Adaptive Computation with Conformal Certification

# Adaptive Computation with Conformal Certification
Source: [https://arxiv.org/html/2610.19189](https://arxiv.org/html/2610.19189)
Aneesh Ragnekar††thanks:Work done while at MSKCC\.Affiliation:Department of Medical Physics,‡Department of Radiation OncologyMemorial Sloan Kettering Cancer CenterNew York, NY, USANancy Lee‡& Harini Veeraraghavan†

###### Abstract

Scientific imaging data involving medical \(e\.g\., 3D radiographic, microscopic images\) and astronomical sensors pose different challenges from natural images\. One key challenge is that these datasets are often orders of magnitude larger than natural images\. Efficient training and inference on such data is challenging due to the increased hardware requirements, often needing bespoke solutions\. Moreover, the relevant objects such as tumors and astronomical phenomena are sparsely distributed with respect to the background information\. Standard efficiency techniques like quantization and pruning apply uniform scaling of all regions, potentially compromising the precision/fidelity of predictions\. Thus, efficient methods that reduce computational requirements without losing resolution for relevant parts of the data and with uncertainty estimates on predictions are essential\. We presentConformal Adaptationwith risk guarantees for uncertainty\-aware sparse computations\. Our formalism is general, spanning dense pixel\-level classification and regression tasks where hierarchical models that predict at low resolution, certify the signal using conformal risk control, and refine the certified predictions\. We benchmark our method on multiple tasks such as 3D medical image segmentation, 2D galaxy flux prediction, and 1D calcium spike rate prediction, achieving2\.52\.5–72×72\\timeswall\-clock speedup across the various tasks, including on commodity CPUs\.

## 1Introduction

Deep learning models have expanded far beyond benchmark vision and natural language processing tasks, becoming foundational components of high\-throughput scientific pipelines ranging from drug discovery and weather forecasting to medical imaging and astronomical surveys\. However, deploying these models in real\-world environments frequently exposes a fundamental tension between predictive capacity and resource constraints\([Jazbec et al\., 2024](https://arxiv.org/html/2610.19189#bib.bib18)\)\. While frontier models deliver state\-of\-the\-art accuracy, their practical utility rapidly degrades when constrained by edge hardware, strict latency budgets, or extreme data scales\. As computational access becomes a bottleneck across disciplines, designing models that are efficient, scalable, and trustworthy has become a central challenge\.

This bottleneck also carries severe environmental and structural costs\. Maximizing benchmark performance via scale demands unsustainable energy and carbon footprints\([Schwartz et al\., 2020](https://arxiv.org/html/2610.19189#bib.bib32);[Mehlin et al\., 2023](https://arxiv.org/html/2610.19189#bib.bib30)\)\. It also raises steep capital barriers, concentrating research within well\-funded enterprise labs while sidelining academic and clinical researchers\([Guan et al\., 2024](https://arxiv.org/html/2610.19189#bib.bib29)\)\. Because hardware scaling yields diminishing returns, algorithmic efficiency is critical for responsible scientific computing\.

Practitioners pursue efficiency along two broad routes\. Hardware specialization and kernel\-level tuning preserve computational fidelity but require long development cycles and capital\-intensive infrastructure\. Algorithmic optimizations such as post\-training quantization\([Liu et al\., 2024](https://arxiv.org/html/2610.19189#bib.bib26);[Chen et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib27)\), structural pruning\([Liu et al\., 2018](https://arxiv.org/html/2610.19189#bib.bib49);[Blalock et al\., 2020](https://arxiv.org/html/2610.19189#bib.bib50)\), low\-rank factorization\([Hu et al\., 2021](https://arxiv.org/html/2610.19189#bib.bib47);[Dettmers et al\., 2023](https://arxiv.org/html/2610.19189#bib.bib48)\), and distillation\([Gou et al\., 2021](https://arxiv.org/html/2610.19189#bib.bib51)\)trade some precision for large runtime reductions on commodity hardware\. On standard vision and NLP benchmarks this tradeoff is usually benign, and sub\-8\-bit representations are now common practice\. These techniques are increasingly applied in safety\-critical settings\([Pang et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib40);[Zeng et al\., 2026](https://arxiv.org/html/2610.19189#bib.bib42);[Alwadee et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib46)\), where they are typically validated by aggregate metrics but can still degrade performance\([Alcaraz\-Ortiz et al\., 2026](https://arxiv.org/html/2610.19189#bib.bib43)\)\. Many dense scientific tasks, however, are characterized by extreme spatial sparsity: tumors in a positron emission tomography \(PET\) volume or rare transients in a sky survey occupy a tiny fraction of the data\. In this regime an aggregate metric can remain high while revealing little about the few regions that matter, and the cost of an error \(a missed lesion, a false discovery\) is concentrated there\. Practitioners therefore benefit from acceleration methods whose risk on the sparse signal is explicitly bounded\.

We approach efficiency along a complementary axis\. Rather than reducing the cost of each evaluation, we reduce the number of locations that require expensive evaluation and certify that decision\. We introduceConformal Adaptation, a framework for allocating compute across dense prediction tasks\. A low\-resolution pass identifies candidate regions of interest, and conformal risk control\([Angelopoulos et al\., 2024](https://arxiv.org/html/2610.19189#bib.bib1)\)calibrates this selection to provide finite\-sample, distribution\-free bounds on the expected fraction of signal missed, at a user\-specified level\. A hierarchy of sparse operators then refines only the selected regions during both training and inference, and we show that the calibrated bound is preserved through refinement under two stated conditions\. As the coarse screening model is an ordinary network,Conformal Adaptationcomposes naturally with quantization, pruning, and other model\-level compression techniques\.

##### Contributions

Our primary contributions are\(a\)an end\-to\-end framework combining conformalized candidate selection with sparse execution, delivering significant training and inference speedups backed by conformal risk control \(CRC\) guarantees,\(b\)a general formulation of admissible label aggregation operators that makes the framework applicable across tasks, and\(c\)case studies spanning multiple scientific domains, data modalities, and sparsity regimes that empirically validate the guarantees and quantify the compute savings\.

The manuscript is organized as follows\. We first establish common notation for dense predictions, sparse refinement solutions, and CRC in Sec\.[2](https://arxiv.org/html/2610.19189#S2)\. We then detail theConformal Adaptationalgorithm in Sec\.[3](https://arxiv.org/html/2610.19189#S3)and define our performance metrics in Sec\.[4](https://arxiv.org/html/2610.19189#S4)\. We showcase the applicability of our method across segmentation \(Sec\.[5](https://arxiv.org/html/2610.19189#S5)\) and regression tasks \(Sec\.[6](https://arxiv.org/html/2610.19189#S6)\), review related work in Sec\.[7](https://arxiv.org/html/2610.19189#S7), and discuss limitations, trade\-offs, and future work in Sec\.[8](https://arxiv.org/html/2610.19189#S8)\.

## 2Background

##### Notation\.

We focus on dense prediction tasks whose domains admit*hierarchical*partitionings\. InputsXXand outputsYYare spatially discretized, withX∈ℝ\|Ω\|×lX\\in\\mathbb\{R\}^\{\|\\Omega\|\\times l\}andY∈ℝ\|Ω\|×kY\\in\\mathbb\{R\}^\{\|\\Omega\|\\times k\}, whereΩ\\Omegais a finite set of*sites*\(pixels in 2D, voxels in 3D, or more generally any indexed cell\) andllandkkare the numbers of input and output channels, respectively\. We say a coarse partitioningΩcoarse\\Omega\_\{\\text\{coarse\}\}*coarsens*a fine partitioningΩfine\\Omega\_\{\\text\{fine\}\}, writtenΩcoarse⪯Ωfine\\Omega\_\{\\text\{coarse\}\}\\preceq\\Omega\_\{\\text\{fine\}\}, when every cell ofΩcoarse\\Omega\_\{\\text\{coarse\}\}is a disjoint union of cells ofΩfine\\Omega\_\{\\text\{fine\}\}; equivalently, the fine partitioning refines the coarse one, and each fine site lies in exactly one coarse cell\.

We seek to learn a non\-linear functionℱθ:𝒳→𝒴\\mathcal\{F\}\_\{\\theta\}:\\mathcal\{X\}\\rightarrow\\mathcal\{Y\}\. Due to hierarchical partitioning, we considerℱθ\\mathcal\{F\}\_\{\\theta\}that decomposes into a coarse model followed by a refinement operator,ℱθ​\(X\)=𝒢⁡\(XΩfine,ℱΩcoarse​\(XΩcoarse\)\)\\mathcal\{F\}\_\{\\theta\}\(X\)=\\mathcal\{G\}\\bigl\(X^\{\\Omega\_\{\\text\{fine\}\}\},\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}\(X^\{\\Omega\_\{\\text\{coarse\}\}\}\)\\bigr\), so that predictions can be performed at multiple levels\. It is far cheaper to predict onΩcoarse\\Omega\_\{\\text\{coarse\}\}and refine than to predict directly onΩfine\\Omega\_\{\\text\{fine\}\}with a full\-resolution modelℱΩfine\\mathcal\{F\}\_\{\\Omega\_\{\\text\{fine\}\}\}\.

##### Conformal Risk Control\.

We turn to CRC\([Angelopoulos et al\., 2024](https://arxiv.org/html/2610.19189#bib.bib1)\)as a distribution\-free black\-box method for providing prediction guarantees at coarse resolution\.

Given a per\-image lossL⁡\(λ,X,Y\)L\(\\lambda;X,Y\)that, for every fixed\(X,Y\)\(X,Y\), is non\-increasing and right\-continuous inλ\\lambda, bounded in\[0,B\]\[0,B\], and satisfiesL⁡\(1,X,Y\)≤αL\(1;X,Y\)\\leq\\alpha, together with calibration data\(Xi,Yi\)i=1n\(X\_\{i\},Y\_\{i\}\)\_\{i=1\}^\{n\}exchangeable with the test point\(Xn\+1,Yn\+1\)\(X\_\{n\+1\},Y\_\{n\+1\}\), the threshold

λ^=inf\{λ:1n\+1​\(∑i=1nL⁡\(λ,Xi,Yi\)\+B\)≤α\}\\hat\{\\lambda\}\\;=\\;\\inf\\Bigl\\\{\\lambda:\\tfrac\{1\}\{n\+1\}\\bigl\(\\textstyle\\sum\_\{i=1\}^\{n\}L\(\\lambda;X\_\{i\},Y\_\{i\}\)\+B\\bigr\)\\leq\\alpha\\Bigr\\\}\(1\)
guarantees,

𝔼⁡\[L⁡\(λ^,Xn\+1,Yn\+1\)\]≤α\\mathbb\{E\}\\bigl\[L\(\\hat\{\\lambda\};X\_\{n\+1\},Y\_\{n\+1\}\)\\bigr\]\\leq\\alpha\(2\)
The goal ofConformal Adaptationis to construct a lossLLand hierarchical refinement operations,𝒢\\mathcal\{G\}, satisfying the CRC requirements at partitioningΩcoarse\\Omega\_\{\\text\{coarse\}\}while presenting the output at partitioningΩfine\\Omega\_\{\\text\{fine\}\}\. Operating on the lower resolutionΩcoarse\\Omega\_\{\\text\{coarse\}\}reduces computation, and sparsifyingΩcoarse\\Omega\_\{\\text\{coarse\}\}by selecting candidate cells for refinement with CRC further increases efficiency without compromising fidelity\.

## 3Conformal Adaptation

Conformal Adaptationproposes a trade\-off between statistical guarantees and spatial fidelity to speed up computation through sparsification\. We are primarily concerned with loss of fidelity due to prediction at the coarser level and seek to close the gap in prediction uncertainty with respect to the finest resolution\. Hence, our current focus does not extend the CRC framework to the finest resolution\. We first establish an abstract formulation to highlight the generalizability of our method, then provide specializations to individual use cases including segmentation and regression\.

![Refer to caption](https://arxiv.org/html/2610.19189v1/Splash-figure.png)Figure 1:The two\-stage process forConformal Adaptationconsists of\(1\)dense predictions at a spatial scale reduced by a factorrrand\(2\)sparse refinement on the selected setSSof coarse cells\. The selection is based on thresholds calibrated using CRC, and the operating regime we target is\|S\|≪\|Ωcoarse\|\|S\|\\ll\|\\Omega\_\{\\text\{coarse\}\}\|\.Exploiting the spatial coherence of adjacent units in an image\-like spatial grid, we reformulate dense prediction as a coarse prediction using an aggregation operatorAAfollowed by*selective*refinement using an operator𝒢\\mathcal\{G\}on relevant regions needing additional computation:

