Modern problems in trustworthy medical imaging

Jeremias Sulam

June 2025

50 years ago ... 

first CT scan
ELECTRIC & MUSICAL INDUSTRIES

50 years ago ... 

imaging
diagnostics

complete hardware & software description

human expert diagnosis and recommendations

imaging was "simple"

... 50 years forward 

Data

Compute & Hardware

Sensors & Connectivity

Research & Engineering

... 50 years forward 

data-driven  imaging
automatic analysis and rec.
societal implications

Data

Compute & Hardware

Sensors & Connectivity

Research & Engineering

Data

Compute & Hardware

Sensors & Connectivity

Research & Engineering

... 50 years forward 

data-driven  imaging
automatic analysis and rec.
societal implications

Problems in trustworthy biomedical imaging

inverse problems

uncertainty quantification

robustness

generalization

policy & regulation

Demographic fairness

hardware & protocol optimization

model-agnostic interpretability

inverse problems

uncertainty quantification

model-agnostic interpretability

robustness

generalization

policy & regulation

demographic fairness

hardware & protocol optimization

data-driven  imaging
automatic analysis and rec.
societal implications

Problems in trustworthy biomedical imaging

Demographic fairness

Inputs (features):                            \(X\in \mathcal X \subset \mathbb R^d\)

Responses (labels):                         \(Y\in \{0,1\}\)

Sensitive attributes:                      \(Z \in \mathbb R^k \)  (sex, race, age, etc)

Random variables sampled:      \((X,Y,Z) \sim \mathcal D\)

Eg: \(Z_1: \) biological sex, \(X_1: \) BMI, then

\( g(Z,X) = \boldsymbol{1}\{Z_1 = 1 ~\texttt{and}~ X_1 > 35 \}: \) women with BMI > 35

Goal: ensure that \(f\) is fair w.r.t groups \(g \in \mathcal G\)

Demographic fairness

Group memberships      \( \mathcal G = \{ g(X,Z) \mapsto \{0,1\} \} \)

Predictor     \( f(X) : \mathcal X \to [0,1]\)  (e.g. likelihood of X having disease Y)

  • Group/Associative Fairness
           Predictors should not have very different (error) rates among groups
         [Calders et al, '09][Zliobaite, '15][Hardt et al, '16]
  • Individual Fairness
           Similar individuals/patients should have similar outputs
           [Dwork et al, '12][Fleisher, '21][Petersen et al, '21]
  • Causal Fairness
           Predictors should be fair in a counter-factual world
          [Nabi & Shpitser, '18][Nabi et al, '19][Plecko & Bareinboim, '22]
  • Multiaccuracy/Multicalibration
           Predictors should be approximately unbiased/calibrated for every group
           [Kim et al, '20][Hebert-Johnson et al, '18][Globus-Harris et at, 22]

Demographic fairness

Demographic fairness

  • Group/Associative Fairness
           Predictors should not have very different (error) rates among groups
     

Equal Opportunity

Equal True Positive Rates (TPR) across groups

\(\mathbb P[\hat Y=1 | Y=1, G_{\texttt{age}\leq 60}=0] = \mathbb P[\hat Y = 1 | Y=1, G_{\texttt{age}>60}=1]\)

\(\Delta \text{TPR}_\text{age} = \left| \text{TPR}_{\texttt{age}\leq 60} - \text{TPR}_{\texttt{age}>60}\right| \leq \alpha \)

Demographic fairness

  • Group/Associative Fairness
           Predictors should not have very different (error) rates among groups
     

Multiaccuracy

Similar accuracy across different groups

  • Multiaccuracy/Multicalibration
           Predictors should be approximately unbiased/calibrated for every group

 \(\text{MA} (f,g) = \big| \mathbb E [ g(X,Z) (f(X) - Y) ] \big|  \)

\(f\) is \(\alpha\)-multiaccurate if   \( \max_{g\in\mathcal G} \text{MA}(f,g) \leq \alpha \)

