With neural diffusion models, we can generate samples from the unconditional distribution $p (x_{τ_{0}}) \approx p (x)$ . To solve inverse problems, however, we need to sample from the posterior $p (x_{τ_{0}} ∣ y)$ .

There are lots of ways we might try to condition, differing sometimes only in emphasis.

1 Notation

First, let us fix notation. I’ll use a slight variant of the notation from the denoising diffusion SDE notebook. Because I need $t$ for other things, we’ll talk about $τ$ for the pseudo-time grid, and use $τ_{0} = 0 < τ_{1} < \dots < τ_{T} = 1$ to index the discrete pseudo-time grid. We write $x_{τ_{i}}$ for the state at time $τ_{i}$ .

For simplicity, we’ll assume a variance-preserving (VP) diffusion SDE. We corrupt data $x_{τ_{0}} = x_{0} \sim p_{d a t a} (x)$ by the variance-preserving SDE

$d x_{τ} = - \frac{1}{2} β (τ) x_{τ} d τ + \sqrt{β (τ)} d W_{τ},$

or in discrete form for each step $τ_{i} \to τ_{i + 1}$ :

$x_{τ_{i + 1}} = \sqrt{1 - β_{i + 1}} x_{τ_{i}} + \sqrt{β_{i + 1}} ε, ε \sim N (0, I) .$ Write $Δ τ := τ_{i} - τ_{i - 1}$ (uniform unless stated otherwise).

We also define the convenience terms $\bar{α} (τ) = \exp (- \int_{0}^{τ} β (s) d s), so σ (τ)^{2} = 1 - \bar{α} (τ) .$ and $β_{i} := 1 - \frac{\bar{α} (τ_{i})}{\bar{α} (τ_{i - 1})} .$

1.1 Score Network & Training

We train $s_{θ} (x, τ)$ to approximate the time-indexed score $\nabla_{x_{τ}} \log p_{τ} (x_{τ})$ by minimizing the denoising loss

$L (θ) = E ‖ \nabla_{x_{τ}} \log p_{τ} (x_{τ}) - s_{θ} (x_{τ}, τ) ‖^{2} .$

where $x_{τ} = \sqrt{\bar{α} (τ)} x_{0} + \sqrt{1 - \bar{α} (τ)} ε$ .

noise vs score

Equivalently, we can parametrize $s_{θ}$ directly to predict the noise. In a VP diffusion the conditional distribution of the noise $ε$ given the noisy point $x_{τ}$ is Gaussian with mean $\begin{aligned} E [ε ∣ x_{τ}] & = - \frac{σ (τ)}{\sqrt{\bar{α} (τ)}} \nabla_{x_{τ}} \log p_{τ} (x_{τ}) \\ \nabla_{x_{τ}} \log p_{τ} (x_{τ}) & = - \frac{\sqrt{\bar{α} (τ)}}{σ (τ)} ε . \end{aligned}$ Hence predicting the noise (the “ $ε_{θ}$ ” parametrisation) or predicting the score (the “ $s_{θ}$ ” parametrisation) carries the same information up to the known factor $\sqrt{\bar{α} (τ)} / σ (τ) = σ (τ)$ .

1.2 Reverse-Time Sampling

The reverse SDE is as follows. Integrate the reverse SDE from $τ_{T} = 1$ down to $τ_{0} = 0$

$d x_{τ} = [- \frac{1}{2} β (τ) x_{τ} - β (τ) \nabla_{x_{τ}} \log p_{τ} (x_{τ})] d τ + \sqrt{β (τ)} d {\bar{W}}_{τ},$ using $s_{θ} (x, τ) \approx \nabla \log p_{τ} (x)$ . ( ${\bar{W}}_{τ}$ is an independent time reversal of the Wiener process $W_{τ}$ .)

Alternatively, we can use the deterministic / probability-flow ODE:

$d x_{τ} = [- \frac{1}{2} β (τ) x_{τ} - β (τ) \nabla_{x_{τ}} \log p_{τ} (x_{τ})] d τ,$

with initial draw $x_{_T}N(0,I). This yields the same marginals $p_{τ}$ without injecting extra noise at each step (which is insane and not at all obvious to me).