ℱΩcoarse\\displaystyle\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}:𝒳Ωcoarse→𝒴Ωcoarse,\\displaystyle:\\mathcal\{X\}^\{\\Omega\_\{\\text\{coarse\}\}\}\\rightarrow\\mathcal\{Y\}^\{\\Omega\_\{\\text\{coarse\}\}\},\(3\)𝒢Ωcoarse→Ωfine\\displaystyle\\mathcal\{G\}\_\{\\Omega\_\{\\text\{coarse\}\}\\rightarrow\\Omega\_\{\\text\{fine\}\}\}:𝒳Ωfine×𝒴Ωcoarse×\{0,1\}Ωcoarse→𝒴Ωfine,Ωcoarse⪯Ωfine\\displaystyle:\\mathcal\{X\}^\{\\Omega\_\{\\text\{fine\}\}\}\\times\\mathcal\{Y\}^\{\\Omega\_\{\\text\{coarse\}\}\}\\times\\\{0,1\\\}^\{\\Omega\_\{\\text\{coarse\}\}\}\\to\\mathcal\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\},\\quad\\Omega\_\{\\text\{coarse\}\}\\preceq\\Omega\_\{\\text\{fine\}\}\(4\)
We define a binary selection functionσ:Ωcoarse→\{0,1\}\\sigma:\\Omega\_\{\\text\{coarse\}\}\\rightarrow\\\{0,1\\\}over the coarse partitionΩcoarse=\{c1,…,cK\}\\Omega\_\{\\text\{coarse\}\}=\\\{c\_\{1\},\\dots,c\_\{K\}\\\}, where a coarse cellck∈Ωcoarsec\_\{k\}\\in\\Omega\_\{\\text\{coarse\}\}is selected for refinement if and only ifσ⁡\(ck\)=1\\sigma\(c\_\{k\}\)=1\. The resulting set of selected cells isSσ=\{ck∈Ωcoarse:σ⁡\(ck\)=1\}⊆ΩcoarseS\_\{\\sigma\}=\\\{c\_\{k\}\\in\\Omega\_\{\\text\{coarse\}\}:\\sigma\(c\_\{k\}\)=1\\\}\\subseteq\\Omega\_\{\\text\{coarse\}\}\.σ\\sigmais derived from the coarse predictionO^Ωcoarse=ℱΩcoarse​\(XΩcoarse\)\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\}=\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}\(X^\{\\Omega\_\{\\text\{coarse\}\}\}\)\. Our framework allows for dense pixel\-level categorical \(binary, multi\-class\) as well as continuous \(regression\) predictions\. The refinement𝒢Ωcoarse→Ωfine​\(XΩfine,O^Ωcoarse,σ\)\\mathcal\{G\}\_\{\\Omega\_\{\\text\{coarse\}\}\\rightarrow\\Omega\_\{\\text\{fine\}\}\}\\bigl\(X^\{\\Omega\_\{\\text\{fine\}\}\},\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\},\\sigma\\bigr\)acts piecewise on the*fine*sites of each*coarse*cell:

𝒢Ωcoarse→Ωfine\|ck=𝒢ck​\(XΩfine\|ck,O^Ωcoarse​\[ck\]\)ck∈Sσ\\mathcal\{G\}\_\{\\Omega\_\{\\text\{coarse\}\}\\rightarrow\\Omega\_\{\\text\{fine\}\}\}\\big\|\_\{c\_\{k\}\}=\\mathcal\{G\}\_\{c\_\{k\}\}\\left\(X^\{\\Omega\_\{\\text\{fine\}\}\}\\big\|\_\{c\_\{k\}\},\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\}\[c\_\{k\}\]\\right\)\\qquad c\_\{k\}\\in S\_\{\\sigma\}\(5\)
Unselected cells ofΩcoarse\\Omega\_\{\\text\{coarse\}\}are filled by the null value or simple upsampling, to preserve the prediction structure at fine level\. The general inference algorithm is visualized in Fig\.[1](https://arxiv.org/html/2610.19189#S3.F1)\. The cost for the refinement scales with\|Sσ\|\|S\_\{\\sigma\}\|rather than\|Ωfine\|\|\\Omega\_\{\\text\{fine\}\}\|, so a smaller selection is cheaper\. The task is thus to makeSσS\_\{\\sigma\}as small as possible without discarding signal that matters, a trade\-off we control with a statistical guarantee as described in the following paragraphs\. The detailed training, calibration, and inference algorithms are provided in Appendix[A](https://arxiv.org/html/2610.19189#A1)\.

##### Conformal cell selection from the coarse prediction\.

The selection set must be a function of information available at test time, i\.e\. of the input through the coarse prediction\. We obtain it from a coarse*score field*s:𝒳Ωcoarse→\[0,1\]Ωcoarses:\\mathcal\{X\}^\{\\Omega\_\{\\text\{coarse\}\}\}\\to\[0,1\]^\{\\Omega\_\{\\text\{coarse\}\}\}, read out fromO^Ωcoarse\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\}, thresholded at a levelλ∈\[0,1\]\\lambda\\in\[0,1\]:

σλ​\(X\)​\[ck\]\\displaystyle\\sigma\_\{\\lambda\}\(X\)\[c\_\{k\}\]=𝕀\[s\(XΩcoarse\)\[ck\]≥1−λ\]\\displaystyle=\\mathbb\{I\}\\left\[s\(X^\{\\Omega\_\{\\text\{coarse\}\}\}\)\[c\_\{k\}\]\\geq 1\-\\lambda\\right\]\(6\)Sλ​\(X\)\\displaystyle S\_\{\\lambda\}\(X\)=\{ck:σλ​\(X\)​\[ck\]=1\}\\displaystyle=\\\{c\_\{k\}:\\sigma\_\{\\lambda\}\(X\)\[c\_\{k\}\]=1\\\}\(7\)
Note that the selection is monotone inλ\\lambda, withλ1≤λ2⇒Sλ1​\(X\)⊆Sλ2​\(X\)\\lambda\_\{1\}\\leq\\lambda\_\{2\}\\Rightarrow S\_\{\\lambda\_\{1\}\}\(X\)\\subseteq S\_\{\\lambda\_\{2\}\}\(X\), andλ=1\\lambda=1selects every cell\. Crucially, the selection flag,σλ\\sigma\_\{\\lambda\}, depends onXXalone, through the coarse prediction\.

##### Cell aggregation from fine labels\.

Since predictions must happen in lower resolution, we must define an*aggregation operator*AAthat maps the fine labels inside a coarse cell to a*cell weight*\. The cell weight scores the ‘amount’ of signal in each coarse cell by a single non\-negative scalar\.

wck​\(Y\)=A⁡\(YΩfine\|ck\)≥0,w\_\{c\_\{k\}\}\(Y\)\\;=\\;A\\\!\\left\(Y^\{\\Omega\_\{\\text\{fine\}\}\}\\big\|\_\{c\_\{k\}\}\\right\)\\;\\geq\\;0,\(8\)
whereYΩfine\|ckY^\{\\Omega\_\{\\text\{fine\}\}\}\|\_\{c\_\{k\}\}denotes the restriction of the fine\-resolution label to the fine sites contained in the coarse cellckc\_\{k\}\. BecauseΩcoarse⪯Ωfine\\Omega\_\{\\text\{coarse\}\}\\preceq\\Omega\_\{\\text\{fine\}\}, each coarse cell is a disjoint union of fine sites\. Larger weights represent more signal within the coarse cell\.

The fine sites of a coarse cellc∈Ωcoarsec\\in\\Omega\_\{\\text\{coarse\}\}are the setΩfine\|c=\{u∈Ωfine:u⊆c\}\\Omega\_\{\\text\{fine\}\}\|\_\{c\}=\\\{u\\in\\Omega\_\{\\text\{fine\}\}:u\\subseteq c\\\}\. Let∅∈𝒴\\varnothing\\in\\mathcal\{Y\}denote the null value, corresponding to background labels in segmentation or zero signal in regression\. We requireAAto satisfy two conditions:

###### Condition 1\(Nullity\)\.

For everyc∈Ωcoarsec\\in\\Omega\_\{\\text\{coarse\}\}and everyy∈𝒴Ωfine\|cy\\in\\mathcal\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\|\_\{c\}\},y≡∅⟹A⁡\(y\)=0y\\equiv\\varnothing\\;\\Longrightarrow\\;A\(y\)=0\.

###### Condition 2\(Occupancy\-preservation\)\.

For everyc∈Ωcoarsec\\in\\Omega\_\{\\text\{coarse\}\}and everyy∈𝒴Ωfine\|cy\\in\\mathcal\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\|\_\{c\}\},∃u∈Ωfine\|c\\exists\\,u\\in\\Omega\_\{\\text\{fine\}\}\|\_\{c\}withy⁡\[u\]≠∅⟹A⁡\(y\)\>0y\[u\]\\neq\\varnothing\\;\\Longrightarrow\\;A\(y\)\>0\.

##### Connecting to CRC\.

We can now define a loss meeting the requirements of Eq\.[1](https://arxiv.org/html/2610.19189#S2.E1)\. Let the lossL⁡\(λ,X,Y\)L\(\\lambda;X,Y\)be the fraction of total signal mass that lies in*unselected*cells\.L⁡\(λ,X,Y\)=∑ck∉Sλ​\(X\)wck​\(Y\)∑ck∈Ωcoarsewck​\(Y\)L\(\\lambda;X,Y\)\\;=\\;\\frac\{\\displaystyle\\sum\_\{c\_\{k\}\\notin S\_\{\\lambda\}\(X\)\}w\_\{c\_\{k\}\}\(Y\)\}\{\\displaystyle\\sum\_\{c\_\{k\}\\in\\Omega\_\{\\text\{coarse\}\}\}w\_\{c\_\{k\}\}\(Y\)\}\(9\)𝔼⁡\[∑ck∉Sλ^​\(X\)wck​\(Y\)∑ck∈Ωcoarsewck​\(Y\)\]≤α\.\\mathbb\{E\}\\\!\\left\[\\frac\{\\displaystyle\\sum\_\{c\_\{k\}\\notin S\_\{\\hat\{\\lambda\}\}\(X\)\}w\_\{c\_\{k\}\}\(Y\)\}\{\\displaystyle\\sum\_\{c\_\{k\}\\in\\Omega\_\{\\text\{coarse\}\}\}w\_\{c\_\{k\}\}\(Y\)\}\\right\]\\;\\leq\\;\\alpha\.\(10\)

Note thatLLis bounded in\[0,1\]\[0,1\], asw≥0w\\geq 0, and0/0:=00/0:=0for examples with zero total ground\-truth signal weight \(or empty predictions\)\. We requireLLto be non\-increasing and right\-continuous inλ\\lambda\. Increasingλ\\lambdaenlargesSλ​\(X\)S\_\{\\lambda\}\(X\)by Eq\.[6](https://arxiv.org/html/2610.19189#S3.E6), which removes non\-negative terms from the numerator\.

Atλ=1\\lambda=1every cell is selected, so the numerator is empty andL⁡\(1,X,Y\)=0≤αL\(1;X,Y\)=0\\leq\\alpha\. All conditions for applying CRC are therefore satisfied\. Substituting Eq\.[9](https://arxiv.org/html/2610.19189#S3.E9)into the calibration of Eq\.[1](https://arxiv.org/html/2610.19189#S2.E1)yields a thresholdλ^\\hat\{\\lambda\}, which gives us the guarantee in Eq\.[2](https://arxiv.org/html/2610.19189#S2.E2)reformulated for signal loss in Eq\.[10](https://arxiv.org/html/2610.19189#S3.E10)\. Under exchangeability with a calibration set, we ensure that the unselected cells contain, in expectation, at most anα\\alphafraction of the true signal mass\. The CRC guarantee is established entirely atΩcoarse\\Omega\_\{\\text\{coarse\}\}as it depends onXXandYYthrough the cell aggregateswck​\(Y\)w\_\{c\_\{k\}\}\(Y\)andSλ^​\(X\)S\_\{\\hat\{\\lambda\}\}\(X\)\. We decouple the presentation of the solution from the certification of the solution\.

### 3\.1Theoretical Guarantees

We call an aggregation operatorAA*admissible*if it is fixed and satisfies non\-negativity \(Eq\.[8](https://arxiv.org/html/2610.19189#S3.E8)\) and Conditions[1](https://arxiv.org/html/2610.19189#Thmcondition1)and[2](https://arxiv.org/html/2610.19189#Thmcondition2)\. We write𝒪\(Y\)=\{c∈Ωcoarse:∃u∈Ωfine\|c,Y\[u\]≠∅\}\\mathcal\{O\}\(Y\)=\\\{c\\in\\Omega\_\{\\text\{coarse\}\}:\\exists\\,u\\in\\Omega\_\{\\text\{fine\}\}\|\_\{c\},\\;Y\[u\]\\neq\\varnothing\\\}for the set of coarse cells*occupied*by the fine labelYY\. These are the coarse cells that have positive signal mass\.Y^σΩfine=𝒢Ωcoarse→Ωfine​\(XΩfine,O^Ωcoarse,σ\)\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\sigma\}=\\mathcal\{G\}\_\{\\Omega\_\{\\text\{coarse\}\}\\to\\Omega\_\{\\text\{fine\}\}\}\(X^\{\\Omega\_\{\\text\{fine\}\}\},\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\},\\sigma\)is the output of the sparse pipeline under selectionσ\\sigma, withσ≡𝟏\\sigma\\equiv\\mathbf\{1\}recovering dense refinement\.

The aforementioned definition of admissible aggregation operators provides sufficient conditions to show that \(i\) the solution sets are nested in coarse resolution and monotone inλ\\lambda, \(ii\) the missed signal is proportional to the weight and bounded, and \(iii\) we obtain a marginal guarantee over discarded signal mass\. Formally, we have:

###### Theorem 3\.1\.

LetAAbe admissible, let\(X1,Y1\),…,\(Xn,Yn\),\(X,Y\)\(X\_\{1\},Y\_\{1\}\),\\dots,\(X\_\{n\},Y\_\{n\}\),\(X,Y\)be exchangeable, where\(X,Y\)=\(Xn\+1,Yn\+1\)\(X,Y\)=\(X\_\{n\+1\},Y\_\{n\+1\}\)is the test pair, and letλ^\\hat\{\\lambda\}be the CRC threshold obtained by calibrating Eq\.[9](https://arxiv.org/html/2610.19189#S3.E9)at levelα\\alphaon the firstnnpairs\. WriteD=Ωcoarse∖Sλ^​\(X\)D=\\Omega\_\{\\text\{coarse\}\}\\setminus S\_\{\\hat\{\\lambda\}\}\(X\)for the discarded cells\. Then:

1. \(i\)Almost surely,Y^σλ^​\(X\)Ωfine\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\sigma\_\{\\hat\{\\lambda\}\}\(X\)\}andY^𝟏Ωfine\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\mathbf\{1\}\}agree at every fine site of every cell inSλ^​\(X\)S\_\{\\hat\{\\lambda\}\}\(X\); the two pipelines can differ only on⋃c∈DΩfine\|c\\bigcup\_\{c\\in D\}\\Omega\_\{\\text\{fine\}\}\|\_\{c\}\.
2. \(ii\)The true signal mass carried by that region is controlled: 𝔼⁡\[∑c∈Dwc​\(Y\)∑c∈Ωcoarsewc​\(Y\)\]≤α\.\\mathbb\{E\}\\\!\\left\[\\frac\{\\sum\_\{c\\in D\}w\_\{c\}\(Y\)\}\{\\sum\_\{c\\in\\Omega\_\{\\text\{coarse\}\}\}w\_\{c\}\(Y\)\}\\right\]\\;\\leq\\;\\alpha\.
3. \(iii\)With probability at least1−α/t1\-\\alpha/tfor anyt∈\(α,1\]t\\in\(\\alpha,1\], the discarded region carries at most att\-fraction of the signal mass; and the event that it carries none is exactly\{𝒪\(Y\)⊆Sλ^\(X\)\}\\\{\\mathcal\{O\}\(Y\)\\subseteq S\_\{\\hat\{\\lambda\}\}\(X\)\\\}, for every admissibleAA\.

