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Beyond DPS — A Landscape of Consistency Methods

DPS enforces measurement consistency via soft guidance (a gradient step at each diffusion iteration). Several alternative methods enforce consistency through different mechanisms: SVD-based null-space preservation (DDRM), range-null space decomposition (DDNM), manifold projection (MCG), and denoising-diffusion restoration (DiffPIR). Each method offers a different tradeoff between reconstruction quality, computational cost, and assumptions on the forward model $\mathbf{A}$ .

Definition:
Denoising Diffusion Restoration Models (DDRM)

DDRM exploits the SVD of the forward model $\mathbf{A} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H$ to enforce exact data consistency in the measurement subspace while allowing the diffusion model to fill in the null space.

At each reverse step, DDRM decomposes the update into:

Range-space component: set to match the measurements via $\boldsymbol{\Sigma}^\dagger \mathbf{U}^H \mathbf{y}$
Null-space component: sampled from the conditional prior via the reverse diffusion process

The result is a reconstruction that satisfies $\mathbf{A}\hat{\mathbf{x}}_0 = \mathbf{y}$ exactly (in the noiseless case), with the prior filling in the missing information.

DDRM requires the SVD of $\mathbf{A}$ , which is tractable for structured operators (convolution, subsampling, inpainting) but expensive for general sensing matrices. For RF imaging, the sensing matrix is often too large for explicit SVD computation.

Definition:
Denoising Diffusion Null-Space Model (DDNM)

DDNM is a zero-shot (no fine-tuning) method that enforces data consistency by replacing the range-space components of the Tweedie estimate at each step:

$\hat{\mathbf{x}}_0^{\text{DDNM}} = \hat{\mathbf{x}}_0 - \mathbf{A}^\dagger(\mathbf{A}\hat{\mathbf{x}}_0 - \mathbf{y}) + \mathbf{A}^\dagger\mathbf{n}_{\text{approx}},$

where $\hat{\mathbf{x}}_0$ is the Tweedie estimate from the unconditional diffusion model, $\mathbf{A}^\dagger$ is the pseudoinverse, and $\mathbf{n}_{\text{approx}}$ accounts for measurement noise.

The key insight: the correction $\mathbf{A}^\dagger(\mathbf{A}\hat{\mathbf{x}}_0 - \mathbf{y})$ only modifies the range space of $\mathbf{A}$ , leaving the null-space content (generated by the prior) untouched.

Definition:
Manifold Constrained Gradients (MCG)

MCG combines the DPS gradient with a hard projection step at each diffusion iteration:

Compute the DPS guidance gradient (soft consistency)
Apply a projection onto the measurement-consistent manifold: $\mathbf{x}_{t-1} \leftarrow \mathbf{x}_{t-1} - \mathbf{A}^{H}(\mathbf{A}\mathbf{A}^{H})^{-1}(\mathbf{A}\mathbf{x}_{t-1} - \mathbf{y})$

The projection ensures exact measurement consistency after each step, while the gradient provides a smooth trajectory on the data manifold.

The projection requires $(\mathbf{A}\mathbf{A}^{H})^{-1}$ , which may not exist or may be ill-conditioned. In practice, MCG uses a regularised pseudoinverse or applies the projection only in the range space.

Definition:
DiffPIR (Diffusion-Based Plug-and-Play Image Restoration)

DiffPIR integrates diffusion models into the half-quadratic splitting (HQS) framework from Chapter 21:

Data-fidelity step: solve $\mathbf{z}_k = \arg\min_\mathbf{z}\;\frac{1}{2\sigma^2_{n}}\|\mathbf{y} - \mathbf{A}\mathbf{z}\|^2 + \frac{\rho}{2}\|\mathbf{z} - \hat{\mathbf{x}}_k\|^2$
Prior step: run a few reverse diffusion steps starting from $\mathbf{z}_k$ to produce $\hat{\mathbf{x}}_{k+1}$

The diffusion model replaces the explicit denoiser in PnP, but operates over multiple noise levels within each HQS iteration. This provides a principled connection between PnP methods (Chapter 21) and diffusion-based reconstruction.

DiffPIR is computationally cheaper than full DPS because it runs only a few diffusion steps per HQS iteration rather than a full $T$ -step reverse process. Typical configurations use 15--20 HQS iterations with 5--10 diffusion steps each.