Demographic fairness

Multiaccuracy

Similar accuracy across different groups

 \(\text{MA} (f,g) = \big| \mathbb E [ g(X,Z) (f(X) - Y) ] \big|  \)

\(f\) is \(\alpha\)-multiaccurate if   \( \max_{g\in\mathcal G} \text{MA}(f,g) \leq \alpha \)

Example: predicting high risk of complications from flu based on clinical features

Demographic fairness

Multiaccuracy

Similar accuracy across different groups

 \(\text{MA} (f,g) = \big| \mathbb E [ g(X,Z) (f(X) - Y) ] \big|  \)

\(f\) is \(\alpha\)-multiaccurate if   \( \max_{g\in\mathcal G} \text{MA}(f,g) \leq \alpha \)

Example: predicting high risk of complications from flu based on clinical features

Observation:
Evaluation for fairness notions requires samples over \((X,Y,Z)\)

Demographic fairness

Multiaccuracy

Similar accuracy across different groups

 \(\text{MA} (f,g) = \big| \mathbb E [ g(X,Z) (f(X) - Y) ] \big|  \)

\(f\) is \(\alpha\)-multiaccurate if   \( \max_{g\in\mathcal G} \text{MA}(f,g) \leq \alpha \)

Example: predicting high risk of complications from flu based on clinical features

Observation:
Evaluation for fairness notions requires samples over \((X,Y,Z)\)

Problem: This is not always possible...

Problem: This is not always possible...

Problem: This is not always possible...

sex and race attributes missing

  • We might want to conceal \(Z\) on purpose, or might need to

?

?

?

?

?

We observe samples  over \((X,Y)\) to obtain \(\hat Y = f(X)\) for \(Y\) 

Fairness in partially observed regimes

\( \text{MSE}(f) = \mathbb E [(Y-f(X))^2 ] \)

A developer provides us with proxies  \( \color{Red} \hat{g} : \mathcal X \to \{0,1\} \)

\( \text{err}(\hat g) = \mathbb P [({\color{Red}\hat g(X)} \neq {\color{blue}g(X,Z)} ] \)

Can we use \(\hat g\) to measure (and correct) for fairness metrics?

 [Awasti et al, '21][Kallus et al, '22][Zhu et al, '23][Bharti et al, '24]

We observe samples  over \((X,Y)\) to obtain \(\hat Y = f(X)\) for \(Y\) 

Fairness in partially observed regimes

\( \text{MSE}(f) = \mathbb E [(Y-f(X))^2 ] \)

A developer provides us with proxies  \( \color{Red} \hat{g} : \mathcal X \to \{0,1\} \)

\( \text{err}(\hat g) = \mathbb P [({\color{Red}\hat g(X)} \neq {\color{blue}g(X,Z)} ] \)

 [Awasti et al, '21][Kallus et al, '22][Zhu et al, '23][Bharti et al, '24]

Can we use \(\hat g\) to measure (and correct) for fairness metrics?

Fairness in partially observed regimes

Theorem [Bharti, Clemens-Sewall, Yi, Sulam]

With access to \((X,Y)\sim \mathcal D_{\mathcal{XY}}\), proxies \( \hat{\mathcal G}\) and predictor \(f\)

 

\[ \max_{\color{Blue}g\in\mathcal G} \text{MA}(f,{\color{blue}g}) ~\leq ~\max_{\color{red}\hat g\in \hat{\mathcal{G}} } \text{MA}(f,{\color{red}\hat{g}}) + B(f,{\color{red}\hat g})  \]

 

with \(B(f,\hat g) = \min \left( \text{err}(\hat g), \sqrt{MSE(f)\cdot \text{err}(\hat g)} \right) \)

true error

worst possible error

  • Practical/computable upper bounds

Fairness in partially observed regimes

  • Practical/computable upper bounds

Theorem [Bharti, Clemens-Sewall, Yi, Sulam]

With access to \((X,Y)\sim \mathcal D_{\mathcal{XY}}\), proxies \( \hat{\mathcal G}\) and predictor \(f\)