###### Proof sketch\.

Informally, \(i\) is a consequence of selectivity of the refinement operation\. Coarsening a sparse refinement is the same as masking the coarsening of a dense prediction\. The statement \(ii\) is the result of the admissible operator being compatible with CRC\. Finally, \(iii\) follows from Markov’s inequality applied to the non\-negative loss\. The detailed proof can be found in Appendix[B](https://arxiv.org/html/2610.19189#A2)\. ∎

## 4Performance Metrics

We focus on both dense classification and regression tasks from varying domains\. Segmentation tasks use Dice similarity coefficient \(DSC\) and lesion detection metrics as primary accuracy targets\. The spike count prediction and galaxy super\-resolution use mean absolute error \(MAE\)\. For performance metrics, we audit both wall\-clock speedup as well as FLOPs\. We use both CPUs and GPUs to obtain timing results\.

## 5Case Study: Conformal Adaptation for Segmentation

###### Definition 1\(Segmentation Cell Aggregation\)\.

We use fine\-grained cell occupation to define our aggregation operator\. Each coarse cellc∈Ωcoarsec\\in\\Omega\_\{\\text\{coarse\}\}contains the set of fine sitesΩfine\|c\\Omega\_\{\\text\{fine\}\}\|\_\{c\}\. We define the aggregation operator for segmentation maps, transporting masks from resolutionΩfine\\Omega\_\{\\text\{fine\}\}toΩcoarse\\Omega\_\{\\text\{coarse\}\}as,

wc​\(YΩfine\)=⋁u∈Ωfine\|cYΩfine​\[u\],c∈Ωcoarse\.w\_\{c\}\(Y^\{\\Omega\_\{\\text\{fine\}\}\}\)=\\bigvee\_\{u\\in\\Omega\_\{\\text\{fine\}\}\|\_\{c\}\}Y^\{\\Omega\_\{\\text\{fine\}\}\}\[u\],\\qquad c\\in\\Omega\_\{\\text\{coarse\}\}\.

In the case of tumor segmentation, the logical OR operator ensures positive tumor cells are not missed during coarsening\. By construction, we can see that the OR\-aggregation operator is zero\-preserving, as it is 0 when all inputs are 0, and monotonically increasing when any cell is nonzero\. We take any off\-the\-shelf segmentation prediction model providing softmax\-like scores per cell, and we can define the selection setSλ​\(X\)S\_\{\\lambda\}\(X\)\.

### 5\.1Brain Tumor Segmentation

The brain tumor dataset from the Medical Segmentation Decathlon \(MSD\)\([Antonelli et al\., 2022](https://arxiv.org/html/2610.19189#bib.bib10)\)consists of 484 cases with brain tumors of varying sizes, occupying at most 3\.5% of the image volume\. Each case is a 4D volume combining pre\- and post\-contrast T1\-weighted, T2\-weighted, and fluid attenuation inverse recovery \(FLAIR\) MRI scans\.

We applyConformal Adaptationto three segmentation architectures: U\-Net\([Ronneberger et al\., 2015](https://arxiv.org/html/2610.19189#bib.bib36)\), Swin\-UNETR\([Hatamizadeh et al\., 2021](https://arxiv.org/html/2610.19189#bib.bib38);[He et al\., 2023](https://arxiv.org/html/2610.19189#bib.bib37)\), and VISTA3D\([He et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib39)\)\. We train each randomly initialized architecture three times at full resolution \(Full\) and at4×4\\times\-downsampled resolution \(Coarse\)\. The coarse model then determines the sparse patches used to train the second\-stage refiner \(Coarse \+ Refine\)\. The data processing, architecture hyperparameters, and training methods are detailed in Appendix[C\.1](https://arxiv.org/html/2610.19189#A3.SS1)\. Across architectures and devices, the method provides substantial FLOPs and wall\-clock reductions while retaining similar DSC \(Fig\.[2](https://arxiv.org/html/2610.19189#S5.F2)\)\. In CPU\-only inference, VISTA3D achieves up to72×72\\timeswall\-clock speedup and an82\.1×82\.1\\timesreduction in FLOPs \(Table[1](https://arxiv.org/html/2610.19189#S5.T1)\)\. We see super\-linear improvement as the data can use fewer forward passes with sliding window inference in coarse resolution\.

Table 1:Performance comparison on MSD Task01\. Values are reported as mean±\\pmstandard deviation over three seeds\. Columns 2\-4 are the DSC for each model arrangement\. Capacity\-constrained models like U\-Net see an improvement in performance due to additional weights\. Larger scans require sliding window inference, increasing the FLOPs required for a single sample\.Figure 2:Left:Empirical verification of the CRC calibration quality\.Middle:mean DSC of the Full baseline against the Coarse \+ Refine model atα=0\.1\\alpha=0\.1\.Right:Corresponding measured wall\-clock speedup over the Full baseline on different devices\.![Refer to caption](https://arxiv.org/html/2610.19189v1/swin-unetr_contour_grid_single.png)Figure 3:Predicted tumor boundaries on a held\-out test scan \(Swin\-UNETR,α=0\.1\\alpha=0\.1\)\. The Coarse \+ Refine model tracks the boundary as closely as the Full baseline at a fraction of the FLOPs \(Table[1](https://arxiv.org/html/2610.19189#S5.T1)\)\.
### 5\.2Head and Neck Cancer and Metastatic Lymph Node Segmentation

Accurate segmentation of primary head and neck cancer \(HNC\) and affected lymph node volumes is essential for radiotherapy planning \(RTP\) as well as for monitoring tumor response to treatments\. Computed tomography \(CT\) and 2\-\[18F\]fluoro\-2\-deoxyglucose \(FDG\) PET are imaging modalities commonly used to localize and delineate tumors for RTP\. We base our study on the multimodal HECKTOR 2026 grand challenge dataset\([Saeed et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib44)\), which provides 782 FDG\-PET and CT volumes spanning from the head to the pelvis\. The size, structure, and sparsity of the HNC scans and tumors make the dataset an ideal candidate forConformal Adaptation, as seen in Fig\.[4](https://arxiv.org/html/2610.19189#S5.F4)\. Each tumor spans significantly less than the total scanned area at the standard1×1×1​mm31\\times 1\\times 1\\,\\mathrm\{mm\}^\{3\}spacing\. We train and evaluate off\-the\-shelf models, including nnU\-Net\([Isensee et al\., 2021](https://arxiv.org/html/2610.19189#bib.bib4)\), an industry standard that establishes a strong baseline, and the transformer\-CNN\-based Swin\-UNETR\([He et al\., 2023](https://arxiv.org/html/2610.19189#bib.bib37)\)\. We use the standard CT and PET data pre\-processing techniques, such as normalization to\[0,1\]\[0,1\]and isotropic resampling\. During training, we achieve a large speedup, as training the coarse model along with a sparse refiner takes15×15\\timesfewer FLOPs than the full\-resolution model \(see Fig\.[4](https://arxiv.org/html/2610.19189#S5.F4)\)\. Full data\-processing and training details are presented in Appendix[C\.2](https://arxiv.org/html/2610.19189#A3.SS2)\.

Figure 4:\(Left\): False negative rates \(FNR\) can rapidly increase depending on chosen threshold, requiring a principled methods like CRC to choose appropriateλ\\lambda\(See Appendix[C\.3](https://arxiv.org/html/2610.19189#A3.SS3)\.\(Middle\)Tumor cells span a small fraction of the total scanned volume in most scans in the dataset\.\(Right\)Training the coarse model on the aggregated samples and a separate refiner to convergence requires15×15\\timesfewer FLOPs\. All models are trained until validation loss stops decreasing with a set patience level\.Once trained and calibrated, our Coarse \+ Refine model achieves results comparable to the Full model \(DSC 0\.72 vs\. 0\.74\) while requiring68×68\\timesfewer FLOPs \(Fig\.[5](https://arxiv.org/html/2610.19189#S5.F5)\)\. Test\-time augmentation with mirroring is used on all dense models\. Furthermore, for lesion\-detection metrics, where the predicted volume is used to detect lesions, our method achieves comparable recall and precision metrics but suffers from missing small adversarial GTVn cases \(see Table[3](https://arxiv.org/html/2610.19189#A3.T3)\)\.

![Refer to caption](https://arxiv.org/html/2610.19189v1/contour_grid.png)
Figure 5:\(Left\) Visualizations of model segmentation with ground truth showing Coarse \+ Refine models with comparable contours\. \(Right\) A coarse inference reduces per\-scan compute cost while a sparse refinement improves prediction quality with little additional expense\.

## 6Case Study: Conformal Adaptation for Regression

We evaluate the generality of the cell weight mechanism defined in Eq\.[8](https://arxiv.org/html/2610.19189#S3.E8)for regression\-type problems, where the prediction target is a continuous, non\-negative signal \(a rate, a count, a flux\) rather than a discrete class label\.

###### Definition 2\(Regression Cell Aggregation\)\.

For a continuous, non\-negative fine\-resolution targetY∈𝒴ΩfineY\\in\\mathcal\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}, we define the aggregation operator as the total signal mass inside the coarse cell,wc​\(YΩfine\)=∑u∈Ωfine\|cYΩfine​\[u\],c∈Ωcoarse\.w\_\{c\}\(Y^\{\\Omega\_\{\\text\{fine\}\}\}\)=\\sum\_\{u\\in\\Omega\_\{\\text\{fine\}\}\|\_\{c\}\}Y^\{\\Omega\_\{\\text\{fine\}\}\}\[u\],\\quad c\\in\\Omega\_\{\\text\{coarse\}\}\.

Summation is zero\-preserving and, sinceY≥0Y\\geq 0, monotonically non\-decreasing in any added signal, satisfying the requirements of Eq\.[8](https://arxiv.org/html/2610.19189#S3.E8)\. The CRC loss of Eq\.[9](https://arxiv.org/html/2610.19189#S3.E9)then bounds, in expectation, the*fraction of total signal mass*that falls in unselected regions\. Regression scores \(e\.g\., a predicted upper quantile of cell mass\) are not confined to\[0,1\]\[0,1\], so we apply Eq\.[6](https://arxiv.org/html/2610.19189#S3.E6)after a fixed, strictly increasing map into\[0,1\]\[0,1\]\. The map preserves score order, so the selected sets stay nested inλ\\lambdaas CRC requires, and we reportλ^\\hat\{\\lambda\}by its equivalent raw\-score cutoff\.

### 6\.1CASCADE: 1D Calcium\-Imaging Spike\-Rate Inference

We use the CASCADE benchmark\([Rupprecht et al\., 2021](https://arxiv.org/html/2610.19189#bib.bib33)\)with 515 samples across 117 sources, in which the task is to convert fluorescence traces from calcium imaging into spike\-rate curves\. The coarse modelℱΩcoarse\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}block\-averages the fluorescence trace by a factor of eight before applying the 1D U\-Net of the full\-resolution baselineℱΩfine\\mathcal\{F\}\_\{\\Omega\_\{\\text\{fine\}\}\}\. Note, we do not use the benchmark’s decoder\-free sliding\-window regressor\. CRC is calibrated on the coarse block counts, and the certified selection is refined with a count\-predicting local refiner combined with a mass\-preserving Gaussian reconstruction kernel, so bounding missed count mass at the coarse resolution also bounds missed mass in the reconstructed rate curve\. An example input and output prediction is provided in Fig\.[6](https://arxiv.org/html/2610.19189#S6.F6)\. Table[6](https://arxiv.org/html/2610.19189#A3.T6)reports the reference operating point\. Acrossα∈\{0\.01,0\.05,0\.1,0\.2\}\\alpha\\in\\\{0\.01,0\.05,0\.1,0\.2\\\}, CRC calibration is empirically valid \(test risk≤α\\leq\\alphaat every level\), and the reconstructed\-curve MAE stays within0\.050\.05–0\.100\.10Hz of the full\-resolution baseline \(median MAE0\.2500\.250Hz\) while the retained fraction of coarse cells falls to8\.4%8\.4\\%atα=0\.1\\alpha\{=\}0\.1, an analytic FLOPs speedup of4\.77×4\.77\\times\. All details of the data pre\-processing, model architecture, and calibration can be found in Appendix[C\.4](https://arxiv.org/html/2610.19189#A3.SS4)\.

![Refer to caption](https://arxiv.org/html/2610.19189v1/figures/cascade/fig_predictions.png)Figure 6:\(a\) Input signal\. \(b\) Example test window: ground\-truth spike rate overlaid with the Full prediction and the Coarse \+ Refine reconstruction, at a retained fraction of8\.4%8\.4\\%\.
### 6\.2Galaxy Flux Field Super\-Resolution

Extending our experiments further, we also study a 2D galaxy\-flux field task of predicting galaxy shape and flux from simulated surface\-brightness maps\. Simulated surface\-brightness fields are foundational to observational astronomy\. As the raw data provide no labeled truth, surveys characterize what their pipelines detect and miss by injecting sources of known flux into images and measuring recovery\([Anbajagane et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib11);[Team et al\., 2026](https://arxiv.org/html/2610.19189#bib.bib12)\)\. The resulting completeness and photometric bias curves are used as inputs for correction terms on downstream detectors\. Fields are generated withGalSim\([Rowe et al\., 2015](https://arxiv.org/html/2610.19189#bib.bib13)\), the community\-standard image simulator validated against the precision requirements of Stage IV surveys\. In practice surveys such as Rubin/LSST process on the order of2020TB per night under a latency budget of seconds per visit\([Ivezić et al\., 2019](https://arxiv.org/html/2610.19189#bib.bib14)\)over fields that are overwhelmingly empty sky, making this setting well suited to our method\.