Theorem: Null-Space Preservation Theorem

Let $\mathbf{A} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H$ be the SVD of the forward model with $\text{rank}(\mathbf{A}) = r < n$ (underdetermined system). Any reconstruction $\hat{\mathbf{x}}$ consistent with the measurements ( $\mathbf{A}\hat{\mathbf{x}} = \mathbf{y}$ ) can be written as:

$\hat{\mathbf{x}} = \underbrace{\mathbf{V}_r\boldsymbol{\Sigma}_r^{-1}\mathbf{U}_r^H\mathbf{y}}_{\text{range-space (determined by data)}} + \underbrace{\mathbf{V}_\perp\boldsymbol{\eta}}_{\text{null-space (free parameter)}},$

where $\mathbf{V}_r$ contains the first $r$ right singular vectors, $\mathbf{V}_\perp$ spans the null space of $\mathbf{A}$ , and $\boldsymbol{\eta} \in \mathbb{R}^{n-r}$ is arbitrary.

The role of the diffusion prior is to provide an informative distribution over $\boldsymbol{\eta}$ , filling in the $(n - r)$ -dimensional null space with plausible content.

The measurements determine $r$ components of $\hat{\mathbf{x}}$ (the range space). The remaining $n - r$ components (the null space) are invisible to the measurements and must be filled by the prior. A stronger prior produces a more informative null-space estimate; a weaker prior produces a blurrier or noisier estimate in those components.

Proof

SVD decomposition

Write $\mathbf{A} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H$ and partition $\mathbf{V} = [\mathbf{V}_r \mid \mathbf{V}_\perp]$ . Any $\hat{\mathbf{x}} \in \mathbb{R}^n$ can be decomposed as $\hat{\mathbf{x}} = \mathbf{V}_r\mathbf{a} + \mathbf{V}_\perp\boldsymbol{\eta}$ .

Impose measurement consistency

$\mathbf{A}\hat{\mathbf{x}} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H(\mathbf{V}_r\mathbf{a} + \mathbf{V}_\perp\boldsymbol{\eta}) = \mathbf{U}_r\boldsymbol{\Sigma}_r\mathbf{a} = \mathbf{y}$ , since $\mathbf{V}^H\mathbf{V}_\perp = \mathbf{0}$ for the null-space vectors. Therefore $\mathbf{a} = \boldsymbol{\Sigma}_r^{-1}\mathbf{U}_r^H\mathbf{y}$ , which is uniquely determined, while $\boldsymbol{\eta}$ is free. $\blacksquare$

Comparison of Measurement-Consistent Diffusion Methods

Method	Consistency Type	Requires SVD	NFEs	Strengths
DPS	Soft (gradient)	No	$\sim 2T$	General $\mathbf{A}$ ; posterior sampling
DDRM	Exact (null-space)	Yes	$\sim T$	Exact consistency; no guidance scale
DDNM	Exact (pseudoinverse)	Yes ( $\mathbf{A}^\dagger$ )	$\sim T$	Zero-shot; no fine-tuning
MCG	Soft + hard projection	Yes ( $\mathbf{A}^{H}$ )	$\sim 2T$	Manifold-aware; strong consistency
DiffPIR	HQS + diffusion prior	No	$\sim 100$ -- $200$	Faster; connects to PnP framework

Example: DDRM for Image Inpainting

Consider image inpainting where $\mathbf{A} = \mathbf{M}$ is a diagonal binary mask ( $M_{ii} = 1$ if pixel $i$ is observed, $0$ otherwise). Describe the DDRM reconstruction.

Solution

SVD of the mask

The mask $\mathbf{M}$ is already diagonal with singular values $\sigma_i \in \{0, 1\}$ . The range space consists of observed pixels; the null space consists of masked pixels.

Range-space component

For observed pixels ( $M_{ii} = 1$ ): set $\hat{x}_i = y_i$ (the observed value).

Null-space component

For masked pixels ( $M_{ii} = 0$ ): the diffusion model generates plausible content conditioned on the surrounding observed pixels. Each run of DDRM produces a different plausible fill-in.

Key advantage

DDRM guarantees that observed pixels are exactly preserved ( $\hat{x}_i = y_i$ for $M_{ii} = 1$ ), while the diffusion prior provides a natural-looking completion of the masked region. No guidance scale $\zeta$ is needed.

