS method approximates the measurement score through the linear combination of Gaussian noise vectors. Can similar effects be achieved via approximation methods such as Hermite polynomials? After all, Hermite polynomials are the optimal basis functions for Gaussian-related processes.
3. The authors state that the NCS method can be integrated into sampling strategies such as DDPM. Is NCS incompatible with DDIM out of inverse problems, a more commonly used deterministic sampling method? Can it be extended to Rectified flow in video generation models?
4. In the description of Figure 1, the authors state that NCS embeds the measurement score into the optimal noise within an ellipsoidal subspace, which is defined by the span of the noise codebook. I confuse that is this definition related to the one in Equation (10)? Why is the noise obtained from a simple weighted sum explained using the concept of an ellipsoidal subspace?
5. When the authors explain the advantages of NCS, they highlight a key feature: it adheres to conditional constraints while ensuring generation stability. However, most of the experimental results focus on image quality metrics such as PSNR. Although these metrics are suitable for evaluating inverse problems, I am also curious about NCS’s performance in terms of conditional compliance—for example, metrics like image-text alignment.