We use a 2D U\-Net as our base model\. The Coarse \+ Refine pipeline is a coarse model sharing the base model’s architecture but applied to an8×8\\times\-downsampled field, CRC certifying coarse cells by flux mass, and a sparse refiner predicting a non\-negative spatial spread within each certified cell\. The normalized refiner output is multiplied by the coarse net’s own predicted median flux, making flux conservation between coarse and refined predictions exact by construction\.

At the calibrated operating point \(α=0\.1\\alpha\{=\}0\.1, score cutoff7\.697\.69\), the pipeline retains only2\.2%2\.2\\%of coarse cells on average, for an analytic FLOPs speedup of61\.6×61\.6\\timesover the full\-resolution baseline, at the cost of a higher per\-pixel MAE \(0\.1340\.134vs\.0\.0460\.046for the full\-resolution baseline\) with total\-flux relative error of5\.6%5\.6\\%vs\.2\.3%2\.3\\%, consistent with the certification\. Figure[7](https://arxiv.org/html/2610.19189#S6.F7)presents example predictions and measured wall\-clock speedups across several devices, illustrating the method’s potential for edge deployment\.

![Refer to caption](https://arxiv.org/html/2610.19189v1/main_nsrc60.png)Figure 7:\(Left\) Achieved real\-world wall\-clock speedups on GPUs and CPU thread counts\. \(Right\) Sample selection and comparison between the Full and Coarse \+ Refine models\. The CRC\-certified Coarse \+ Refine predictions have little discrepancy while costing a fraction of the compute and time\.

## 7Related Work

Static compression reduces inference cost by modifying the*model*independently of its input\. Common approaches include pruning\([Yu et al\., 2017](https://arxiv.org/html/2610.19189#bib.bib15);[Osawa et al\., 2017](https://arxiv.org/html/2610.19189#bib.bib16)\), architectural simplification\([Pang et al\., 2023](https://arxiv.org/html/2610.19189#bib.bib41);[Pang et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib40)\), static activation sparsification\([Alanova et al\., 2026](https://arxiv.org/html/2610.19189#bib.bib22)\), and quantization\([Gupta et al\., 2015](https://arxiv.org/html/2610.19189#bib.bib28);[Sun et al\., 2020](https://arxiv.org/html/2610.19189#bib.bib31);[Liu et al\., 2024](https://arxiv.org/html/2610.19189#bib.bib26);[Chen et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib27)\)\.[Zeng et al\. \(2026\)](https://arxiv.org/html/2610.19189#bib.bib42)propose token sparsification for efficient 3D segmentation, while static resolution scaling accelerates training by operating at fixed lower resolutions\([Zhang et al\., 2026](https://arxiv.org/html/2610.19189#bib.bib6)\)\. Unlike our method, these approaches evaluate error post hoc and provide no accuracy guarantees\. Their gains are largely orthogonal to our input\-adaptive sparsification\.

Input\-adaptive methods allocate compute based on the sample\. Examples include token sparsification in vision transformers\([Rao et al\., 2021](https://arxiv.org/html/2610.19189#bib.bib23);[Fayyaz et al\., 2022](https://arxiv.org/html/2610.19189#bib.bib25);[Uddin et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib21)\), region\-wise token granularity selection\([Havtorn et al\., 2023](https://arxiv.org/html/2610.19189#bib.bib5)\), early exiting\([He et al\., 2021](https://arxiv.org/html/2610.19189#bib.bib19)\), and adaptive pixel refinement in PointRend\([Kirillov et al\., 2020](https://arxiv.org/html/2610.19189#bib.bib7)\)\. While these methods focus computation on informative regions, they rely on learned or heuristic selection mechanisms without guarantees on discarded information, making them susceptible to failure modes and adversarial attacks\([Goyal et al\., 2026](https://arxiv.org/html/2610.19189#bib.bib17)\)\. In contrast, our calibrated selection rule provably bounds discarded signal mass at risk levelα\\alpha\. Conformal prediction has been extended to structured outputs through unified task formulations\([Zhang et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib8)\), segmentation\-specific spatial modeling\([Viti et al\., 2026](https://arxiv.org/html/2610.19189#bib.bib9);[Angelopoulos et al\., 2024](https://arxiv.org/html/2610.19189#bib.bib1)\), and input\-conditional coverage\([Luo and Zhou, 2025](https://arxiv.org/html/2610.19189#bib.bib24)\)\. These methods treat the predictor as fixed and use conformal prediction only to construct uncertainty sets, while we use the conformal certificate to guide computation\.[Wang et al\. \(2026\)](https://arxiv.org/html/2610.19189#bib.bib34),[Schuster et al\. \(2022\)](https://arxiv.org/html/2610.19189#bib.bib45),[Schuster et al\. \(2021\)](https://arxiv.org/html/2610.19189#bib.bib20), and[Jazbec et al\. \(2024\)](https://arxiv.org/html/2610.19189#bib.bib18)apply distribution\-free guarantees for adaptive inference and calibrate early\-exit strategies to match full\-model performance under compute constraints\.

## 8Discussion, Limitations, and Conclusion

Although we evaluate on uniform grids, the derivation in Sec\.[3](https://arxiv.org/html/2610.19189#S3)is domain\-agnostic and extends to non\-uniform domains \(e\.g\., graphs\) whenever the refinement operators preserve the hierarchical partition\. We use identical backbones for the native and coarse passes to isolate the gains due to conformalized selection itself\.Conformal Adaptationdecides*where*to compute, whereas quantization, pruning, sparse architectures, and token selection reduce the cost of*each*computation, so the two are orthogonal\. Any stage may use quantized or pruned backbones\([Liu et al\., 2024](https://arxiv.org/html/2610.19189#bib.bib26);[Yu et al\., 2017](https://arxiv.org/html/2610.19189#bib.bib15)\), sparse architectures\([Pang et al\., 2023](https://arxiv.org/html/2610.19189#bib.bib41);[Pang et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib40);[Zeng et al\., 2026](https://arxiv.org/html/2610.19189#bib.bib42)\), or token selection\([Rao et al\., 2021](https://arxiv.org/html/2610.19189#bib.bib23);[Uddin et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib21)\)without changing the calibration procedure or the guarantee; whether their savings compound with ours is promising future work\. We acknowledge that the guarantee has three limitations\. First, it bounds the expected fraction of true signal in discarded cells, not the error of the refined prediction\. Second, it holds marginally, so an individual scan may exceedα\\alpha, and small lesions that are faint at coarse resolution may be missed\. Third, it assumes exchangeability between calibration and test data and is therefore sensitive to covariate shift, e\.g\., across scanners or sites\. Weighted CRC[Tibshirani et al\. \(2019\)](https://arxiv.org/html/2610.19189#bib.bib3), high\-probability \(PAC\-style\) bounds\([Angelopoulos et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib2)\), and per\-lesion risk control are natural extensions\.Conformal Adaptationoffers a simple, calibrated way to spend compute only where it is required\.

### Aknowledgements

Research reported was supported by NCI of the National Institutes of Health under grant number R01CA258821, P30CA008748, and R01EB032825\. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health\. This work utilized resources from the High Performance Computing Group at Memorial Sloan Kettering Cancer Center\.

### AI use statement

In this work, we used generative AI tools to help refine and audit mathematical claims, assist in writing proofs, and assist in writing code to implement methods\. We have not used generative AI tools for generating synthetic datasets, help develop theoretical models or conceptual frameworks, propose or refine hypotheses, design or provide feedback on research methodology and experiments, assist with translation, clean and reformat datasets, support qualitative and thematic data analysis, and interpret results\.

Additionally, we used generative AI tools to edit code, improve readability, check grammar, and help identify relevant literature\. We have reviewed all AI\-assisted work\. Author 1 has reviewed the implementation assisted with AI as well as the mathematical claims that were refined using AI\. Authors 2 and 3 have further verified the notation and edits that were aided by generative AI, including checking mathematical claims, references, and manuscript edits\. We take responsibility for the final content of this work, including text, claims or artifacts produced with the aid of generative AI\.

### Reproducibility statement

We take multiple steps to ensure all our results and analysis are reproducible\. Our analysis uses publicly available data that can be freely downloaded from the cited sources\. Although open to the public, the HECKTOR 2026 dataset requires signing a data use agreement and cannot be shared under its license terms\. We detail the required data pre\-processing in Appendices[C\.1](https://arxiv.org/html/2610.19189#A3.SS1),[C\.2](https://arxiv.org/html/2610.19189#A3.SS2),[C\.4](https://arxiv.org/html/2610.19189#A3.SS4), and[C\.5](https://arxiv.org/html/2610.19189#A3.SS5)\. The model definitions, training algorithms, and post\-processing are also detailed in these appendices\. Our main algorithmic contribution is detailed in Appendix[A](https://arxiv.org/html/2610.19189#A1)\. To aid in reviewing mathematical correctness, the full proof is given in Appendix[B](https://arxiv.org/html/2610.19189#A2)\. Finally, we provide all anonymized pre\- and post\-processing, training, and visualization code, along with all seeds used, as supplementary material\.

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## Appendix AConformal Adaptation: Training, Calibration, Inference

### A\.1Training

Given an admissible operatorAA, a coarse predictor modelℱΩcoarse\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}is trained together with the refinement operator𝒢\\mathcal\{G\}\. In general, training the two models is a three\-step process:

1. 1\.Train the coarse model: - •Downsample the input to the coarse partitioning, and aggregate the labels into coarse cell weights with an admissible operatorAA
2. 2\.Extract sparse training data: - •Rather than the complete fine\-grained samples, the sparse refiner is trained on small patches \(that will be selected by the coarse model\) - •The sparse models can be trained either with ‘simulated’ sparse data that is selected by an oracle or it can be selected using the now trained coarse modelℱΩcoarse\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}at a training selection levelλtrain\\lambda\_\{\\text\{train\}\}\. Settingλtrain=λ^\\lambda\_\{\\text\{train\}\}=\\hat\{\\lambda\}\(Algorithm[2](https://arxiv.org/html/2610.19189#alg2)\) trains the refiner on the certified distribution it will see at inference, at the cost of one refiner per calibratedα\\alpha
3. 3\.Train the refiner model - •The refiner model is trained on the extracted sparse patch\-level data\.

Throughout this section we use𝒢\\mathcal\{G\}for𝒢Ωcoarse→Ωfine\\mathcal\{G\}\_\{\\Omega\_\{\\text\{coarse\}\}\\rightarrow\\Omega\_\{\\text\{fine\}\}\}for conciseness\.

Algorithm 1Training the Coarse and Selective Refinement Models1:Training dataset

𝒟train=\{\(XjΩfine,YjΩfine\)\}j=1N\\mathcal\{D\}\_\{\\text\{train\}\}=\\\{\(X\_\{j\}^\{\\Omega\_\{\\text\{fine\}\}\},Y\_\{j\}^\{\\Omega\_\{\\text\{fine\}\}\}\)\\\}\_\{j=1\}^\{N\}, downsampling operator

Downsample⁡\(⋅\)\\operatorname\{Downsample\}\(\\cdot\), admissible aggregation operator

AA, scoring function

s⁡\(⋅\)s\(\\cdot\), training selection level

λtrain∈\[0,1\]\\lambda\_\{\\text\{train\}\}\\in\[0,1\]\.

2:Trained coarse model

ℱΩcoarse\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}, trained refinement model

𝒢\\mathcal\{G\}\.

3:// Step 1: Coarse Data Preparation & Training

4:Construct downsampled coarse dataset:

5:

𝒟Ωcoarse←\{\(Downsample⁡\(XjΩfine\),w⁡\(YjΩfine\)\)\}j=1N\\mathcal\{D\}\_\{\\Omega\_\{\\text\{coarse\}\}\}\\leftarrow\\bigl\\\{\\bigl\(\\operatorname\{Downsample\}\(X\_\{j\}^\{\\Omega\_\{\\text\{fine\}\}\}\),\\,w\(Y\_\{j\}^\{\\Omega\_\{\\text\{fine\}\}\}\)\\bigr\)\\bigr\\\}\_\{j=1\}^\{N\}, where

w⁡\(Y\)=\(wc​\(Y\)\)c∈Ωcoarsew\(Y\)=\(w\_\{c\}\(Y\)\)\_\{c\\in\\Omega\_\{\\text\{coarse\}\}\}
6:Train

ℱΩcoarse:𝒳Ωcoarse→𝒴Ωcoarse\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}\\colon\\mathcal\{X\}^\{\\Omega\_\{\\text\{coarse\}\}\}\\to\\mathcal\{Y\}^\{\\Omega\_\{\\text\{coarse\}\}\}by minimizing standard empirical risk on

𝒟Ωcoarse\\mathcal\{D\}\_\{\\Omega\_\{\\text\{coarse\}\}\}
7:// Step 2: Refinement Data Extraction via Coarse Selection

8:Initialize refinement dataset

𝒟refine←∅\\mathcal\{D\}\_\{\\text\{refine\}\}\\leftarrow\\emptyset
9:foreach sample

\(XjΩfine,YjΩfine\)∈𝒟train\(X\_\{j\}^\{\\Omega\_\{\\text\{fine\}\}\},Y\_\{j\}^\{\\Omega\_\{\\text\{fine\}\}\}\)\\in\\mathcal\{D\}\_\{\\text\{train\}\}do

10:Compute coarse prediction:

O^jΩcoarse←ℱΩcoarse​\(XjΩcoarse\)\\hat\{O\}\_\{j\}^\{\\Omega\_\{\\text\{coarse\}\}\}\\leftarrow\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}\(X\_\{j\}^\{\\Omega\_\{\\text\{coarse\}\}\}\)
11:foreach coarse cell

ck∈Ωcoarsec\_\{k\}\\in\\Omega\_\{\\text\{coarse\}\}do

12:Select cell by Eq\.[6](https://arxiv.org/html/2610.19189#S3.E6):

