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Minimax Multi-Target Conformal Prediction with Applications to Imaging Inverse Problems

Jeffrey Wen, Rizwan Ahmad, Philip Schniter

TL;DR

This work tackles uncertainty quantification in ill-posed imaging inverse problems by introducing a minimax-based multi-target conformal prediction framework that guarantees joint marginal coverage at level $1-\alpha$ while balancing single-target intervals. It develops a finite-sample, split-conformal procedure and proves asymptotic minimax convergence, then demonstrates applications to multi-metric FRIQ assessment, multi-task uncertainty quantification, and multi-round measurement acquisition in imaging. Through synthetic experiments and accelerated MRI (fastMRI) data, the method shows improved balance across targets and competitive interval lengths compared to existing multi-target approaches, with practical gains in acquisition efficiency. The approach provides a principled, distribution-free uncertainty quantification tool for complex imaging pipelines and downstream tasks, with public code and clear pathways for extension to clinical workflows.

Abstract

In ill-posed imaging inverse problems, uncertainty quantification remains a fundamental challenge, especially in safety-critical applications. Recently, conformal prediction has been used to quantify the uncertainty that the inverse problem contributes to downstream tasks like image classification, image quality assessment, fat mass quantification, etc. While existing works handle only a scalar estimation target, practical applications often involve multiple targets. In response, we propose an asymptotically minimax approach to multi-target conformal prediction that provides tight prediction intervals while ensuring joint marginal coverage. We then outline how our minimax approach can be applied to multi-metric blind image quality assessment, multi-task uncertainty quantification, and multi-round measurement acquisition. Finally, we numerically demonstrate the benefits of our minimax method, relative to existing multi-target conformal prediction methods, using both synthetic and magnetic resonance imaging (MRI) data. Code is available at https://github.com/jwen307/multi_target_minimax.

Minimax Multi-Target Conformal Prediction with Applications to Imaging Inverse Problems

TL;DR

This work tackles uncertainty quantification in ill-posed imaging inverse problems by introducing a minimax-based multi-target conformal prediction framework that guarantees joint marginal coverage at level while balancing single-target intervals. It develops a finite-sample, split-conformal procedure and proves asymptotic minimax convergence, then demonstrates applications to multi-metric FRIQ assessment, multi-task uncertainty quantification, and multi-round measurement acquisition in imaging. Through synthetic experiments and accelerated MRI (fastMRI) data, the method shows improved balance across targets and competitive interval lengths compared to existing multi-target approaches, with practical gains in acquisition efficiency. The approach provides a principled, distribution-free uncertainty quantification tool for complex imaging pipelines and downstream tasks, with public code and clear pathways for extension to clinical workflows.

Abstract

In ill-posed imaging inverse problems, uncertainty quantification remains a fundamental challenge, especially in safety-critical applications. Recently, conformal prediction has been used to quantify the uncertainty that the inverse problem contributes to downstream tasks like image classification, image quality assessment, fat mass quantification, etc. While existing works handle only a scalar estimation target, practical applications often involve multiple targets. In response, we propose an asymptotically minimax approach to multi-target conformal prediction that provides tight prediction intervals while ensuring joint marginal coverage. We then outline how our minimax approach can be applied to multi-metric blind image quality assessment, multi-task uncertainty quantification, and multi-round measurement acquisition. Finally, we numerically demonstrate the benefits of our minimax method, relative to existing multi-target conformal prediction methods, using both synthetic and magnetic resonance imaging (MRI) data. Code is available at https://github.com/jwen307/multi_target_minimax.

Paper Structure

This paper contains 23 sections, 7 theorems, 75 equations, 9 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Single-target conformal prediction
  4. Application to imaging inverse problems
  5. Multi-target conformal prediction
  6. Minimax multi-target conformal prediction
  7. Random variable perspective
  8. Finite-sample case
  9. Asymptotically minimax
  10. Applications of multi-target conformal prediction in imaging
  11. Multi-metric blind FRIQ assessment
  12. Multi-task uncertainty quantification
  13. Multi-round measurement acquisition
  14. Numerical experiments
  15. Synthetic data
  16. ...and 8 more sections

Key Result

Lemma 1

For any $\alpha\in (0,1)$, the prediction intervals $\{\mathcal{C}_{\widehat{\lambda}(D_{\mathsf{cal}}),k}(\widehat{Z}_0)\}_{k=1}^K$ from (eq:prediction_set_joint) obey when $(\widehat{Z}_0,Z_0), (\widehat{Z}_1,Z_1),\dots,(\widehat{Z}_n,Z_n)$ are statistically exchangeable.

Figures (9)

  • Figure 1: For the synthetic experiment, (a) shows $\min_k\mathop{\mathrm{ESC}}\nolimits_k$ and $\max_k\mathop{\mathrm{ESC}}\nolimits_k$ versus desired joint coverage $1\!-\!\alpha$ for the independent-noise case and (b) shows the same for the correlated-noise case. Then for the correlated-noise case, (c) shows $\min_k\mathop{\mathrm{ESC}}\nolimits_k$ and $\max_k\mathop{\mathrm{ESC}}\nolimits_k$ versus $n_{\mathsf{train}}$ at $1\!-\!\alpha=0.9$ and (d) shows the same versus $n_{\mathsf{tune}}$. The traces in (c) and (d) represent the average across 5 draws of training and tuning data. In some cases the green curves are hidden behind the red curves.
  • Figure 2: Min and max $\mathop{\mathrm{ESC}}\nolimits_k$ versus $1\!-\!\alpha$ for multi-metric MRI at acceleration $R=8$.
  • Figure 3: $\mathop{\mathrm{ESC}}\nolimits_k$ for each FRIQ metric $k$ and desired coverage level $1\!-\!\alpha$ in MRI at acceleration $R=8$.
  • Figure 4: MIL versus $1\!-\!\alpha$ for the multi-metric MRI experiments at acceleration $R=8$.
  • Figure 5: Min and max $\mathop{\mathrm{ESC}}\nolimits_k$ versus $1\!-\!\alpha$ for multi-task MRI at acceleration $R=8$.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Lemma 1
  • Theorem 2
  • Definition 1: Almost-sure convergence
  • Theorem 3: Glivenko-Cantelli Fristedt:Book:13
  • Theorem 4
  • proof
  • Lemma 5
  • proof : Proof
  • Lemma 6
  • proof : Proof
  • ...and 2 more