Aim 3: Posterior Sampling and Uncertainty
December 12, 2023
Testing DPS Step Size
non-Bayesian (i.e., \(g(t)^2\) is not applied to MLE step) update
\[x_{t-1} = x_t - [f(x,t) - (g(t)^2s_{\hat{\theta}}(x_t, t) + \eta\nabla_{x_t}\text{MLE}(y, x_t, t))] + g(t)dw\]
Testing DPS Step Size
\(\eta(t) = 0.1\)
\(\eta(t) = 1.0\)
\(\eta(t) = 8.0\)
Samples (\(I_0 = 1024\))
UQ
Faster Sampling with DDIM
DDIM update
\[x_s^{\text{DDIM}} = \alpha_s \hat{x}_0(x_t) + \sqrt{b^2_s - \sigma^2} \cdot \epsilon_{\theta}(x_t, t) + \sigma z,~z \sim \mathcal{N}(0, \mathbb{I})\]
with
\[\sigma = \eta^{\text{DDIM}} \frac{b_s^2}{b_t^2}\sqrt{1 - \frac{\alpha_t^2}{\alpha_s^2}}\]
\[x_t \mid x_0 \sim \mathcal{N}(\alpha_t x_0, b^2_t\mathbb{I})\]
DDIM-based DPS
\[x_s = x_s^{\text{DDIM}} + \eta^{\text{DPS}}\nabla_{x_t} \text{MLE}(y, x_t, t)\]
Samples with 200 Steps \((I_0 = 1024)\)
\(x\)
\(y\)
samples
\(\hat{u} - \hat{l}\)
Samples with 200 Steps \((I_0 = 1024)\)
\(x\)
\(y\)
samples
\(\hat{u} - \hat{l}\)
Samples with 200 Steps \((I_0 = 1024)\)
\(x\)
\(y\)
samples
\(\hat{u} - \hat{l}\)
Baselines: FBP, FBP + QR U-net
\(x\)
FBP(\(x\))
\(\hat{x}\)
\(\hat{l}\)
\(\hat{u}\)
\(\hat{u} - \hat{l}\)
Quantitative Comparison
Calibration Results
U-net
Gaussian
Poisson
Semantic Calibration
From
\[\mathcal{I}(y)_j = [\hat{l}_j - \lambda_j, \hat{u}_j + \lambda_{j}]\]
to
\[\mathcal{I}(y)_j = [\hat{l}_j - \lambda_{c_j}, \hat{u}_j + \lambda_{c_j}]\]
where
\[c_j = \underset{k \in [C]}{\arg\max}~p(k_j \mid y)\]
is the posterior semantic segmentation of pixel \(j\)
\[\downarrow\]
testing with background, body, and lungs
Semantic Calibration
conformalized uncertainty maps
\(\lambda\)
RCPS
K-RCPS
sem-RCPS
background
body
lungs
Beyond Gaussian Diffusion
Poisson DPS
\[p(y \mid x_t) \approx p(y \mid \hat{x}_0(x_t)) = \text{Pois}(y;I_0e^{-A\hat{x}_0(x_t)})\]
\[\downarrow\]
QUESTION
\(I_0 < I_{\text{max}}\)
\(y_0\) measured at \(I_0\)
can sample \(y_{\text{max}}\) at \(I_{\text{max}}\)?
\[\downarrow\]
\[y_t \mid \mu \sim \text{Pois}(I(t)e^{-A\mu})\]
is a Poisson Point Process
Stochastic Localization [Montanari, '23]
If
\[(Y_{t})_{t \geq 0}\big\lvert_{x} \sim \text{PPP}(x~\text{d}t) \implies Y_t \sim \text{Pois}(tx)\]
probability of not adding 1 is
\[\mathbb{P}[Y_{t + \delta} = y \mid Y_t = y] = 1 - \delta m(y,t) + o(\delta)\]
probability of adding 1 is
\[\mathbb{P}[Y_{t + \delta} = y + 1 \mid Y_t = t] = \delta m(y,t) + o(\delta)\]
then
and
where
\[m(y,t) = \mathbb{E}[X \mid Y_t = y]\]
Stochastic Localization [Montanari, '23]
Learn \(m_{\theta}(y,t)\) with
\[\hat{\theta} = \underset{\theta}{\arg\min}~\mathbb{E}_{\mu, t, y \sim \text{Pois}(te^{-A\mu})}[\| e^{-A\mu} - m_{\theta}(y,t)\|^2]\]
given \(y_0\) sample \(y_T\) with
\[y_{t + \text{d}t} = y_t + \text{Pois}(m_{\hat{\theta}}(y_t, t)~\text{d}t)\]
obtain \(x_T\) via FBP on \(y_T\)
🤔
Stochastic Localization [Montanari, '23]
Possible things that are going wrong:
1. Exponential range makes optimization problem hard
2. PPP is a discrete process, once a count has appeared, it is difficult for it to go away
(ideas to fix: add and remove counts? smooth process by adding additive Gaussian noise?)
3. Tweedie estimate of Poisson lives in log space, so things are not as nice as in Gaussian diffusion
(there is no score here)
[12/12/23] Aim 3: Posterior Sampling and Uncertainty
By Jacopo Teneggi