On our $τ$ grid, the discrete time DDPM reverse update becomes, for $i = T, \dots, 1$ :

$x_{τ_{i - 1}} = \frac{1}{\sqrt{1 - β_{i}}} (x_{τ_{i}} - β_{i} s_{θ} (x_{τ_{i}}, τ_{i})) + \sqrt{{\tilde{β}}_{i}} ζ, ζ \sim N (0, I) .$ We introduced here ${\tilde{β}}_{i} := β_{i} (\frac{1}{\bar{α} (τ_{i - 1})} - \frac{1}{\bar{α} (τ_{i})})$ (the “posterior” variance).

The DDIM variant removes $ζ$ for a deterministic two-step inversion.

2 Generic conditioning

Here is a quick rewrite of Rozet and Louppe (2023b). Note I have updated the notation to match the rest of this notebook.

We could train a conditional score network $s_{ϕ} (x_{τ_{i}}, τ_{i} ∣ y)$ to approximate the posterior score $\nabla_{x_{τ_{i}}} \log p (x_{τ_{i}} ∣ y)$ and plug it into the reverse SDE. But this requires $(x, y)$ pairs during training and re‐training whenever the observation model $p (y ∣ x)$ changes.

Instead, many have observed (Song, Sohl-Dickstein, et al. 2022; Adam et al. 2022; Chung et al. 2023; Kawar, Vaksman, and Elad 2021; Song, Shen, et al. 2022) that by Bayes’ rule the posterior score decomposes as $\nabla_{x_{τ_{i}}} \log p (x_{τ_{i}} ∣ y) = \nabla_{x_{τ_{i}}} \log p (x_{τ_{i}}) + \nabla_{x_{τ_{i}}} \log p (y ∣ x_{τ_{i}}) .$ Since the prior score $\nabla_{x_{τ_{i}}} \log p (x_{τ_{i}})$ is well‐approximated by a single score network $s_{ϕ} (x_{τ_{i}}, τ_{i})$ , the remaining task is to estimate the likelihood score $\nabla_{x_{τ_{i}}} \log p (y ∣ x_{τ_{i}})$ .

Assuming a differentiable measurement operator $A$ and Gaussian observations $p (y ∣ x) = N (y ∣ A (x), Σ_{y})$ , Chung et al. (2023) propose approximating $p (y ∣ x_{τ_{i}}) = \int p (y ∣ x) p (x ∣ x_{τ_{i}}) d x \approx N (y ∣ A (\hat{x} (x_{τ_{i}})), Σ_{y}),$ where the denoised mean $\hat{x} (x_{τ_{i}}) = E [x ∣ x_{τ_{i}}]$ is given by Tweedie’s formula (Efron 2011; Kim and Ye 2021): $\begin{aligned} E [x ∣ x_{τ_{i}}] & = E [x_{0} ∣ x_{τ}] = \frac{x_{τ} - \sqrt{1 - \bar{α} (τ)} s_{θ} (x_{τ}, τ)}{\sqrt{\bar{α} (τ)}} \end{aligned}$ Because the log‐likelihood of a multivariate Gaussian is analytic and $s_{ϕ} (x_{τ_{i}}, τ_{i})$ is differentiable, we can compute $\nabla_{x_{τ_{i}}} \log p (y ∣ x_{τ_{i}})$ in a zero‐shot fashion—without training any network beyond the unconditional score model $s_{ϕ}$ .

Note that this last assumption is strong; probably too strong for the models I would bother using diffusions on. Don’t worry we can get fancier and more effective.

3 Ensemble Score Conditioning

A simple trick that sometimes works (F. Bao, Zhang, and Zhang 2024a) but is biased. TBC.

4 Sequential Monte Carlo

This seems to be SOTA?