σj\[ck\]←𝕀\[s\(XjΩcoarse\)\[ck\]≥1−λtrain\]\\sigma\_\{j\}\[c\_\{k\}\]\\leftarrow\\mathbb\{I\}\[s\(X\_\{j\}^\{\\Omega\_\{\\text\{coarse\}\}\}\)\[c\_\{k\}\]\\geq 1\-\\lambda\_\{\\text\{train\}\}\]
13:\(oracle variant:σj\[ck\]←𝕀\[wck\(Yj\)\>0\]\\sigma\_\{j\}\[c\_\{k\}\]\\leftarrow\\mathbb\{I\}\[w\_\{c\_\{k\}\}\(Y\_\{j\}\)\>0\]\)

14:if

σj​\[ck\]=1\\sigma\_\{j\}\[c\_\{k\}\]=1then

15:Extract fine patch

xj,kΩfine←XjΩfine\|ckx\_\{j,k\}^\{\\Omega\_\{\\text\{fine\}\}\}\\leftarrow X\_\{j\}^\{\\Omega\_\{\\text\{fine\}\}\}\\big\|\_\{c\_\{k\}\}
16:Extract ground\-truth fine label

yj,kΩfine←YjΩfine\|cky\_\{j,k\}^\{\\Omega\_\{\\text\{fine\}\}\}\\leftarrow Y\_\{j\}^\{\\Omega\_\{\\text\{fine\}\}\}\\big\|\_\{c\_\{k\}\}
17:

𝒟refine←𝒟refine∪\{\(\(xj,kΩfine,O^jΩcoarse​\[ck\]\),yj,kΩfine\)\}\\mathcal\{D\}\_\{\\text\{refine\}\}\\leftarrow\\mathcal\{D\}\_\{\\text\{refine\}\}\\cup\\left\\\{\\left\(\(x\_\{j,k\}^\{\\Omega\_\{\\text\{fine\}\}\},\\hat\{O\}\_\{j\}^\{\\Omega\_\{\\text\{coarse\}\}\}\[c\_\{k\}\]\),\\,y\_\{j,k\}^\{\\Omega\_\{\\text\{fine\}\}\}\\right\)\\right\\\}
18:endif

19:endfor

20:endfor

21:// Step 3: Train Refinement Model

22:Train local refinement operator

𝒢:𝒳Ωfine\|ck×𝒴Ωcoarse​\[ck\]→𝒴Ωfine\|ck\\mathcal\{G\}\\colon\\mathcal\{X\}^\{\\Omega\_\{\\text\{fine\}\}\}\\big\|\_\{c\_\{k\}\}\\times\\mathcal\{Y\}^\{\\Omega\_\{\\text\{coarse\}\}\}\[c\_\{k\}\]\\to\\mathcal\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\\big\|\_\{c\_\{k\}\}on

𝒟refine\\mathcal\{D\}\_\{\\text\{refine\}\}
23:return

ℱΩcoarse,𝒢\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\},\\mathcal\{G\}

### A\.2Calibration

The CRC algorithm is run on the low\-resolution partitioning as seen in Algorithm[2](https://arxiv.org/html/2610.19189#alg2)\. It uses the missed\-signal\-mass loss of Eq\.[9](https://arxiv.org/html/2610.19189#S3.E9), which depends only on the coarse selection and the cell weights, so the refiner𝒢\\mathcal\{G\}plays no part in calibration\.

Algorithm 2Conformal Risk Calibration \(Threshold Selection\)1:Calibration set

𝒟cal=\{\(XiΩfine,YiΩfine\)\}i=1n\\mathcal\{D\}\_\{\\text\{cal\}\}=\\\{\(X\_\{i\}^\{\\Omega\_\{\\text\{fine\}\}\},Y\_\{i\}^\{\\Omega\_\{\\text\{fine\}\}\}\)\\\}\_\{i=1\}^\{n\}, trained coarse model

ℱΩcoarse\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}, downsampling operator

Downsample⁡\(⋅\)\\operatorname\{Downsample\}\(\\cdot\), admissible aggregation operator

AA, scoring function

s⁡\(⋅\)s\(\\cdot\), risk level

α∈\(0,1\)\\alpha\\in\(0,1\), grid of candidates

Λ⊆\[0,1\]\\Lambda\\subseteq\[0,1\]with

1∈Λ1\\in\\Lambda\.

2:Calibrated selection threshold

λ^\\hat\{\\lambda\}\.

3:for

i=1i=1to

nndo

4:

XiΩcoarse←Downsample⁡\(XiΩfine\)X\_\{i\}^\{\\Omega\_\{\\text\{coarse\}\}\}\\leftarrow\\operatorname\{Downsample\}\(X\_\{i\}^\{\\Omega\_\{\\text\{fine\}\}\}\)
5:

O^iΩcoarse←ℱΩcoarse​\(XiΩcoarse\)\\hat\{O\}\_\{i\}^\{\\Omega\_\{\\text\{coarse\}\}\}\\leftarrow\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}\(X\_\{i\}^\{\\Omega\_\{\\text\{coarse\}\}\}\)
6:Cell weights \(Eq\.[8](https://arxiv.org/html/2610.19189#S3.E8)\):

wck​\(Yi\)←A⁡\(YiΩfine\|ck\)w\_\{c\_\{k\}\}\(Y\_\{i\}\)\\leftarrow A\\bigl\(Y\_\{i\}^\{\\Omega\_\{\\text\{fine\}\}\}\\big\|\_\{c\_\{k\}\}\\bigr\)for each

ck∈Ωcoarsec\_\{k\}\\in\\Omega\_\{\\text\{coarse\}\}
7:endfor

8:foreach candidate threshold

λ∈Λ\\lambda\\in\\Lambdado

9:for

i=1i=1to

nndo

10:Form selection vector

σλ∈\{0,1\}Ωcoarse\\sigma\_\{\\lambda\}\\in\\\{0,1\\\}^\{\\Omega\_\{\\text\{coarse\}\}\}:

11:

σλ\[ck\]←𝕀\[s\(XiΩcoarse\)\[ck\]≥1−λ\]\\sigma\_\{\\lambda\}\[c\_\{k\}\]\\leftarrow\\mathbb\{I\}\[s\(X\_\{i\}^\{\\Omega\_\{\\text\{coarse\}\}\}\)\[c\_\{k\}\]\\geq 1\-\\lambda\]
12:Missed signal mass \(Eq\.[9](https://arxiv.org/html/2610.19189#S3.E9),

0/0:=00/0:=0\):

13:

Li\(λ\)←∑k:σλ​\[ck\]=0wck\(Yi\)/∑kwck\(Yi\)L\_\{i\}\(\\lambda\)\\leftarrow\\sum\_\{k:\\,\\sigma\_\{\\lambda\}\[c\_\{k\}\]=0\}w\_\{c\_\{k\}\}\(Y\_\{i\}\)\\,\\big/\\,\\sum\_\{k\}w\_\{c\_\{k\}\}\(Y\_\{i\}\)
14:endfor

15:Compute empirical risk:

R^n​\(λ\)←1n​∑i=1nLi​\(λ\)\\widehat\{R\}\_\{n\}\(\\lambda\)\\leftarrow\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}L\_\{i\}\(\\lambda\)
16:endfor

17:Select threshold guaranteeing controlled risk \(

L∈\[0,1\]L\\in\[0,1\], so

B=1B=1\):

18:

λ^←inf\{λ∈Λ:1n\+1​\(n​R^n​\(λ\)\+1\)≤α\}\\hat\{\\lambda\}\\leftarrow\\inf\\left\\\{\\lambda\\in\\Lambda:\\tfrac\{1\}\{n\+1\}\\bigl\(n\\,\\widehat\{R\}\_\{n\}\(\\lambda\)\+1\\bigr\)\\leq\\alpha\\right\\\}
19:return

λ^\\hat\{\\lambda\}

### A\.3Inference

Once trained and calibrated, the coarse and refiner modelsℱΩcoarse\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}and𝒢\\mathcal\{G\}can be used for inference along with the calibrated thresholdλ^\\hat\{\\lambda\}\. The inference algorithm is detailed in Algorithm[3](https://arxiv.org/html/2610.19189#alg3)\.

Algorithm 3Conformal Adaptation Inference1:Test input

XΩfineX^\{\\Omega\_\{\\text\{fine\}\}\}, models

ℱΩcoarse,𝒢\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\},\\mathcal\{G\}, calibrated threshold

λ^\\hat\{\\lambda\}, downsampling operator

Downsample⁡\(⋅\)\\operatorname\{Downsample\}\(\\cdot\), scoring function

s⁡\(⋅\)s\(\\cdot\)\.

2:Sparsely refined prediction

Y^Ωfine\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\.

3:Downsample coarse input:

XΩcoarse←Downsample⁡\(XΩfine\)X^\{\\Omega\_\{\\text\{coarse\}\}\}\\leftarrow\\operatorname\{Downsample\}\(X^\{\\Omega\_\{\\text\{fine\}\}\}\)
4:Run coarse model:

O^Ωcoarse←ℱΩcoarse​\(XΩcoarse\)\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\}\\leftarrow\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}\(X^\{\\Omega\_\{\\text\{coarse\}\}\}\)
5:Initialize fine canvas:

Y^Ωfine←Upsample⁡\(O^Ωcoarse\)\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\\leftarrow\\operatorname\{Upsample\}\(\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\}\)
6:Determine sparse refinement set:

7:

S←\{ck∈Ωcoarse:s⁡\(XΩcoarse\)​\[ck\]≥1−λ^\}S\\leftarrow\\left\\\{c\_\{k\}\\in\\Omega\_\{\\text\{coarse\}\}:s\(X^\{\\Omega\_\{\\text\{coarse\}\}\}\)\[c\_\{k\}\]\\geq 1\-\\hat\{\\lambda\}\\right\\\}
8:foreach coarse cell

ck∈Sc\_\{k\}\\in Sdo⊳\\trianglerightExecute refinement strictly on sparse locations

9:

Y^Ωfine\|ck←𝒢⁡\(XΩfine\|ck,O^Ωcoarse​\[ck\]\)\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\\big\|\_\{c\_\{k\}\}\\leftarrow\\mathcal\{G\}\\left\(X^\{\\Omega\_\{\\text\{fine\}\}\}\\big\|\_\{c\_\{k\}\},\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\}\[c\_\{k\}\]\\right\)
10:endfor

11:return

Y^Ωfine\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}

## Appendix BDerivation of Guarantees

###### Lemma 1\.

For every\(X,Y\)\(X,Y\), the loss of Eq\.[9](https://arxiv.org/html/2610.19189#S3.E9)satisfiesL⁡\(λ,X,Y\)∈\[0,1\]L\(\\lambda;X,Y\)\\in\[0,1\], is non\-increasing and right\-continuous inλ∈\[0,1\]\\lambda\\in\[0,1\], and satisfiesL⁡\(1,X,Y\)=0L\(1;X,Y\)=0\.

###### Proof\.

Non\-negativity ofAAmakes everywc​\(Y\)≥0w\_\{c\}\(Y\)\\geq 0; the numerator sums a subset of the terms of the denominator, soL∈\[0,1\]L\\in\[0,1\]\(with0/0:=00/0:=0\)\. The denominator does not depend onλ\\lambda\. Ifλ1≤λ2\\lambda\_\{1\}\\leq\\lambda\_\{2\}thenSλ1​\(X\)⊆Sλ2​\(X\)S\_\{\\lambda\_\{1\}\}\(X\)\\subseteq S\_\{\\lambda\_\{2\}\}\(X\)by Eq\.[6](https://arxiv.org/html/2610.19189#S3.E6), so passing fromλ1\\lambda\_\{1\}toλ2\\lambda\_\{2\}deletes non\-negative terms from the numerator andL⁡\(λ2,X,Y\)≤L⁡\(λ1,X,Y\)L\(\\lambda\_\{2\};X,Y\)\\leq L\(\\lambda\_\{1\};X,Y\)\. SinceΩcoarse\\Omega\_\{\\text\{coarse\}\}is finite, the score fieldsstakes finitely many values, and the closed threshold in Eq\.[6](https://arxiv.org/html/2610.19189#S3.E6)makesλ↦Sλ​\(X\)\\lambda\\mapsto S\_\{\\lambda\}\(X\)right\-continuous; hence so isLL\. Atλ=1\\lambda=1every cell satisfiess⁡\(XΩcoarse\)​\[c\]≥0s\(X^\{\\Omega\_\{\\text\{coarse\}\}\}\)\[c\]\\geq 0, soS1​\(X\)=ΩcoarseS\_\{1\}\(X\)=\\Omega\_\{\\text\{coarse\}\}, the numerator is an empty sum, andL⁡\(1,X,Y\)=0L\(1;X,Y\)=0\. ∎

###### Lemma 2\.

Under Conditions[1](https://arxiv.org/html/2610.19189#Thmcondition1)and[2](https://arxiv.org/html/2610.19189#Thmcondition2), for everyY∈𝒴ΩfineY\\in\\mathcal\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}and everyc∈Ωcoarsec\\in\\Omega\_\{\\text\{coarse\}\},

wc​\(Y\)\>0⇔c∈𝒪⁡\(Y\)\.w\_\{c\}\(Y\)\>0\\iff c\\in\\mathcal\{O\}\(Y\)\.Consequently, for everyλ∈\[0,1\]\\lambda\\in\[0,1\],

L⁡\(λ,X,Y\)=0⇔𝒪⁡\(Y\)⊆Sλ​\(X\),L\(\\lambda;X,Y\)=0\\iff\\mathcal\{O\}\(Y\)\\subseteq S\_\{\\lambda\}\(X\),and the right\-hand event depends on neitherAAnorλ\\lambdaexcept throughSλ​\(X\)S\_\{\\lambda\}\(X\)\. In particular all admissible operators induce losses with the same zero set\.