 

\[ \max_{\color{Blue}g\in\mathcal G} MA(f,{\color{blue}g}) ~\leq ~\max_{\color{red}\hat g\in \hat{\mathcal{G}} } MA(f,{\color{red}\hat{g}}) + B(f,{\color{red}\hat g})  \]

 

with \(B(f,\hat g) = \min \left( \text{err}(\hat g), \sqrt{MSE(f)\cdot \text{err}(\hat g)} \right) \)

true error

worst possible error

Fairness in partially observed regimes

  • Correcting w.r.t \(\hat{\mathcal G}\) provably improves upper bound

                                                               [Gopalan et al. (2022)][Roth (2022)][Bharti et al (2025)]

  • Practical/computable upper bounds

Theorem [Bharti, Clemens-Sewall, Yi, Sulam]

With access to \((X,Y)\sim \mathcal D_{\mathcal{XY}}\), proxies \( \hat{\mathcal G}\) and predictor \(f\)

 

\[ \max_{\color{Blue}g\in\mathcal G} \text{MA}(f,{\color{blue}g}) ~\leq ~\max_{\color{red}\hat g\in \hat{\mathcal{G}} } \text{MA}(f,{\color{red}\hat{g}}) + B(f,{\color{red}\hat g})  \]

 

with \(B(f,\hat g) = \min \left( \text{err}(\hat g), \sqrt{MSE(f)\cdot \text{err}(\hat g)} \right) \)

true error

worst possible error

Fairness in partially observed regimes

  • Correcting w.r.t \(\hat{\mathcal G}\) provably improves upper bound

                                                               [Gopalan et al. (2022)][Roth (2022)][Bharti et al (2025)]

  • Practical/computable upper bounds

Theorem [Bharti, Clemens-Sewall, Yi, Sulam]

With access to \((X,Y)\sim \mathcal D_{\mathcal{XY}}\), proxies \( \hat{\mathcal G}\) and predictor \(f\)

 

\[ \max_{\color{Blue}g\in\mathcal G} \text{MA}(f,{\color{blue}g}) ~\leq ~\max_{\color{red}\hat g\in \hat{\mathcal{G}} } \text{MA}(f,{\color{red}\hat{g}}) + B(f,{\color{red}\hat g})  \]

 

with \(B(f,\hat g) = \min \left( \text{err}(\hat g), \sqrt{MSE(f)\cdot \text{err}(\hat g)} \right) \)

true error

worst possible error

Fairness in partially observed regimes

CheXpert:  Predicting abnormal findings in chest X-rays
(not accessing race or biological sex)

\(f(X): \) likelihood of \(X\) having  \(\texttt{pleural effusion}\)

Demographic fairness

Take-home message

  • Proxies can be very useful in certifying max. fairness violations
  • Can allow for simple post-processing corrections

Fairness in partially observed regimes

CheXpert:  Predicting abnormal findings in chest X-rays
(not accessing race or biological sex)

\(f(X): \) likelihood of \(X\) having  \(\texttt{pleural effusion}\)

Demographic fairness

Take-home message

  • Proxies can be very useful in certifying max. fairness violations
  • Can allow for simple post-processing corrections

inverse problems

uncertainty quantification

model-agnostic interpretability

robustness

generalization

policy & regulation

demographic fairness

hardware & protocol optimization

data-driven  imaging
automatic analysis and rec.
societal implications

Problems in trustworthy biomedical imaging

"The biggest lesson that can be read from 70 years of AI research is that general methods that leverage computation are ultimately the most effective, and by a large margin. [...] Seeking an improvement that makes a difference in the shorter term, researchers seek to leverage their human knowledge of the domain, but the only thing that matters in the long run is the leveraging of computation. [...]
We want AI agents that can discover like we can, not which contain what we have discovered."

The Bitter Lesson, Rich Sutton 2019

model-agnostic interpretability

"The biggest lesson that can be read from 70 years of AI research is that general methods that leverage computation are ultimately the most effective, and by a large margin. [...] Seeking an improvement that makes a difference in the shorter term, researchers seek to leverage their human knowledge of the domain, but the only thing that matters in the long run is the leveraging of computation. [...]
We want AI agents that can discover like we can, not which contain what we have discovered."

