保形椭球体的尾部感知几何学习

# Tail-Aware Geometry Learning for Conformal Ellipsoids
Source: [https://arxiv.org/html/2609.27221](https://arxiv.org/html/2609.27221)
Affiliation:School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore

###### Abstract

This paper studies multivariate conformal prediction \(CP\), a distribution\-free uncertainty quantification framework with finite\-sample coverage guarantees\. The efficiency of multivariate prediction sets hinges critically on the residual geometry encoded by the nonconformity score, while existing minimum\-volume methods rely on quantile thresholds that ignore tail residual severity and implicitly bind geometry learning to coverage level\. We propose a tail\-aware geometry learning framework for conformal ellipsoids that decouples tail sensitivity in geometry learning from the final coverage guarantee\. Using a two\-split design, we learn the metric matrix via volume minimization under a CVaR constraint on an estimation split, then apply standard conformal calibration on a held\-out calibration split\. The resulting problem is convex and admits a bounded\-reweighting interpretation that prioritizes high\-residual samples\. Moreover, we theoretically characterize the trade\-off between ellipsoidal volume and tail severity\. Experimental results demonstrate the effectiveness of the proposed method\.

Keywords: Conformal prediction, uncertainty quantification, multivariate data, ellipsoids learning

## 1Introduction

Conformal prediction \(CP\) is a model\-agnostic, distribution\-free uncertainty quantification framework\[[1](https://arxiv.org/html/2609.27221#bib.bib8),[2](https://arxiv.org/html/2609.27221#bib.bib9)\]that yields prediction sets with finite\-sample coverage under exchangeability\. It has been applied in numerous signal processing tasks, including time series prediction\[[2](https://arxiv.org/html/2609.27221#bib.bib9)\]and anomaly detection\[[2](https://arxiv.org/html/2609.27221#bib.bib9)\]\. A core design choice is the nonconformity score, which quantifies how atypical a prediction is relative to observed data\. In one dimension, scalar residuals admit a natural total order, so quantiles are well defined\. In multivariate regression, however, score design is nontrivial because the score must encode the joint geometry of vector\-valued residuals\. Consequently, conformal procedures with the same coverage guarantee can produce prediction regions with substantially different shapes and volumes\.

Ellipsoidal scores provide a convenient way to model such residue geometry\. Common choices include Mahalanobis scores derived from estimated residual covariance\[[3](https://arxiv.org/html/2609.27221#bib.bib3),[4](https://arxiv.org/html/2609.27221#bib.bib13),[5](https://arxiv.org/html/2609.27221#bib.bib15)\]and its structured variants\[[6](https://arxiv.org/html/2609.27221#bib.bib7),[7](https://arxiv.org/html/2609.27221#bib.bib14)\], while recent methods\[[8](https://arxiv.org/html/2609.27221#bib.bib12),[9](https://arxiv.org/html/2609.27221#bib.bib10)\]learn the geometry via minimum\-volume objectives\[[10](https://arxiv.org/html/2609.27221#bib.bib22),[11](https://arxiv.org/html/2609.27221#bib.bib23)\]\. Such formulations are typically governed by an order statistic of the residual scores and can be interpreted as value\-at\-risk \(VaR\)\-type criteria\[[12](https://arxiv.org/html/2609.27221#bib.bib24)\]\. Beyond inducing nonconvex optimization, VaR criteria merely delineate a tail boundary: residual severity beyond the active quantile has no impact on the learned geometry\. This is limiting when upper\-tail residuals exhibit systematic directional structure\.

Our key observation is that geometry learning and conformal coverage need not be controlled by the same criterion\. The former determines which residual directions shape the score, whereas the latter is enforced afterwards by conformal calibration\. We thus use conditional value\-at\-risk \(CVaR\) for ellipsoidal geometry learning, which aggregates upper\-tail score magnitudes rather than relying solely on the quantile boundary\. A tail parameterτ\\taucontrols which residuals are emphasized during learning, while an independent miscoverage levelα\\alphadetermines final coverage, allowing separate tuning of tail sensitivity and coverage\. The resulting problem is convex and admits a bounded\-reweighting interpretation, where large\-score residuals exert greater influence on the learned metric\.