###### Proof\.

Ifc∉𝒪⁡\(Y\)c\\notin\\mathcal\{O\}\(Y\)thenYΩfine\|c≡∅Y^\{\\Omega\_\{\\text\{fine\}\}\}\|\_\{c\}\\equiv\\varnothingand Condition[1](https://arxiv.org/html/2610.19189#Thmcondition1)giveswc​\(Y\)=0w\_\{c\}\(Y\)=0\. Ifc∈𝒪⁡\(Y\)c\\in\\mathcal\{O\}\(Y\)then some fine site ofcccarries a non\-null label and Condition[2](https://arxiv.org/html/2610.19189#Thmcondition2)giveswc​\(Y\)\>0w\_\{c\}\(Y\)\>0\. This proves the equivalence\.

For the second claim, suppose first that𝒪⁡\(Y\)=∅\\mathcal\{O\}\(Y\)=\\emptyset\. Then all weights vanish, the convention0/0:=00/0:=0givesL=0L=0, and the inclusion holds trivially\. Otherwise the denominator of Eq\.[9](https://arxiv.org/html/2610.19189#S3.E9)is strictly positive, soL⁡\(λ,X,Y\)=0L\(\\lambda;X,Y\)=0iff∑c∉Sλ​\(X\)wc​\(Y\)=0\\sum\_\{c\\notin S\_\{\\lambda\}\(X\)\}w\_\{c\}\(Y\)=0, iffwc​\(Y\)=0w\_\{c\}\(Y\)=0for everyc∉Sλ​\(X\)c\\notin S\_\{\\lambda\}\(X\)\(all terms being non\-negative\), iff no occupied cell lies outsideSλ​\(X\)S\_\{\\lambda\}\(X\)by the equivalence just proved\. ∎

###### Lemma 3\.

Letσ∈\{0,1\}Ωcoarse\\sigma\\in\\\{0,1\\\}^\{\\Omega\_\{\\text\{coarse\}\}\}with selected setSS, and writeY^σΩfine:=𝒢Ωcoarse→Ωfine​\(XΩfine,O^Ωcoarse,σ\)\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\sigma\}:=\\mathcal\{G\}\_\{\\Omega\_\{\\text\{coarse\}\}\\to\\Omega\_\{\\text\{fine\}\}\}\(X^\{\\Omega\_\{\\text\{fine\}\}\},\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\},\\sigma\)\. Then

Y^σΩfine\|c=Y^𝟏Ωfine\|cfor every​c∈S,Y^σΩfine\|c=Y^𝟎Ωfine\|cfor every​c∉S,\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\sigma\}\\big\|\_\{c\}=\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\mathbf\{1\}\}\\big\|\_\{c\}\\quad\\text\{for every \}c\\in S,\\qquad\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\sigma\}\\big\|\_\{c\}=\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\mathbf\{0\}\}\\big\|\_\{c\}\\quad\\text\{for every \}c\\notin S,whereY^𝟎Ωfine=Upsample⁡\(O^Ωcoarse\)\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\mathbf\{0\}\}=\\operatorname\{Upsample\}\(\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\}\)\. Consequently, for any cellwise functionalϕ\\phi\(i\.e\.ϕ​\(Y\)​\[c\]\\phi\(Y\)\[c\]depends only onY\|cY\|\_\{c\}\),

ϕ⁡\(Y^σΩfine\)​\[c\]=σ⁡\[c\]​ϕ​\(Y^𝟏Ωfine\)​\[c\]\+\(1−σ⁡\[c\]\)​ϕ​\(Y^𝟎Ωfine\)​\[c\],\\phi\\bigl\(\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\sigma\}\\bigr\)\[c\]=\\sigma\[c\]\\,\\phi\\bigl\(\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\mathbf\{1\}\}\\bigr\)\[c\]\+\\bigl\(1\-\\sigma\[c\]\\bigr\)\\,\\phi\\bigl\(\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\mathbf\{0\}\}\\bigr\)\[c\],and in particular

𝒪⁡\(Y^σΩfine\)=\(S∩𝒪⁡\(Y^𝟏Ωfine\)\)∪\(𝒪⁡\(Y^𝟎Ωfine\)∖S\)\.\\mathcal\{O\}\\bigl\(\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\sigma\}\\bigr\)=\\bigl\(S\\cap\\mathcal\{O\}\(\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\mathbf\{1\}\}\)\\bigr\)\\,\\cup\\,\\bigl\(\\mathcal\{O\}\(\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\mathbf\{0\}\}\)\\setminus S\\bigr\)\.Sparse refinement is a cellwise selection between two fields, neither of which depends onσ\\sigma\.

###### Proof\.

By Eq\.[5](https://arxiv.org/html/2610.19189#S3.E5), the output of𝒢Ωcoarse→Ωfine\\mathcal\{G\}\_\{\\Omega\_\{\\text\{coarse\}\}\\to\\Omega\_\{\\text\{fine\}\}\}on a selected cellccis𝒢c​\(XΩfine\|c,O^Ωcoarse​\[c\]\)\\mathcal\{G\}\_\{c\}\(X^\{\\Omega\_\{\\text\{fine\}\}\}\|\_\{c\},\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\}\[c\]\)\. Its arguments do not depend onσ\\sigma, so it coincides with the value onccwhenσ=𝟏\\sigma=\\mathbf\{1\}\. On an unselected cell,ccis never overwritten, so the output retains the initial canvasUpsample⁡\(O^Ωcoarse\)\|c\\operatorname\{Upsample\}\(\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\}\)\|\_\{c\}\(Algorithm[3](https://arxiv.org/html/2610.19189#alg3), line 3\)\. This canvas is computed fromO^Ωcoarse\\hat\{O\}^\{\\Omega\_\{\\text\{coarse\}\}\}alone, so it is independent ofσ\\sigmaand equalsY^𝟎Ωfine\|c\\hat\{Y\}^\{\\Omega\_\{\\text\{fine\}\}\}\_\{\\mathbf\{0\}\}\|\_\{c\}\. The formula for a cellwiseϕ\\phifollows by evaluatingϕ\\phion each cell in the two cases\. The occupancy identity is the instanceϕ=𝕀\[c∈𝒪\(⋅\)\]\\phi=\\mathbb\{I\}\[c\\in\\mathcal\{O\}\(\\cdot\)\]\. ∎

See[3\.1](https://arxiv.org/html/2610.19189#S3.Thmtheorem1)

###### Proof\.

For \(ii\), we see that Lemma[1](https://arxiv.org/html/2610.19189#Thmlemma1)verifies the hypotheses of the CRC guarantee \(Eq\.[1](https://arxiv.org/html/2610.19189#S2.E1)\) for the family\{L⁡\(λ,⋅,⋅\)\}λ∈\[0,1\]\\\{L\(\\lambda;\\cdot,\\cdot\)\\\}\_\{\\lambda\\in\[0,1\]\}, which under exchangeability yields𝔼⁡\[L⁡\(λ^,X,Y\)\]≤α\\mathbb\{E\}\[L\(\\hat\{\\lambda\};X,Y\)\]\\leq\\alpha\. Claim \(i\) is Lemma[3](https://arxiv.org/html/2610.19189#Thmlemma3)applied withσ=σλ^​\(X\)\\sigma=\\sigma\_\{\\hat\{\\lambda\}\}\(X\), and it identifies⋃c∈DΩfine\|c\\bigcup\_\{c\\in D\}\\Omega\_\{\\text\{fine\}\}\|\_\{c\}as the exact support of the discrepancy between the sparse and dense pipelines, which is the region whose mass \(ii\) bounds\. The tail bound in \(iii\) is Markov’s inequality applied to the non\-negative variableL⁡\(λ^,X,Y\)L\(\\hat\{\\lambda\};X,Y\), and the characterization of its zero set is Lemma[2](https://arxiv.org/html/2610.19189#Thmlemma2)\. The discarded mass fraction is precisely the loss at the calibrated threshold:

L⁡\(λ^,X,Y\)=∑c∈Dwc​\(Y\)∑c∈Ωcoarsewc​\(Y\),D=Ωcoarse∖Sλ^​\(X\),L\(\\hat\{\\lambda\};X,Y\)=\\frac\{\\sum\_\{c\\in D\}w\_\{c\}\(Y\)\}\{\\sum\_\{c\\in\\Omega\_\{\\text\{coarse\}\}\}w\_\{c\}\(Y\)\},\\qquad D=\\Omega\_\{\\text\{coarse\}\}\\setminus S\_\{\\hat\{\\lambda\}\}\(X\),with0/0:=00/0:=0\. By Lemma[1](https://arxiv.org/html/2610.19189#Thmlemma1),L⁡\(λ^,X,Y\)≥0L\(\\hat\{\\lambda\};X,Y\)\\geq 0, and by \(ii\),𝔼⁡\[L⁡\(λ^,X,Y\)\]≤α\\mathbb\{E\}\[L\(\\hat\{\\lambda\};X,Y\)\]\\leq\\alpha\. Fort∈\(0,1\]t\\in\(0,1\], Markov’s inequality gives

Pr⁡\(L⁡\(λ^,X,Y\)≥t\)≤𝔼⁡\[L⁡\(λ^,X,Y\)\]t≤αt,\\Pr\\bigl\(L\(\\hat\{\\lambda\};X,Y\)\\geq t\\bigr\)\\;\\leq\\;\\frac\{\\mathbb\{E\}\[L\(\\hat\{\\lambda\};X,Y\)\]\}\{t\}\\;\\leq\\;\\frac\{\\alpha\}\{t\},and therefore

Pr⁡\(L⁡\(λ^,X,Y\)≤t\)≥Pr⁡\(L⁡\(λ^,X,Y\)<t\)=1−Pr⁡\(L⁡\(λ^,X,Y\)≥t\)≥1−αt\.\\Pr\\bigl\(L\(\\hat\{\\lambda\};X,Y\)\\leq t\\bigr\)\\;\\geq\\;\\Pr\\bigl\(L\(\\hat\{\\lambda\};X,Y\)<t\\bigr\)\\;=\\;1\-\\Pr\\bigl\(L\(\\hat\{\\lambda\};X,Y\)\\geq t\\bigr\)\\;\\geq\\;1\-\\frac\{\\alpha\}\{t\}\.That is, with probability at least1−α/t1\-\\alpha/t, the cells inDDcarry at most att\-fraction of the total signal mass∑c∈Ωcoarsewc​\(Y\)\\sum\_\{c\\in\\Omega\_\{\\text\{coarse\}\}\}w\_\{c\}\(Y\)\.

For the zero set, Lemma[2](https://arxiv.org/html/2610.19189#Thmlemma2)holds for everyλ∈\[0,1\]\\lambda\\in\[0,1\], so it applies to each realisation ofλ^\\hat\{\\lambda\}:

L⁡\(λ^,X,Y\)=0⇔𝒪⁡\(Y\)⊆Sλ^​\(X\)\.L\(\\hat\{\\lambda\};X,Y\)=0\\iff\\mathcal\{O\}\(Y\)\\subseteq S\_\{\\hat\{\\lambda\}\}\(X\)\.This event depends onAAonly throughSλ^​\(X\)S\_\{\\hat\{\\lambda\}\}\(X\)\. Its form is therefore the same for every admissibleAA, although the event itself can still vary withAAthroughλ^\\hat\{\\lambda\}\. ∎

## Appendix CExperiment Details

### C\.1MSD Brain Tumor Segmentation

The MSD experiment’s data, model, and calibration details are presented in this section\.

#### C\.1\.1Dataset Details

We use the MSD*Task01: Brain Tumour*dataset\([Antonelli et al\., 2022](https://arxiv.org/html/2610.19189#bib.bib10)\), comprising 484 multimodal brain MRI scans, each with four co\-registered240×240×155240\\times 240\\times 155\-voxel sequences \(FLAIR, T1\-weighted, contrast\- enhanced T1\-weighted, and T2\-weighted\) and a voxel\-wise annotation into three positive tumor subregions \(edema, non\-enhancing tumor, and enhancing tumor\) plus background\. The official MSD test split ships without public labels, so, following standard practice, we instead use MONAI’s deterministic 80/20 split of the released*training*set: the resulting 96\-scan validation subset is held out entirely and used as our test set, while the remaining 388\-scan training subset is further split into a 320\-scan model\-fitting set and a 68\-scan \(17\.5%\) calibration set, sampled with a fixed random seed independent of the 80/20 split\. The calibration set is held out of all first\-stage model training; it is used exclusively to calibrate the conformal risk\-control thresholdλ^\\hat\{\\lambda\}\.

#### C\.1\.2Data Preprocessing

Each modality is independently normalized by clipping its non\-zero voxels to their own 99\.5th percentile and rescaling to\[0,1\]\[0,1\]; background \(zero\) voxels are left untouched\. Volumes are then spatially padded to the nearest multiple of the coarse downsampling factorr=4r=4on every axis, so that the coarse pass’s4×4×44\\times 4\\times 4average/max pooling divides the volume exactly\. The coarse \(low\-resolution\) image is obtained by trilinear downsampling of the padded volume byrr; the corresponding coarse label is obtained by non\-overlapping max\-pooling over the same4×4×44\\times 4\\times 4blocks, so that a positive \(tumor\) voxel is never averaged away at low resolution\. For training the first\-stage \(coarse and full\-resolution\) segmentation models, we sample fixed64×64×3264\\times 64\\times 32\-voxel crops per scan, biased 1:1 toward foreground\-containing regions, followed by random flips along each of the three spatial axes \(p=0\.5p=0\.5each\) and a random90∘90^\{\\circ\}in\-plane rotation \(p=0\.5p=0\.5\), then pad back to the fixed crop size\. Evaluation, calibration, and the refinement stage instead operate on the full padded volume via sliding\-window inference with the same64×64×3264\\times 64\\times 32window\.

#### C\.1\.3Model Architecture

We compare three segmentation architectures, all trained from scratch \(no pretrained weights\) and used identically as the coarse\-pass model and as the full\-resolution baseline – only the spatial resolution of the input differs between the two, isolating the effect of input resolution from any change in model capacity\.