The Bitter Lesson, Rich Sutton 2019

model-agnostic interpretability

{f}\huge(
{\huge)} = \text{\texttt{sick}}

  • What parts of the image are important for this prediction?

  • What are the subsets of the input \(S\in[n]\) so that \(f(x_S) \approx f(x)\)?

Interpretability in Image Classification

Predictor \(f(x)\) trained to predict \(\texttt{sick/healthy}\)

  • Sensitivity or Gradient-based perturbations

  • Shapley coefficients

  • Variational formulations

  • Counterfactual & causal explanations

LIME [Ribeiro et al, '16], CAM [Zhou et al, '16], Grad-CAM [Selvaraju et al, '17]

Shap [Lundberg & Lee, '17], ...

RDE [Macdonald et al, '19], ...

[Sani et al, 2020] [Singla et al '19],..

Post-hoc Interpretability in Image Classification

  • Sensitivity or Gradient-based perturbations

  • Shapley coefficients

  • Variational formulations

  • Counterfactual & causal explanations

LIME [Ribeiro et al, '16], CAM [Zhou et al, '16], Grad-CAM [Selvaraju et al, '17]

Shap [Lundberg & Lee, '17], ...

RDE [Macdonald et al, '19], ...

[Sani et al, 2020] [Singla et al '19],..

Post-hoc Interpretability in Image Classification

Post-hoc Interpretability Methods

Interpretable by

construction

  • Sensitivity or Gradient-based perturbations

  • Shapley coefficients

  • Variational formulations

  • Counterfactual & causal explanations

LIME [Ribeiro et al, '16], CAM [Zhou et al, '16], Grad-CAM [Selvaraju et al, '17]

Shap [Lundberg & Lee, '17], ...

RDE [Macdonald et al, '19], ...

[Sani et al, 2020] [Singla et al '19],..

Post-hoc Interpretability in Image Classification

Post-hoc Interpretability Methods

Interpretable by

construction

  • efficiency

  • nullity

  • symmetry

  • exponential complexity

Lloyd S Shapley. A value for n-person games. Contributions to the Theory of Games, 2(28):307–317, 1953.

Let  \(G = ([n],f)\) be an \(n\)-person cooperative game with characteristic function \(f:\mathcal P([n])\to \mathbb R\)

How important is each player for the outcome of the game?

\displaystyle \phi_i = \sum_{S_j\subseteq [n]\setminus \{i\} } w_{S_j} \left[ f(S_j\cup \{i\}) - f(S_j) \right]

marginal contribution of player i with coalition S

Shapley values

We focus on data with certain structure:

f\huge(
{\huge)} = 0
{f}\huge(
{\huge)} = 1
{f}\huge(
{\huge)} = 0

Example:

f(x) = 1

if       contains a sick cell

x

Hierarchical Shap (h-Shap)

\text{\textbf{Assumption 1:}}~ f(x) = 1 \Leftrightarrow \exist~ i: f(\tilde X_i) = 1

Can we resolve the computational bottleneck (and when) ?

Theorem (informal)

  • h-Shap runs in linear time

  • Under A1, h-Shap \(\to\) Shapley

\mathcal O(2^\gamma k \log n)

\(\tilde{X}_i \sim \mathcal D_{X|X_i=x_i}\)

Hierarchical Shap (h-Shap)

Hierarchical Shap (h-Shap)

Fast hierarchical games for image explanations, Teneggi, Luster & S., IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022

Cheaper predictors via Interpretability

Cheaper predictors via Interpretability

Hemorrhage detection in head CT

Image-by-image supervision (strong learner)
true/false

one label per image

Cheaper predictors via Interpretability

Image-by-image supervision (strong learner)

one label per image

one label per study

true/false
Study/volume supervision (weak learner)
true/false

Cheaper predictors via Interpretability

Cheaper predictors via Interpretability

  • Both methods do as well for case screaning

training labels

Cheaper predictors via Interpretability

  • Both methods do as well for case screaning

  • Weak learner is more label-efficient for detecting positive slices

Teneggi, J., Yi, P. H., & Sulam, J. (2023). Examination-level supervision for deep learning–based intracranial hemorrhage detection at head CT. Radiology: Artificial Intelligence.