CVaR\-based minimum\-volume ellipsoids have classical precedents\[[13](https://arxiv.org/html/2609.27221#bib.bib27)\]\. Our contribution is instead to use CVaR as a conformal score\-learning mechanism followed by an independent calibration step\. This also differs from conformal risk\-control methods that apply CVaR to downstream losses\[[14](https://arxiv.org/html/2609.27221#bib.bib28),[15](https://arxiv.org/html/2609.27221#bib.bib26)\]: here CVaR shapes prediction\-set geometry, with coverage provided by conformal calibration\. Finally, our approach differs from existing multivariate CP methods\[[16](https://arxiv.org/html/2609.27221#bib.bib6)\]—including hyperrectangle\-based\[[17](https://arxiv.org/html/2609.27221#bib.bib21)\], density\-based\[[18](https://arxiv.org/html/2609.27221#bib.bib19),[19](https://arxiv.org/html/2609.27221#bib.bib1)\], distance\-based\[[3](https://arxiv.org/html/2609.27221#bib.bib3)\], copula\-based\[[20](https://arxiv.org/html/2609.27221#bib.bib4),[21](https://arxiv.org/html/2609.27221#bib.bib20)\], and optimal\-transport\-based approaches\[[22](https://arxiv.org/html/2609.27221#bib.bib5)\]—in that it employs ellipsoidal scores to encode the joint geometry of vector residuals\.

Our contributions are threefold\. \(i\) We formulate tail\-aware conformal geometry learning that decouples the tail\-sensitivity parameter from the target miscoverage level\. \(ii\) We characterize how upper\-tail residuals shape the learned geometry, and derive a convex formulation of the proposed model along with its bounded\-reweighting interpretation\. \(iii\) We conduct extensive experiments to demonstrate the effectiveness of the proposed method\.

## 2Conformal Ellipsoids: Geometry, Calibration and Coverage

Letf^\\widehat\{f\}be a predictor trained independently of the calibration data, and let𝒟est=\{\(Xm,Ym\)\}m=1M\\mathcal\{D\}\_\{\\mathrm\{est\}\}=\\\{\(X\_\{m\},Y\_\{m\}\)\\\}\_\{m=1\}^\{M\}and𝒟cal=\{\(Xn,Yn\)\}n=1N\\mathcal\{D\}\_\{\\mathrm\{cal\}\}=\\\{\(X\_\{n\},Y\_\{n\}\)\\\}\_\{n=1\}^\{N\}denote disjoint geometry\-estimation and calibration sets\. We consider ellipsoidal nonconformity function of the formSΩ​\(X,Y\)=r⊤​Ω​rS\_\{\\Omega\}\(X,Y\)=r^\{\\top\}\\Omega r, wherer=Y−f^​\(X\)r=Y\-\\widehat\{f\}\(X\)is the residue vector of a sample\(X,Y\)\(X,Y\)\. The metric matrixΩ\\Omegadetermines the relative scaling and orientation of the residual space and is estimated using only𝒟est\\mathcal\{D\}\_\{\\mathrm\{est\}\}\. For any matrixΩ^\\widehat\{\\Omega\}estimated from𝒟est\\mathcal\{D\}\_\{\\mathrm\{est\}\}and fixed before calibration, define the calibration scoress^n​\(Ω^\)=SΩ^​\(Xn,Yn\)\\widehat\{s\}\_\{n\}\(\\widehat\{\\Omega\}\)=S\_\{\\widehat\{\\Omega\}\}\(X\_\{n\},Y\_\{n\}\)\. Letkα=⌈\(N\+1\)​\(1−α\)⌉k\_\{\\alpha\}=\\left\\lceil\(N\+1\)\(1\-\\alpha\)\\right\\rceiland letq^α\\widehat\{q\}\_\{\\alpha\}be thekαk\_\{\\alpha\}\-th order statistic of\{s^1​\(Ω^\),…,s^N​\(Ω^\),\+∞\}\\\{\\widehat\{s\}\_\{1\}\(\\widehat\{\\Omega\}\),\\ldots,\\widehat\{s\}\_\{N\}\(\\widehat\{\\Omega\}\),\+\\infty\\\}\. The corresponding conformal prediction ellipsoid set is