- •U\-Net\([Ronneberger et al\., 2015](https://arxiv.org/html/2610.19189#bib.bib36)\): a 3\-level 3D U\-Net with channel widths\(16,32,64\)\(16,32,64\)and stride\-2 downsampling at each level\.
- •Swin\-UNETR\([Hatamizadeh et al\., 2021](https://arxiv.org/html/2610.19189#bib.bib38);[He et al\., 2023](https://arxiv.org/html/2610.19189#bib.bib37)\): a Swin Transformer encoder with a convolutional U\-Net\-style decoder, at its default configuration\.
- •VISTA3D\([He et al\., 2025](https://arxiv.org/html/2610.19189#bib.bib39)\): a segmentation foundation\-model architecture composed of a SegResNet\-based image encoder \(downsampling blocks\(1,2,2,4,4\)\(1,2,2,4,4\), instance normalization, 48\-channel embedding\) and a class\-mapping classification head\. We run VISTA3D in its automatic \(class\-vector\) segmentation mode only, without any point prompts, so that it can be invoked identically to the other two architectures\.

The region\-refinement stage uses a separate, lightweight two\-branch 3D convolutional network, common to all three first\-stage architectures\. Each selected coarse cell is a4×4×44\\times 4\\times 4block; adding a one\-voxel halo on every side gives a6×6×66\\times 6\\times 6high\-resolution window\. One branch encodes this high\-resolution image window and the other encodes the upsampled coarse\-model mask over the same window \(two3×3×33\\times 3\\times 3convolutions, 16 channels, GELU activations\); the two branches are fused by addition and passed through two further convolutions before a1×1×11\\times 1\\times 1output head\. The refiner’s full6×6×66\\times 6\\times 6output is center\-cropped to the central4×4×44\\times 4\\times 4block, which is what is scored against the ground truth and scattered back into the full\-resolution prediction; the halo voxels serve only as context for the convolutions\.

#### C\.1\.4Training and Calibration Details

Each of the three first\-stage architectures is trained at both full and4×4\\times\-downsampled resolution, for three random seeds, with Adam \(learning rate10−410^\{\-4\}, batch size 8, up to 100 epochs\), minimizing an equally weighted sum of Dice loss and cross\-entropy\. A single region\-refinement network, shared by all architectures and seeds, is trained for one epoch with Adam \(learning rate10−310^\{\-3\}, batch size 256\), minimizing cross\-entropy on the central4×4×44\\times 4\\times 4block only\. Its training windows are the cells the coarse model selects at itsα=0\.1\\alpha=0\.1calibrated threshold on the 388\-scan training subset, with that model’s softmax as the coarse\-mask input\. CRC is calibrated on each architecture’s coarse\-pass foreground probability, per training seed, using the missed\-positive\-cell\-fraction loss of Eq\.[9](https://arxiv.org/html/2610.19189#S3.E9)on the 68\-scan calibration set described above; the calibrated thresholdλ^\\hat\{\\lambda\}, the resulting retained cell fraction, and downstream DSC and FLOPs are then evaluated on the fully held\-out 96\-scan test set\.

#### C\.1\.5Qualitative Examples

Fig\.[3](https://arxiv.org/html/2610.19189#S5.F3)in the main text \(Sec\.[5](https://arxiv.org/html/2610.19189#S5)\) compares predicted tumor boundaries directly; Fig\.[8](https://arxiv.org/html/2610.19189#A3.F8)here shows, for the same three test scans, how the region CRC sends to refinement \(Eq\.[1](https://arxiv.org/html/2610.19189#S2.E1)\) shrinks toward the tumor as the target miss rateα\\alphagrows, together with the resulting measured per\-scan cell miss rate\. Thus, the coverage guarantee behind Table[1](https://arxiv.org/html/2610.19189#S5.T1)’s cell miss rate column is visible at the level of a single slice, not only as an aggregate statistic\.

![Refer to caption](https://arxiv.org/html/2610.19189v1/figures/decathlon/threshold_sweep_scan0.png)

![Refer to caption](https://arxiv.org/html/2610.19189v1/figures/decathlon/threshold_sweep_scan1.png)

Figure 8:Selected \(refined\) region against the calibrated threshold for two held\-out test scans \(Swin\-UNETR, seed 0\)\. ”Full refinement” is the reference at foreground\-probability threshold00, i\.e\.λ=1\\lambda=1in Eq\.[6](https://arxiv.org/html/2610.19189#S3.E6)\(every foreground\-probable cell refined\); atα=1%\\alpha=1\\%and2%2\\%, CRC’s calibrated threshold is itself00\(λ^=1\\hat\{\\lambda\}=1\) for this architecture/seed\. Asα\\alpharelaxes, the selected region shrinks toward the tumor boundary and the measured cell miss rate rises toward its target\.

### C\.2HECKTOR 2026 Head and Neck Tumor Segmentation

#### C\.2\.1Data Preprocessing

There are three NIfTI volumes per case: the original CT, the FDG\-PET, and the segmentation label\. The labels are included as 0 for background, 1 for GTVp \(primary\), and 2 for GTVn \(node\)\.

The 782 cases are split across 531 train, 125 validation, and 126 test samples\. The 125\-case validation split is theCRC calibration set, not a model\-selection set\. Note that validation splits are obtained randomly from the training set for the Coarse and Refine models\.

#### C\.2\.2Results

Full test\-set results for the HNC study are provided in Tables[2](https://arxiv.org/html/2610.19189#A3.T2)and[3](https://arxiv.org/html/2610.19189#A3.T3)\. It is important to note that the there is a drop in accuracy especially for GTVn predictions when using the Coarse \+ Refine mechanism\. This is particularly due to the fact that there small lesions that are being missed by the coarse model and being enabled by the CRC loss on cell\-level misses\. We acknowledge this as a negative result for our method as it is a particularly adversarial task given many small lesions in HNCs\. A custom designed coarse model and a lesion\-based CRC loss can potentially significantly improve our current lesion\-based results as highlighted before, is a significant area of future work that can be pursued\.

Table 2:DSC and FLOPs speedup across models\. All rows except nnU\-Net are Swin\-UNETR\. Mean DSC is the unweighted average of GTVp and GTVn; speedup is the ratio of Full\-model FLOPs to pipeline FLOPs\.Table 3:Lesion\-specific results for our HNC models\. All rows except nnU\-Net are Swin\-UNETR\. Unlike the volumetric DSC, we perform post\-processing to detect lesions from predicted volumes to calculate lesion detection metrics\.Qualitative results for the predictions are presented in Fig\.[9](https://arxiv.org/html/2610.19189#A3.F9)\.

![Refer to caption](https://arxiv.org/html/2610.19189v1/contours_nnu_net_notitles_rot180.png)

![Refer to caption](https://arxiv.org/html/2610.19189v1/contours_one_stage_swin_unetr_native_notitles_rot180.png)

![Refer to caption](https://arxiv.org/html/2610.19189v1/contours_one_stage_swin_unetr_4x_down_notitles_rot180.png)

![Refer to caption](https://arxiv.org/html/2610.19189v1/contours_two_stage_swin_unetr_notitles_rot180.png)

Figure 9:Grid of HECKTOR predictions\. From top to bottom: nnU\-Net, Swin\-UNETR native resolution, Swin\-UNETR Coarse, and Swin\-UNETR Coarse \+ Refine\.

### C\.3Threshold selection for Segmentation

Depending on the model and dataset, the threshold can result in significantly varied FNR as seen in Tab[4](https://arxiv.org/html/2610.19189#A3.T4)\. This highlights the need for using an established threshold finding algorithm like CRC\. Naively choosingλ\\lambdacan either result in selecting too many cells, diminishing the speedup achieved, or too few cells, resulting an underestimated error\. Furthermore, as Tab\.[4](https://arxiv.org/html/2610.19189#A3.T4)shows, different models and tasks have different thresholds\. Thus effectively selecting a threshold requires a held\-out calibration set, and CRC is a natural algorithm for choosing such a threshold\. Also, using CRC allows us to expand our choices beyond solely threshold based methods\. The pre\-requisite for using CRC is monotonically increasing prediction sets which can be accomplished by other selection criterion as well\.

Table 4:A naively chosen like threshold can result in high FNR as the predicted scores are non\-linear compared to the selection\. CRC
### C\.4CASCADE Calcium Spike Prediction

The CASCADE experiment data processing, model design, and calibration methods are presented in this section\.

#### C\.4\.1Dataset Details

We use the*primary*group of the CASCADE calcium\-imaging benchmark: datasets DS30, DS31, and DS32, recorded with the GCaMP8f/8m/8s indicators respectively\. Each dataset file contains one or more trials, where a trial pairs a fluorescence \(Δ​F/F\\Delta F/F\) trace with ground\-truth spike times obtained from simultaneous electrophysiological recording; frame rates vary by recording session, ranging from 8–60 Hz across the benchmark\. The primary group comprises 117 source\.matfiles spanning 515 valid trials \(trials with fewer than 2 recorded spike events or fewer than 10 fluorescence samples are discarded\), at a mean spike\-mass occupancy \(fraction of non\-empty coarse blocks atr=8r\{=\}8\) of3\.23\.2–4\.0%4\.0\\%\.

Because repeated trials from the same source file are highly dependent draws from the same underlying neuron \(identical indicator, noise level, and firing statistics\), we split*source files*, not individual windows or trials, into train/calibration/test partitions at a60/20/2060/20/20ratio, stratified per dataset so all three indicators appear in every partition\. This file\-level split is the exchangeable unit for the CRC calibration; every downstream step \(coarse model training, CRC calibration, refiner training, evaluation\) consumes the same fixed seed\-0 split, and fluorescence windows are unique between splits\. This split rules out test\-set leakage; the standard sample\-level calibration algorithm is then applied unchanged\.

#### C\.4\.2Data Preprocessing

Each trial’s raw spike times are binned onto the trial’s own fluorescence frame timestamps via a histogram, and the resulting spike\-count sequence is convolved with a Gaussian kernel \(standard deviation200200ms, the CASCADE benchmark’s own default\) and rescaled to spikes/second to obtain the fine\-resolution regression targetYΩfineY^\{\\Omega\_\{\\text\{fine\}\}\}\. The coarse targetYΩcoarseY^\{\\Omega\_\{\\text\{coarse\}\}\}is obtained by*sum*\-pooling \(not averaging\) the raw spike counts by the coarse block factorr=8r\{=\}8, so that each coarse cell’s value is exactly the expected spike count inside it, matching the aggregation operator of Eq\.[8](https://arxiv.org/html/2610.19189#S3.E8)for regression\. The CRC loss is the fraction of missed spike\-count mass\. The fluorescence trace itself iszz\-score normalized per source file \(median and1\.4826×1\.4826\\timesMAD, computed once across all of a file’s trials to avoid leaking window\-local statistics\), then windowed into fixed\-length segments of10241024fine samples \(128128coarse blocks atr=8r\{=\}8\), with stride512512for training, calibration, and test\.

#### C\.4\.3Model Architecture

Both the baseline and coarse networks in the pipeline are 1D convolutional nets sharing a common building block \(ConvBlock: twoConv1d⁡\(k=3\)\\mathrm\{Conv1d\}\(k\{=\}3\)\+GroupNorm\+GELU layers\) and a common output head design: aQuantileHeadpredictingQ=4Q\{=\}4quantiles \(τ∈\{0\.05,0\.5,0\.9,0\.95\}\\tau\\in\\\{0\.05,0\.5,0\.9,0\.95\\\}\) by regressing theτ=0\.05\\tau\{=\}0\.05quantile plus non\-negative \(softplus\) increments, cumulatively summed to guarantee monotonicity\.

- •Full baselineℱΩfine\\mathcal\{F\}\_\{\\Omega\_\{\\text\{fine\}\}\}\(FineUNet1D\): a44\-level 1D U\-Net, base width6464, withMaxPool1d⁡\(2\)\\mathrm\{MaxPool1d\}\(2\)downsampling and nearest\-neighbor upsample \+ skip\-concatenation on the decoder side, applied to the full10241024\-sample window\.
- •Coarse modelℱΩcoarse\\mathcal\{F\}\_\{\\Omega\_\{\\text\{coarse\}\}\}\(FineOnLowRes\): reusesℱΩfine\\mathcal\{F\}\_\{\\Omega\_\{\\text\{fine\}\}\}’s architecture and parameter count*exactly*, but is fed the trace block\-averaged down towindow/r=128\\mathrm\{window\}/r=128samples before the first convolution, rather than a separate, narrower network\. This isolates the coarse model’s speedup to resolution reduction alone: at identical capacity toℱΩfine\\mathcal\{F\}\_\{\\Omega\_\{\\text\{fine\}\}\}, this gives an8\.0×8\.0\\timesFLOPs reduction for the coarse pass \(exactlyrr, since a fixed\-width conv net’s FLOPs scale linearly with sequence length\), so no reviewer can attribute the pipeline’s speedup to simply swapping in a smaller\-capacity model\.
- •Refiner𝒢Ωcoarse→Ωfine\\mathcal\{G\}\_\{\\Omega\_\{\\text\{coarse\}\}\\to\\Omega\_\{\\text\{fine\}\}\}\(LocalRefinerUNet1D\): a22\-level 1D U\-Net \(base width3232\) over a context\-padded crop of lengthL=r⁡\(1\+2×context​\_​blocks\)=136L=r\(1\+2\\times\\mathrm\{context\\\_blocks\}\)=136samples \(context​\_​blocks=8\\mathrm\{context\\\_blocks\}\{=\}8\) around each retained coarse block, whose full\-length decoder output is center\-cropped to the middler=8r\{=\}8samples\. Rather than predicting the fine\-resolution rate curveYΩfineY^\{\\Omega\_\{\\text\{fine\}\}\}directly \(which would hard\-zero discarded neighboring blocks regardless of proximity to a confirmed detection\), the refiner uses aCountHead\(global\-average\-pool \+ linear projection\) to predict a single non\-negative spike\-*count*estimate per retained block, matchingYΩcoarseY^\{\\Omega\_\{\\text\{coarse\}\}\}’s own resolution\. The full continuous rate curve is then reconstructed by convolving this sparse, mostly\-zero per\-block count estimate with the same Gaussian kernel used to buildYΩfineY^\{\\Omega\_\{\\text\{fine\}\}\}from raw spike counts\. Since that kernel is mass\-preserving, bounding missed count mass at the CRC levelα\\alpha\(calibrated againstYΩcoarseY^\{\\Omega\_\{\\text\{coarse\}\}\}\) also bounds the missed mass of the reconstructed continuous curve at the sameα\\alpha\. A detection’s estimated count can be smeared physically into its discarded neighbors rather than being hard\-zeroed there during post processing and presentation\.Note that the certification region is decoupled from the presentation in high resolution\.