Cheaper predictors via Interpretability

Teneggi, J., Yi, P. H., & Sulam, J. (2023). Examination-level supervision for deep learning–based intracranial hemorrhage detection at head CT. Radiology: Artificial Intelligence.

Cheaper predictors via Interpretability

inverse problems

uncertainty quantification

model-agnostic interpretability

robustness

generalization

policy & regulation

demographic fairness

hardware & protocol optimization

data-driven  imaging
automatic analysis and rec.
societal implications

Problems in trustworthy biomedical imaging

inverse problems

uncertainty quantification

model-agnostic interpretability

robustness

generalization

policy & regulation

demographic fairness

hardware & protocol optimization

Problems in trustworthy biomedical imaging

data-driven  imaging
automatic analysis and rec.
societal implications

inverse problems

uncertainty quantification

robustness

generalization

policy & regulation

hardware & protocol optimization

Problems in trustworthy biomedical imaging

model-agnostic interpretability

demographic fairness

data-driven  imaging
automatic analysis and rec.
societal implications

Thank you for hosting me

inverse problems

uncertainty quantification

model-agnostic interpretability

robustness

generalization

policy & regulation

Demographic fairness

hardware & protocol optimization

data-driven  imaging
automatic analysis and rec.
societal implications

Problems in trustworthy biomedical imaging

y = A x^* + v
measurements
\hat x = \arg\min_x \frac 12 \| y - A x \|^2_2 + R(x)
reconstruction

inverse problems

y = A x^* + v
measurements
\hat x = \arg\min_x \frac 12 \| y - A x \|^2_2 + R(x)
reconstruction
= \arg\min_x ~-\log p(y|x) - \log p(x)
= \arg\max_x~ p(x|y)
\text{MAP estimate when }R(x) \propto -~p_x(x):\text{ prior}

inverse problems

\hat x = \arg\min_x \frac 12 \| y - A x \|^2_2 + R(x)

Proximal Gradient Descent:       \( x^{t+1} = \text{prox}_R \left(x^t - \eta  A^\top(Ax^t-y)\right) \)

\text{prox}_R \left( u \right) = \arg\min_x \frac 12 \|u - x\|_2^2 + R(x)
= \texttt{MAP}(x|u), \qquad u = x + v

... a denoiser

\({\color{red}f_\theta}\): off-the-shelf denoiser

[Venkatakrishnan et al., 2013; Zhang et al., 2017b; Meinhardt et al., 2017; Zhang et al., 2021; Gilton, Ongie, Willett, 2019; Kamilov et al., 2023b; Terris et al., 2023; S Hurault et al. 2021, Ongie et al, 2020; ...]

Plug and Play: Implicit Priors

\hat x = \arg\min_x \frac 12 \| y - A x \|^2_2 + R(x)

Proximal Gradient Descent:       \( x^{t+1} = \text{prox}_R \left(x^t - \eta  A^\top(Ax^t-y)\right) \)

\text{prox}_R \left( u \right) = \arg\min_x \frac 12 \|u - x\|_2^2 + R(x)
= \texttt{MAP}(x|u), \qquad u = x + v

... a denoiser

\({\color{red}f_\theta}\): off-the-shelf denoiser

[Venkatakrishnan et al., 2013; Zhang et al., 2017b; Meinhardt et al., 2017; Zhang et al., 2021; Gilton, Ongie, Willett, 2019; Kamilov et al., 2023b; Terris et al., 2023; S Hurault et al. 2021, Ongie et al, 2020; ...]

Plug and Play: Implicit Priors

Question 1)

What are these black-box functions computing? and what have they learned about the data?