LLM-aided summary of Wu et al. (2024):

We recall standard SMC / Particle Filtering:

Goal: sample from a sequence of distributions ${ν_{i}}_{i = 0}^{T}$ , ending in some target $ν_{0}$ .
Particles: maintain $K$ samples (particles) ${x_{i}^{k}}_{k = 1}^{K}$ with weights ${w_{i}^{k}}$ .
Iterate for $i = T \to 0$ :
- Resample particles according to $w_{i + 1}^{k}$ to focus on high-probability regions.
- Propose $x_{i}^{k} \sim r_{i} (x_{i} ∣ x_{i + 1}^{k})$ .
- Weight each by
  
  $w_{i}^{k} = \underset{importance weight}{\underset{⏟}{\frac{target density at x_{i}^{k}}{proposal density at x_{i}^{k}}}} .$
Convergence: as $K \to \infty$ , the weighted ensemble approximates the true target exactly.

In a diffusion model, we can view the reverse noising chain

$p_{θ} (x_{0 : T}) = p (x_{T}) \prod_{i = 1}^{T} p_{θ} (x_{τ_{i - 1}} ∣ x_{τ_{i}})$

as exactly such a sequential model over $x_{τ_{T}} \to x_{0}$ , where $ν_{i}$ is the joint $p_{θ} (x_{0}, \dots, x_{1})$ marginalized forward to $τ_{i}$ .

To sample conditionally $p_{θ} (x_{0} ∣ y)$ , we treat the conditioning as part of the final target and apply SMC. However, if we naïvely run SMC with the unconditional transition kernels

$r_{i} (x_{τ_{i - 1}} ∣ x_{τ_{i}}) = p_{θ} (x_{τ_{i - 1}} ∣ x_{τ_{i}})$

and only tack on a final weight $w_{0} \propto p (y ∣ x_{0})$ , we need an astronomical number of particles, since most will get near-zero weight whenever the prior $p_{θ} (x_{0})$ is substantially unlikely compared to the conditional $p_{θ} (x_{0} ∣ y)$ .

Twisting is a classic SMC technique which solves this problem, introducing a sequence of auxiliary functions ${{\tilde{p}}_{θ} (y ∣ x_{τ_{i}})}_{i = 0}^{T}$ to re-weight proposals at every time-step, not just at the end. The ideal optimal choice at step $i$ would be

$r_{i}^{*} (x_{τ_{i - 1}} ∣ x_{τ_{i}}) \propto p_{θ} (x_{τ_{i - 1}} ∣ x_{τ_{i}}) p_{θ} (y ∣ x_{τ_{i - 1}}),$

which—if we could sample it—would make SMC exact with a single particle. However, $p_{θ} (y ∣ x_{τ_{i - 1}})$ is itself intractable.

TDS replaces the optimal twisting $p_{θ} (y ∣ x_{i})$ with a tractable surrogate based on the denoising network ${\hat{x}}_{0} (x_{i})$ , which estimates the denoised $x_{0}$ from the noisy $x_{i}$ .

${\tilde{p}}_{θ} (y ∣ x_{τ_{i}}) = p (y ∣ {\hat{x}}_{0} (x_{τ_{i}})),$

i.e. we evaluate the observation-likelihood at the diffusion denoiser’s one-step posterior mean estimate ${\hat{x}}_{0}$ . Since ${\hat{x}}_{0} (x_{τ}) \approx E [x_{0} ∣ x_{τ}]$ , this becomes increasingly accurate as $τ \to 0$ . Define $σ_{i}^{2} := σ (τ_{i})^{2} = 1 - \bar{α} (τ_{i}) .$

Twisted proposal from $τ_{i} \to τ_{i - 1}$ :