𝒬α,Ω^​\(X\)=\{Y:SΩ^​\(X,Y\)≤q^α\}\.\\displaystyle\\mathcal\{Q\}\_\{\\alpha,\\widehat\{\\Omega\}\}\(X\)=\\left\\\{Y:S\_\{\\widehat\{\\Omega\}\}\(X,Y\)\\leq\\widehat\{q\}\_\{\\alpha\}\\right\\\}\.\(1\)The prediction set \([1](https://arxiv.org/html/2609.27221#S2.E1)\) can provide the following coverage validity for test sample\(XN\+1,YN\+1\)\(X\_\{N\+1\},Y\_\{N\+1\}\)\.

###### Proposition 1\.

Letℱ\\mathcal\{F\}contains all information fixed before accessing the calibration and test observations, includingf^\\widehat\{f\}and the information used to estimateΩ^\\widehat\{\\Omega\}\. If the calibration and test observations are exchangeable conditional onℱ\\mathcal\{F\},

ℙ⁡\(YN\+1∈𝒬α,Ω^​\(XN\+1\)\|ℱ\)≥1−α\.\\displaystyle\\mathbb\{P\}\\left\(Y\_\{N\+1\}\\in\\mathcal\{Q\}\_\{\\alpha,\\widehat\{\\Omega\}\}\(X\_\{N\+1\}\)\\,\\middle\|\\,\\mathcal\{F\}\\right\)\\geq 1\-\\alpha\.\(2\)

This proposition follows directly from the standard split\-conformal argument, and states that a calibration procedure delivers coverage validity once the estimatedΩ^\\widehat\{\\Omega\}is fixed\. Importantly, conformal validity does not require the geometry\-learning objective itself to enforce empirical1−α1\-\\alphacoverage\.

A second useful property is scale invariance\. For everyc\>0c\>0,Sc​Ω​\(X,Y\)=c​SΩ​\(X,Y\)S\_\{c\\Omega\}\(X,Y\)=c\\,S\_\{\\Omega\}\(X,Y\)andq^α​\(c​Ω\)=c​q^α​\(Ω\)\\widehat\{q\}\_\{\\alpha\}\(c\\Omega\)=c\\,\\widehat\{q\}\_\{\\alpha\}\(\\Omega\)\. Hence,𝒬α,c​Ω​\(X\)=𝒬α,Ω​\(X\)\.\\mathcal\{Q\}\_\{\\alpha,c\\Omega\}\(X\)=\\mathcal\{Q\}\_\{\\alpha,\\Omega\}\(X\)\.Thus, conformal calibration makes the final prediction set invariant to positive global rescaling ofΩ\\Omega\. Only the relative directional scaling and orientation encoded byΩ\\Omegaaffect its geometry\. In the geometry\-learning stage below, the unit threshold therefore serves only as a normalization\.

## 3Tail\-Aware Geometry Learning

This section first introduces the formulation of order\-statistic geometry learning for CP\. We then present our proposed method and conclude with the conformal recalibration\.

### 3\.1Order\-Statistic Geometry Learning

Given𝒟est\\mathcal\{D\}\_\{\\mathrm\{est\}\}, letrm=Ym−f^​\(Xm\)r\_\{m\}=Y\_\{m\}\-\\widehat\{f\}\(X\_\{m\}\)andsm​\(Ω\)=rm⊤​Ω​rms\_\{m\}\(\\Omega\)=r\_\{m\}^\{\\top\}\\Omega r\_\{m\}\. The normalized unit\-threshold ellipsoid has volume