Theorem [Fang, Buchanan, S.]

When will \(f_\theta(x)\) compute a \(\text{prox}_R(x)\), and for what \(R(x)\)?

Let \(f_\theta : \mathbb R^n\to\mathbb R^n\) be a network : \(f_\theta (x) = \nabla \psi_\theta (x)\),

where \(\psi_\theta : \mathbb R^n \to \mathbb R,\) convex and differentiable (ICNN).

 

Then,

1. Existence of regularizer
\(\exists ~R_\theta : \mathbb R^n \to \mathbb R\) not necessarily convex : \(f_\theta(x) \in \text{prox}_{R_\theta}(x),\)

2. Computability
We can compute \(R_{\theta}(x)\) by solving a convex problem

Learned Proximals

: revisiting PnP

\text{Let } y = x+v , \quad ~ x\sim p_x, ~~v \sim \mathcal N(0,\sigma^2I)

How do we find \(f(x) = \text{prox}_R(x)\) for the "correct" \(R(x) \propto -\log p_x(x)\)?

Learned Proximals

: revisiting PnP

Theorem [Fang, Buchanan, S.]

f^* = \arg\min_{f} \lim_{\gamma \searrow 0}~ \mathbb E_{x,y} \left[ \ell^\gamma_\text{PM}(f_\theta(y),x)\right]
f^*(y) = \arg\max_c p_{x|y}(c) \triangleq \text{prox}_{-\sigma^2\log p_x}(y)
\ell^\gamma_\text{PM} (f_\theta(y),x) = 1- \frac{c}{\gamma^{2n}} \exp\left( -\frac{\|f(y)-x\|_2^2}{\gamma} \right) 
Proximal Matching Loss:

\(\gamma\)

Goal: train a denoiser \(f(y)\approx x\)

Let

Then,

a.s.

Learned Proximal Networks

\text{Sample } y = x+v,~ \text{ with } x \sim \text{Laplace}(0,1) \text{ and } v \sim \mathcal N(0,\sigma^2)

Example: recovering a prior

Fang, Buchanan & S. What's in a Prior? Learned Proximal Networks for Inverse Problems, ICLR 2024.

Learned Proximal Networks in

\hat x = \arg\min_x \frac 12 \| y - A x \|^2_2 + \hat{R}_\theta(x)

Convergence guarantees

inverse problems

Fang, Buchanan & S. What's in a Prior? Learned Proximal Networks for Inverse Problems, ICLR 2024.

Learned Proximal Networks in

\(R_\theta(x) = 0.0\)

\(R_\theta(x) = 127.37\)

\(R_\theta(x) = 274.13\)

\(R_\theta(x) = 290.45\)

Understanding the learned model provides new insights:

inverse problems

Take-home message 1

  • Learned Proximal Networks (LPNs) provide data-dependent proximal operators
  • Allow characterization of the learned priors. 

Learned Proximal Networks

Example 2: a prior for CT

Learned Proximal Networks

Example 2: a prior for CT

Learned Proximal Networks

Example 2: a prior for CT

Learned Proximal Networks

Example 2: a prior for CT

Learned Proximal Networks

\(R(\tilde{x})\)

Example 2: priors for images

Learned Proximal Networks

Example 2: priors for images

x^{t+1} = \text{prox}_{\hat R} \left(x^t - \eta A^T(Ax^t - y)\right)
\hat x = \arg\min_x \frac 12 \| y - A x \|^2_2 + \hat{R}(x)

Learned Proximal Networks

via

Convergence Guarantees

Theorem (PGD with Learned Proximal Networks)

x^{t+1} = \text{prox}_{\hat R} \left(x^t - \eta A^T(Ax^t - y)\right)
\hat x = \arg\min_x \frac 12 \| y - A x \|^2_2 + \hat{R}(x)

Let \(f_\theta = \text{prox}_{\hat{R}} {\color{grey}\text{ with } \alpha>0}, \text{ and } 0<\eta<1/\sigma_{\max}(A) \) with smooth activations