${\tilde{r}}_{i} (x_{τ_{i - 1}} ∣ x_{τ_{i}}, y) = N (x_{τ_{i - 1}}; \underset{mean}{\underset{⏟}{x_{τ_{i}} + \underset{“guided” drift}{\underset{⏟}{σ_{i}^{2} s_{i} (x_{τ_{i}}, y)}}}}, σ_{i}^{2} I),$

where

$s_{i} (x_{τ_{i}}, y) = s_{θ} (x_{τ_{i}}, τ_{i}) + \nabla_{x_{τ_{i}}} \log {\tilde{p}}_{θ} (y ∣ x_{τ_{i}}) .$
Twisted weight for each particle:

$w_{τ_{i - 1}} = \frac{p_{θ} (x_{τ_{i - 1}} ∣ x_{τ_{i}}) {\tilde{p}}_{θ} (y ∣ x_{τ_{i - 1}})}{{\tilde{p}}_{θ} (y ∣ x_{τ_{i}}) {\tilde{r}}_{i} (x_{τ_{i - 1}} ∣ x_{τ_{i}}, y)} .$
Twisted Sister:

This corrects for using the surrogate twisting and ensures asymptotic exactness as $K \to \infty$ .

In early steps ( $i \approx T$ ) the surrogate ${\tilde{p}}_{θ} (y ∣ x_{τ_{i}})$ may be very broad — twisting is mild. Then, in late steps ( $i \to 0$ ) ${\hat{x}}_{0} (x_{τ_{i}})$ is accurate, so ${\tilde{p}}_{θ} (y ∣ x_{τ_{i}}) \approx p_{θ} (y ∣ x_{τ_{i}})$ and the proposals are nearly optimal. Resampling in between keeps the particle cloud focused on regions consistent with both the diffusion prior and the conditioning $y$ .

In practice, we often need a surprisingly small number of particles; even 2–8 particles often suffice to outperform heuristic conditional samplers (like plain classifier guidance or “replacement” inpainting).

5 (Conditional) Schrödinger Bridge

Shi et al. (2022) introduced the Conditional Schrödinger Bridge (CSB), which is a natural extension of the Schrödinger Bridge.

We seek a path-measure $π^{*} (x_{0 : T} ∣ y)$ minimising

$KL (π (\cdot ∣ y) ∥ p (x_{0 : T}))$

subject to

Start at $τ_{T}$ : $π_{τ_{T}} (x_{T} ∣ y) = N (x_{T}; 0, I)$ ,
End at $τ_{0}$ : $π_{τ_{0}} (x_{0} ∣ y) = p (x_{0} ∣ y)$ .

Here $p (x_{0 : T}) = p (x_{T}) \prod_{i} p (x_{τ_{i - 1}} ∣ x_{τ_{i}})$ is the unconditional forward noising chain.

5.1 Amortized IPF Algorithm

We parameterize two families of drift networks that take $y$ as input:

$B_{i}^{n} (x, y) and F_{i}^{n} (x, y) for i = 1, \dots, T, n = 0, \dots, L .$

We alternate two KL-projection steps:

Backward half-step ( $τ_{T} \to τ_{0}$ , enforce prior):

$π^{2 n + 1} (\cdot ∣ y) = \arg min_{π} KL (π (\cdot ∣ y) ∥ π^{2 n} (\cdot ∣ y)) s.t. π_{τ_{T}} (x_{T} ∣ y) = N (0, I) .$

Fit $B_{i}^{n + 1} (x, y)$ by matching the backward SDE induced by $π^{2 n + 1}$ .
Forward half-step ( $τ_{0} \to τ_{T}$ , enforce posterior):

$π^{2 n + 2} (\cdot ∣ y) = \arg min_{π} KL (π (\cdot ∣ y) ∥ π^{2 n + 1} (\cdot ∣ y)) s.t. π_{τ_{0}} (x_{0} ∣ y) = p (x_{0} ∣ y) .$

Fit $F_{i}^{n + 1} (x, y)$ by matching the forward SDE induced by $π^{2 n + 2}$ .

After $L$ IPF iterations we obtain networks $B_{i}^{L} (x, y), F_{i}^{L} (x, y)$ whose composed bridge $π^{*} (\cdot ∣ y)$ exactly matches both endpoints for any $y$ .