Table[5](https://arxiv.org/html/2610.19189#A3.T5)summarizes parameter counts and analytic FLOPs \(atwindow=1024\\mathrm\{window\}\{=\}1024\) for the coarse model and full\-resolution baseline, computed with each model’s default constructor arguments\.

Table 5:CASCADE model architectures, parameter counts, and analytic FLOPs\. The total FLOPs for the refiner depends on the number of selected windows and is therefore omitted\.
#### C\.4\.4Training and Calibration Details

All three networks \(baseline, coarse model, refiner\) are trained with Adam \(learning rate10−310^\{\-3\}\) for3030epochs, minimizing the quantile pinball loss summed overτ∈\{0\.05,0\.5,0\.9,0\.95\}\\tau\\in\\\{0\.05,0\.5,0\.9,0\.95\\\}for the baseline and coarse model, and mean\-squared error against the block spike count for the refiner’s count head\. CRC is calibrated on the coarse model’s predictedτ=0\.9\\tau\{=\}0\.9quantile score field \(empirically the best\-performing of the median/q90/ q95 candidates by screening ROC AUC\), using the missed\-signal\-mass loss of Eq\.[9](https://arxiv.org/html/2610.19189#S3.E9), with a retention halo of11coarse block on either side of every selected block \(chosen heuristically\)\. For the reported operating point \(α=0\.1\\alpha\{=\}0\.1\), CRC yields a calibratedλ^\\hat\{\\lambda\}with a raw\-score cutoff of0\.8770\.877and a retained fraction of8\.4%8\.4\\%of coarse blocks on held\-out test data, corresponding to the4\.77×4\.77\\timesFLOPs speedup reported in Table[6](https://arxiv.org/html/2610.19189#A3.T6)\. A separate refiner is trained per calibratedα\\alpha\(each refiner’s training crops are drawn from thatα\\alpha’s own retention distribution\), and reconstructed\-curve accuracy is reported as mean absolute error \(MAE\) againstYΩfineY^\{\\Omega\_\{\\text\{fine\}\}\}on the test split, compared against the full\-resolution baseline’s own median \(τ=0\.5\\tau\{=\}0\.5\) prediction on the same windows\.

#### C\.4\.5Results

The reconstruction results with varyingα\\alpha, their corresponding retained fraction, and final error is shown in Tab\.[6](https://arxiv.org/html/2610.19189#A3.T6)\.

Table 6:Reconstructed MAE is reported in Hz, with the gap to the full\-resolution baseline \(0\.2500\.250Hz\) in parentheses\.

### C\.5Galaxy Flux Field Super\-Resolution

The galaxy flux field experiment’s data generation, model design, training, and calibration methods are presented in this section\.

#### C\.5\.1Dataset Details

Each sample is a simulated1024×10241024\\times 1024sky field at a pixel scale of0\.2′′0\.2^\{\\prime\\prime\}, covering roughly3\.4′×3\.4′3\.4^\{\\prime\}\\times 3\.4^\{\\prime\}\. The number of sources per field is Poisson\-distributed with mean6060, so the realized count varies field to field and occupancy is a statistic over fields rather than a fixed per\-field quantity\. We generate30003000fields in total, partitioned2000/500/5002000/500/500into train, calibration, and test splits\. One field is one exchangeable unit for conformal risk control: unlike the calcium\-imaging study there is no windowing, so a field is never split across partitions\.

Sources are real galaxies drawn from the COSMOS25\.225\.2magnitude\-limited sample \(81,49981\{,\}499objects\) rather than point sources\. We use the analytic Sérsic fits to the catalog objects rather than the raw telescope pixel data, so each source carries the real size, shape, orientation, and Sérsic\-index distribution of an observed galaxy but is a smooth, noiseless profile\.

We avoid the raw pixel data because it carries the original exposure’s own correlated noise: rescaling a source to this experiment’s flux prior can require a factor of more than four orders of magnitude, which would amplify that embedded noise by the same factor and introduce spatially non\-stationary contamination unrelated to the sky and read\-noise model the experiment is built around\.

The three splits draw from*disjoint contiguous slices*of the catalog, in the proportions of the split sizes\. We ensure the same galaxy shape \(same catalog object, differing only in flux, position, rotation, and noise\) cannot appear in both a training field and a test field, since every field otherwise samples uniformly from the whole catalog\. This is a known failure mode for this catalog, and holding out objects rather than only images follows standard practice for COSMOS\-based generative and deblending studies\([Arcelin et al\., 2021](https://arxiv.org/html/2610.19189#bib.bib35)\)\.

#### C\.5\.2Data Generation Pipeline

Every field is reproducible from a single integer seed; all randomness is drawn from one seeded generator, so caching amounts to calling the generator once per seed\. For each field we draw the source count and then, for each source, draw a flux log\-uniformly on\[102,105\]\[10^\{2\},10^\{5\}\]counts, a position uniform in the field interior \(a2020\-pixel margin is excluded so sources are not truncated at the boundary\), a catalog object from that split’s slice, and a uniform rotation\. Each source is rendered into a65×6565\\times 65pixel stamp centered on its position\.

The same object is rendered*twice*, in two different roles, and the two blurring operators are kept strictly separate:

- •Input image\.The source is convolved with the instrument point spread function, a Kolmogorov profile of full width at half maximum0\.8′′0\.8^\{\\prime\\prime\}, i\.e\. the forward model of a real observation\.
- •Regression target\.The same source is convolved instead with a small idealized Gaussian kernel with a standard deviation of1\.51\.5pixels \(0\.3′′0\.3^\{\\prime\\prime\}\), and no noise is added\.

Neither kernel is derived from the other\. The target is therefore a deconvolved, noise\-free surface\-brightness map of the same scene: the task is simultaneously deblurring and denoising, not merely upsampling\.

After all sources are drawn, synthetic detector noise is added to the input image only, as Poisson noise about a sky level of5050counts plus Gaussian read noise of55counts at unit gain\. This is the only noise in the experiment\. The known sky level is subtracted from the image before it is presented to any network, so a network’s input is a background\-subtracted, noisy, PSF\-convolved field and its target is the same field’s noiseless, sharply rendered flux\.

The coarse partition is the8×88\\times 8\-pixel cell grid, giving128×128=16,384128\\times 128=16\{,\}384coarse cells per field\. The coarse target is obtained by*sum*\-pooling the fine target over each cell, not averaging, so a cell’s value is exactly the total flux inside it and matches the regression aggregation operator of Eq\.[8](https://arxiv.org/html/2610.19189#S3.E8); the CRC loss is then literally the fraction of missed flux\. The coarse model’s input is obtained by*average*\-pooling the observed image by the same factor, because the image is a surface brightness rather than a total\.

#### C\.5\.3Model Architecture

All three networks share a common convolutional block \(two3×33\\times 3convolutions, each with group normalization and a GELU nonlinearity\) and, for the two predicting networks, a common output head predictingQ=4Q=4quantiles \(τ∈\{0\.05,0\.5,0\.9,0\.95\}\\tau\\in\\\{0\.05,0\.5,0\.9,0\.95\\\}\) of the log\-transformed flux\. The head regresses theτ=0\.05\\tau\{=\}0\.05quantile plus non\-negative softplus increments, cumulatively summed, so predicted quantiles are monotone by construction and can never cross\. All flux regression is performed inlog\(1\+⋅\)\\log\(1\+\\cdot\)space, which is necessary given the three\-decade flux prior\.

- •Full baseline\.A four\-level two\-dimensional U\-Net of base width6464, with max\-pool downsampling and nearest\-neighbor upsampling with skip concatenation, applied to the full1024×10241024\\times 1024field\.
- •Coarse pass\.Reuses the full\-resolution baseline’s architecture and parameter count*exactly*, applied to the8×8\\timesaverage\-pooled128×128128\\times 128field, rather than a separate narrower network\. This isolates the coarse pass’s savings to resolution alone: at identical capacity it is a64×64\\timesreduction in floating\-point operations \(exactly the square of the downsampling factor, since a fixed\-width convolutional network’s cost scales with the number of sites\), so the speedup cannot be attributed to substituting a smaller\-capacity model\. Its output is one predicted flux total per coarse cell\.
- •Refiner\.A small local operator applied independently to each certified cell, taking the12×1212\\times 12full\-resolution window formed by the8×88\\times 8cell plus a two\-pixel context border, together with the coarse model’s own predicted median flux for that cell as a scalar conditioning input, and returning an8×88\\times 8flux patch\. Rather than re\-predicting*how much*flux the cell contains, which the coarse model already estimates in a calibrated way, the refiner predicts only*where within the cell*the flux lies\. Its output is rectified, normalized to sum to one over the cell, and then multiplied by the coarse model’s predicted total\. Flux conservation between the coarse and refined stages is therefore exact by construction rather than something the refiner must learn, and the refiner is left with the purely spatial distribution of the problem, which is what makes a network of17,80917\{,\}809parameters sufficient\. Uncertified cells are filled by smoothly upsampling the coarse prediction, so a selection miss degrades to coarse\-model quality rather than to blank sky\.

Table[7](https://arxiv.org/html/2610.19189#A3.T7)summarizes parameter counts and analytic operation counts for the three networks\.

Table 7:Galaxy flux field model sizes and analytic floating\-point operation counts, per102421024^\{2\}field for the full\-resolution baseline and the coarse model, and per certified cell for the refiner\.
#### C\.5\.4Training and Calibration Details

All three networks are trained with Adam at a learning rate of10−310^\{\-3\}for3030epochs on the20002000training fields, holding out a fixed10%10\\%of them as an internal validation split, with all randomness seeded\. The full\-resolution baseline and the coarse model minimize the quantile pinball loss summed overτ∈\{0\.05,0\.5,0\.9,0\.95\}\\tau\\in\\\{0\.05,0\.5,0\.9,0\.95\\\}, against the log\-transformed fine target and the log\-transformed sum\-pooled coarse target respectively\. The refiner minimizes anL1L\_\{1\}loss inlog\(1\+⋅\)\\log\(1\+\\cdot\)space between its predicted patch and the true full\-resolution flux patch\. The refiner’s output is anchored to the coarse model’s calibrated total\.

The refiner is trained on the*certified*distribution rather than on uniformly sampled cells\. We run the deployed, already\-calibrated coarse model over the training split, apply the calibrated threshold and halo, and draw the refiner’s training crops from the resulting certified regions; this yields716,629716\{,\}629cells over20002000fields,2\.19%2\.19\\%of all cells\. Halo cells are deliberately included, since they are near\-empty by construction and are where boundary truncation errors appear\. Each cell’s coarse\-predicted median flux is cached at the same time\.

CRC is calibrated on the500500calibration fields\. The score field is the coarse model’s predictedτ=0\.9\\tau\{=\}0\.9quantile of cell flux\. We use an upper quantile rather than the median so that a cell whose median flux is low but whose upper tail is high is still certified\. Cell weights are flux mass, the aggregation operator of Eq\.[8](https://arxiv.org/html/2610.19189#S3.E8), so the calibrated loss of Eq\.[9](https://arxiv.org/html/2610.19189#S3.E9)is the fraction of a field’s true flux that falls in uncertified cells\. The certified set is dilated by a halo of one coarse cell \(chosen heuristically\) in an eight\-connected neighborhood before the loss is evaluated\. Calibration scans an ascending grid of500500candidate thresholds and takes the largest whose finite\-sample corrected risk\(n​R^\+1\)/\(n\+1\)\(n\\hat\{R\}\+1\)/\(n\+1\)remains at or belowα\\alpha\.

At the reported operating pointα=0\.1\\alpha\{=\}0\.1, calibration returns aλ^\\hat\{\\lambda\}with a raw\-score cutoff of7\.697\.69inlog⁡\(1\+flux\)\\log\(1\+\\mathrm\{flux\}\)units, a flux total of approximately21752175counts per cell\. The calibration risk is0\.0400\.040\(0\.0420\.042after the finite\-sample correction\), and on the held\-out test split the empirical risk is0\.0400\.040\(0\.0420\.042corrected\), comfortably withinα\\alpha, at a certified fraction of2\.18%2\.18\\%of coarse cells\. Certifying2\.18%2\.18\\%of cells leaves the pipeline at61\.6×61\.6\\timesfewer floating\-point operations than the full\-resolution baseline, against a ceiling of64×64\\timesset by the coarse pass alone; the refinement of certified cell in an average field costs about4%4\\%of what the coarse pass costs, so the coarse pass, not the refinement, bounds the achievable speedup\.

As a reference for how tight the conformal selection is, we also compute anα\\alpha\-matched oracle, the threshold applied directly to the*true*coarse flux totals that attains exactly riskα\\alphaunder the same halo\. That oracle certifies1\.43%1\.43\\%of cells, so the calibrated coarse model certifies1\.53×1\.53\\timesthe cells an oracle selector would need at the same risk level\.

#### C\.5\.5Results

We provide an assortment of predictions from our model in Fig\.[10](https://arxiv.org/html/2610.19189#A3.F10)\.

![Refer to caption](https://arxiv.org/html/2610.19189v1/predictions_nsrc60_field0134.png)

![Refer to caption](https://arxiv.org/html/2610.19189v1/predictions_nsrc60_field0255.png)

![Refer to caption](https://arxiv.org/html/2610.19189v1/predictions_nsrc60_field0317.png)

![Refer to caption](https://arxiv.org/html/2610.19189v1/predictions_nsrc60_field0422.png)

Figure 10:Grid of galaxy flux field predictions\.