\text{Then } \exists x^* : \lim_{k\to\infty} x^t = x^* \text{ and }
f_\theta(x^* - \eta A^T(Ax^*-y)) = x^*

(Analogous results hold for ADMM)

Learned Proximal Networks

Convergence guarantees for PnP

in a box

Denoiser

diffusion

Measurements

\[y = Ax + \epsilon,~\epsilon \sim \mathcal{N}(0, \sigma^2\mathbb{I})\]

\[\hat{x} = F(y) \sim \mathcal{P}_y\]

Hopefully \(\mathcal{P}_y \approx p(x \mid y)\), but not needed!

Reconstruction

Question 3)

How much uncertainty is there in the samples \(\hat x \sim \mathcal P_y?\)

Question 4)

How far will the samples \(\hat x \sim \mathcal P_y\) be from the true \(x\)?

Conformal guarantees for diffusion models

Lemma

Given \(m\) samples from \(\mathcal P_y\), let

\[\mathcal{I}(y)_j = \left[ Q_{y_j}\left(\frac{\lfloor(m+1)\alpha/2\rfloor}{m}\right), Q_{y_j}\left(\frac{\lceil(m+1)(1-\alpha/2)\rceil}{m}\right)\right]\]

Then \(\mathcal I(y)\) provides entriwise coverage for a new sample \(\hat x \sim \mathcal P_y\), i.e.

\[\mathbb{P}\left[\hat{x}_j \in \mathcal{I}(y)_j\right] \geq 1 - \alpha\]

\(0\)

\(1\)

low: \( l(y) \)

\(\mathcal{I}(y)\)

up: \( u(y) \)

Question 3)

How much uncertainty is there in the samples \(\hat x \sim \mathcal P_y?\)

(distribution free)
 cf [Feldman, Bates, Romano, 2023]

\(y\)

lower

upper

intervals
\(|\mathcal I(y)_j|\)

Conformal guarantees for diffusion models

\(0\)

\(1\)

ground-truth is

contained

\(\mathcal{I}(y_j)\)

\(x_j\)

Conformal guarantees for diffusion models

Question 4)

How far will the samples \(\hat x \sim \mathcal P_y\) be from the true \(x\)?

Conformal guarantees for diffusion models

[Angelopoulos et al, 2022]

[Angelopoulos et al, 2022]

Risk Controlling Prediction Set

      For risk level \(\epsilon\), failure probability \(\delta\), \(\mathcal{I}(y_j) \) is a RCPS if

\[\mathbb{P}\left[\mathbb{E}\left[\text{fraction of pixels not in intervals}\right] \leq \epsilon\right] \geq 1 - \delta\]

[Angelopoulos et al, 2022]

Question 4)

How far will the samples \(\hat x \sim \mathcal P_y\) be from the true \(x\)?

\(0\)

\(1\)

ground-truth is

contained

\(\mathcal{I}(y_j)\)

\(x_j\)

Conformal guarantees for diffusion models

[Angelopoulos et al, 2022]

ground-truth is

contained

\(0\)

\(1\)

\(\mathcal{I}(y_j)\)

\(\lambda\)

\(x_j\)

Procedure:

\[\hat{\lambda} = \inf\{\lambda \in \mathbb{R}:~ \hat{\text{risk}}_{(\mathcal S_{cal})} \leq \epsilon,~\forall \lambda' \geq \lambda \}\]

[Angelopoulos et al, 2022]

single \(\lambda\) for all \(\mathcal I(y_j)\)!

Risk Controlling Prediction Set

      For risk level \(\epsilon\), failure probability \(\delta\), \(\mathcal{I}(y_j) \) is a RCPS if

\[\mathbb{P}\left[\mathbb{E}\left[\text{fraction of pixels not in intervals}\right] \leq \epsilon\right] \geq 1 - \delta\]

[Angelopoulos et al, 2022]

Question 4)

How far will the samples \(\hat x \sim \mathcal P_y\) be from the true \(x\)?