We can imagine that the backward step “pins” the Gaussian prior; the forward step “pins” the conditional $p (x_{0} ∣ y)$ . We now define the composed conditional bridge as the concatenation of both of the aforeπiclrmentioned:

Initial draw at $τ_{T}$ :

$x_{τ_{T}} \sim p (x_{τ_{T}}) = N (0, I) .$
Backward transitions for $i = T, T - 1, \dots, 1$ :

$x_{τ_{i - 1}} \sim \underset{π_{back, i}^{*} (x_{τ_{i - 1}} ∣ x_{τ_{i}}, y)}{\underset{⏟}{N (x_{τ_{i}} + Δ τ B_{i}^{L} (x_{τ_{i}}, y), Δ τ I)}} .$
Forward transitions for $i = 1, 2, \dots, T$ :

$x_{τ_{i}} \sim \underset{π_{for, i}^{*} (x_{τ_{i}} ∣ x_{τ_{i - 1}}, y)}{\underset{⏟}{N (x_{τ_{i - 1}} + Δ τ F_{i}^{L} (x_{τ_{i - 1}}, y), Δ τ I)}} .$

Hence the full path-measure is

$π^{*} (x_{0 : T} ∣ y) = p (x_{τ_{T}}) \prod_{i = T}^{1} π_{back, i}^{*} (x_{τ_{i - 1}} ∣ x_{τ_{i}}, y) \times \prod_{i = 1}^{T} π_{for, i}^{*} (x_{τ_{i}} ∣ x_{τ_{i - 1}}, y),$

which by construction satisfies both endpoint constraints for any $y$ . CSB finds the “smoothest” (in the sense of being entropy-regularized) stochastic flow between noise and data-posterior. This seems intuitively fine, but my reasoning here is vibes-based. I need to read the paper better to get a proper grip on it.

5.2 Sampling with the Learned Conditional Bridge

To draw $x_{0} \sim p (x_{0} ∣ y)$ :

Initialize $x_{τ_{T}} \sim N (0, I)$ .
Backward integrate the learned SDE (or its probability-flow ODE) from $τ = T$ down to $0$ :

$d x_{τ} = B_{i}^{L} (x_{τ}, y) d τ + \sqrt{Δ τ} d W_{τ} for τ \in [τ_{i - 1}, τ_{i}] .$
(Optionally) Forward integrate with $F_{i}^{L} (x, y)$ to refine or to compute likelihoods.

Because $B_{i}^{L}$ and $F_{i}^{L}$ both depend on $y$ , the same trained model applies to arbitrary observations.

Amortized CSB encodes the observation $y$ directly into every drift net $B_{i} (x, y)$ and $F_{i} (x, y)$ . There is no per-instance retraining or importance weights — once the joint IPF training over $(x_{0}, y)$ is done, we can plug in any new $y$ and run the sampler.

6 Computational Trade-offs Of Those Last Two

Aspect	Twisted SMC (TDS)	Amortized CSB
Training	Only train denoiser ${\hat{x}}_{0}$	Train $2 \times L$ drift nets on $(x, y)$
Inference cost	$K$ particles × $T$ steps	Single trajectory over $T$ steps
Exactness	As $K \to \infty$ , exact	as IPF perfectly trained, exact

7 Consistency models

Song et al. (2023)

8 Inpainting

If we want coherence with part of an existing image, we call that inpainting and there are specialized methods for it (Ajay et al. 2023; Grechka, Couairon, and Cord 2024; Liu, Niepert, and Broeck 2023; Lugmayr et al. 2022; Sharrock et al. 2022; Wu et al. 2024; Zhang et al. 2023).

9 Reconstruction/inversion

Perturbed and partial observations; misc methods therefor (Choi et al. 2021; Kawar et al. 2022; Nair, Mei, and Patel 2023; Peng et al. 2024; Xie and Li 2022; Zhao et al. 2023; Song, Shen, et al. 2022; Zamir et al. 2021; Chung et al. 2023; Sui et al. 2024).

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