Reconstruction Quality vs. Consistency Strength

Compare the reconstruction quality (PSNR) and measurement consistency ( $\|\mathbf{A}\hat{\mathbf{x}} - \mathbf{y}\|$ ) for different methods as a function of the measurement noise level. DPS with strong guidance achieves perfect consistency but may sacrifice image quality; DDRM achieves exact consistency by construction; DiffPIR provides a computationally efficient middle ground.

Parameters

Noise std

\sigma_n

0.05

Measurement ratio

m/n

0.5

Common Mistake: Hidden Cost of SVD-Based Methods

Mistake:

Assuming DDRM and DDNM are always more efficient than DPS because they avoid backpropagation through the score network.

Correction:

DDRM and DDNM require the SVD of $\mathbf{A}$ (or at least $\mathbf{A}^\dagger$ ), which costs $O(\min(m,n)^2\max(m,n))$ . For large-scale RF imaging problems where $\mathbf{A} \in \mathbb{C}^{m \times n}$ with $m, n > 10^4$ , this SVD is prohibitively expensive. In such cases, DPS (which requires only matrix-vector products $\mathbf{A}\mathbf{v}$ and $\mathbf{A}^{H}\mathbf{u}$ ) is the more practical choice.

Common Mistake: Hard Projection Can Disrupt the Diffusion Process

Mistake:

Applying hard measurement projection at every diffusion step to ensure $\mathbf{A}\mathbf{x}_t = \mathbf{y}$ throughout the reverse process.

Correction:

At intermediate diffusion times, $\mathbf{x}_t$ is a noisy version of $\mathbf{x}_0$ , and enforcing $\mathbf{A}\mathbf{x}_t = \mathbf{y}$ is not meaningful — the measurements correspond to $\mathbf{x}_0$ , not to $\mathbf{x}_t$ . Hard projection at intermediate steps can push $\mathbf{x}_t$ off the noisy data manifold, causing artefacts or divergence. MCG addresses this by combining soft gradient guidance with a projection applied only in the estimated clean-image space.

Quick Check

In DDRM, the diffusion model's role is to fill in which component of the reconstruction?

The range space of $\mathbf{A}$

The null space of $\mathbf{A}$

Both the range and null spaces equally

The measurement noise component

Correction:

The null space of

\mathbf{A}

The range-space component is determined by the measurements via $\boldsymbol{\Sigma}_r^{-1}\mathbf{U}_r^H\mathbf{y}$ . The null-space component ( $\mathbf{V}_\perp\boldsymbol{\eta}$ ) is invisible to the measurements and is filled by the diffusion prior.

DDRM (Denoising Diffusion Restoration Models)

A diffusion-based reconstruction method that uses the SVD of the forward model to separate the reconstruction into range-space (determined by measurements) and null-space (filled by the prior) components.

Key Takeaway

The landscape of measurement-consistent diffusion methods offers a spectrum from soft guidance (DPS) to exact consistency (DDRM/DDNM) to hybrid approaches (MCG, DiffPIR). The choice depends on the structure of $\mathbf{A}$ : SVD-based methods excel when the SVD is cheap (convolution, masking), while gradient-based methods (DPS) are preferred when $\mathbf{A}$ is a general large-scale operator — the typical situation in RF imaging.

Measurement-Consistent Diffusion Methods

Beyond DPS — A Landscape of Consistency Methods

Definition: Denoising Diffusion Restoration Models (DDRM)

Definition: Denoising Diffusion Null-Space Model (DDNM)

Definition: Manifold Constrained Gradients (MCG)

Definition: DiffPIR (Diffusion-Based Plug-and-Play Image Restoration)

Theorem: Null-Space Preservation Theorem

SVD decomposition

Impose measurement consistency

Comparison of Measurement-Consistent Diffusion Methods

Example: DDRM for Image Inpainting

SVD of the mask

Range-space component

Null-space component

Key advantage

Reconstruction Quality vs. Consistency Strength

Parameters

Common Mistake: Hidden Cost of SVD-Based Methods

Common Mistake: Hard Projection Can Disrupt the Diffusion Process

Quick Check

DDRM (Denoising Diffusion Restoration Models)

Key Takeaway

Definition:
Denoising Diffusion Restoration Models (DDRM)

Definition:
Denoising Diffusion Null-Space Model (DDNM)

Definition:
Manifold Constrained Gradients (MCG)

Definition:
DiffPIR (Diffusion-Based Plug-and-Play Image Restoration)