\(\mathcal{I}_{\bm{\lambda}}(y)_j = [l_\text{low,j} - \lambda, l_\text{up,j} + \lambda]\)

Conformal guarantees for diffusion models

\(K\)-RCPS: High-dimensional Risk Control

 

\[\tilde{{\lambda}}_K = \underset{\lambda \in \mathbb R^K}{\arg\min}~\sum_{k \in [K]}\lambda_k~\quad \text{s.t. }\quad \mathcal I_{\lambda_j}(y) : \text{RCPS}\]

scalar \(\lambda \in \mathbb{R}\)

vector \(\bm{\lambda} \in \mathbb{R}^d\)

\(\mathcal{I}_{\lambda}(y)_j = [\text{low}_j - \lambda, \text{up}_j + \lambda]\)

\(\mathcal{I}_{\bm{\lambda}}(y)_j = [\text{low}_j - \lambda_j, \text{up}_j + \lambda_j]\)

\(\rightarrow\)

\(\rightarrow\)

Procedure:

1. Find anchor point

\[\tilde{\bm{\lambda}}_K = \underset{\bm{\lambda}}{\arg\min}~\sum_{k \in [K]}\lambda_k~\quad\text{s.t.}~~~\hat{\text{risk}}^+(\bm{\lambda})_{(S_{opt})} \leq \epsilon\]

2. Choose

\[\hat{\beta} = \inf\{\beta \in \mathbb{R}:~\hat{\text{risk}}_{S_{cal}}^+(\tilde{\bm{\lambda}}_K + \beta'\bf{1}) \leq \epsilon,~\forall~ \beta' \geq \beta\}\]

\(\tilde{\bm{\lambda}}_K\)

Conformal guarantees for diffusion models

\(K\)-RCPS: High-dimensional Risk Control

 

\[\tilde{{\lambda}}_K = \underset{\lambda \in \mathbb R^K}{\arg\min}~\sum_{k \in [K]}\lambda_k~\quad \text{s.t. }\quad \mathcal I_{\lambda_j}(y) : \text{RCPS}\]

scalar \(\lambda \in \mathbb{R}\)

vector \(\bm{\lambda} \in \mathbb{R}^d\)

\(\rightarrow\)

\(\rightarrow\)

Procedure:

1. Find anchor point

\[\tilde{\bm{\lambda}}_K = \underset{\bm{\lambda}}{\arg\min}~\sum_{k \in [K]}\lambda_k~\quad\text{s.t.}~~~\hat{\text{risk}}^+(\bm{\lambda})_{(S_{opt})} \leq \epsilon\]

2. Choose

\[\hat{\beta} = \inf\{\beta \in \mathbb{R}:~\hat{\text{risk}}_{S_{cal}}^+(\tilde{\bm{\lambda}}_K + \beta'\bf{1}) \leq \epsilon,~\forall~ \beta' \geq \beta\}\]

\(\hat{R}^{\gamma}(\bm{\lambda}_{S_{opt}})\leq \epsilon\)

Guarantee:    \(\mathcal{I}_{\bm{\lambda}_K,\hat{\beta}}(y)_j \) are \((\epsilon,\delta)\)-RCPS 

\(\tilde{\bm{\lambda}}_K\)

\(\mathcal{I}_{\lambda}(y)_j = [\text{low}_j - \lambda, \text{up}_j + \lambda]\)

\(\mathcal{I}_{\bm{\lambda}}(y)_j = [\text{low}_j - \lambda_j, \text{up}_j + \lambda_j]\)

\(\hat{\lambda}_K\)

conformalized uncertainty maps

\(K=4\)

\(K=8\)

\[\mathbb{P}\left[\mathbb{E}\left[\text{fraction of pixels not in intervals}\right] \leq \epsilon\right] \geq 1 - \delta\]

Conformal guarantees for diffusion models

c.f. [Kiyani et al, 2024]
Teneggi, Tivnan, Stayman, S. How to trust your diffusion model: A convex optimization approach to conformal risk control. ICML 2023