Vol\(ℰΩ\)=λ\(𝔹d\)det\(Ω\)−1/2,ℰΩ=\{r:r⊤Ωr≤1\},\\displaystyle\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\Omega\}\)=\\lambda\(\\mathbb\{B\}\_\{d\}\)\\det\(\\Omega\)^\{\-1/2\},\\quad\\mathcal\{E\}\_\{\\Omega\}=\\\{r:r^\{\\top\}\\Omega r\\leq 1\\\},\(3\)where𝔹d\\mathbb\{B\}\_\{d\}denotes the unit Euclidean ball inℝd\\mathbb\{R\}^\{d\}andλ\\lambdadenotes the Lebesgue measure\. To distinguish geometry learning from conformal calibration, we introduce a tail fractionτ∈\(0,1\)\\tau\\in\(0,1\), which is independent of the final miscoverage levelα\\alpha\. After removing constant terms, traditional minimum\-volume geometry learning\[[9](https://arxiv.org/html/2609.27221#bib.bib10)\]learnsΩ\\Omegaby minimizing the log\-volume of this normalized ellipsoid:

minΩ\\displaystyle\\min\_\{\\Omega\}\\,−12logdetΩ\\displaystyle\-\\frac\{1\}\{2\}\\log\\det\\Omega\(4\)s\.t\.\\displaystyle\\mathrm\{s\.t\.\}\\,Card⁡\{m∈\[M\]:sm​\(Ω\)≤1\}≥⌈\(1−τ\)​M⌉,Ω≻0,\\displaystyle\\operatorname\{Card\}\\\{m\\in\[M\]:s\_\{m\}\(\\Omega\)\\leq 1\\\}\\geq\\left\\lceil\(1\-\\tau\)M\\right\\rceil,\\;\\Omega\\succ 0,whereCard⁡\(⋅\)\\operatorname\{Card\}\(\\cdot\)denotes the cardinality functions, and\[M\]:=\{1,⋯,M\}\[M\]:=\\\{1,\\cdots,M\\\}\. The first constraint is an order\-statistic to ensure an empirical1−τ1\-\\tauinclusion requirement on𝒟est\\mathcal\{D\}\_\{\\mathrm\{est\}\}\. Thus, problem \([4](https://arxiv.org/html/2609.27221#S3.E4)\) is an empirical quantile\-constrained minimum\-volume geometry learner\. Conventional methods\[[8](https://arxiv.org/html/2609.27221#bib.bib12),[9](https://arxiv.org/html/2609.27221#bib.bib10)\]typically setτ=α\\tau=\\alpha, which implicitly couples the geometry learning to the final coverage level via the order\-statistic constraint\. However, Proposition[1](https://arxiv.org/html/2609.27221#Thmproposition1)shows that this is not required for conformal validity: the geometry parameter and the final coverage level play different roles\.

We next show that problem \([4](https://arxiv.org/html/2609.27221#S3.E4)\) can be reformulated as a VaR constrained problem\. Define the empirical VaR as

Vτ​\(Ω\)=VaR^1−τ​\(s1​\(Ω\),…,sM​\(Ω\)\)=s\(kτ\)​\(Ω\)\\displaystyle V\_\{\\tau\}\(\\Omega\)=\\widehat\{\\operatorname\{VaR\}\}\_\{1\-\\tau\}\\left\(s\_\{1\}\(\\Omega\),\\ldots,s\_\{M\}\(\\Omega\)\\right\)=s\_\{\(k\_\{\\tau\}\)\}\(\\Omega\)\(5\)wherekτ=⌈\(1−τ\)​M⌉k\_\{\\tau\}=\\left\\lceil\(1\-\\tau\)M\\right\\rceilands\(kτ\)​\(Ω\)s\_\{\(k\_\{\\tau\}\)\}\(\\Omega\)is thekτ−k\_\{\\tau\}\-th smallest value of all scores of𝒟est\\mathcal\{D\}\_\{\\mathrm\{est\}\}\. Then, the empirical inclusion constraint in \([4](https://arxiv.org/html/2609.27221#S3.E4)\) is equivalent toVτ​\(Ω\)≤1V\_\{\\tau\}\(\\Omega\)\\leq 1, leading to the following reformulation

minΩ−12logdetΩ,s\.t\.Vτ\(Ω\)≤1,Ω≻0\.\\displaystyle\\min\_\{\\Omega\}\\quad\-\\frac\{1\}\{2\}\\log\\det\\Omega,\\quad\\mathrm\{s\.t\.\}\\;V\_\{\\tau\}\(\\Omega\)\\leq 1,\\;\\Omega\\succ 0\.\(6\)Nevertheless, VaR retains two limitations for geometry learning\. First,Vτ​\(Ω\)V\_\{\\tau\}\(\\Omega\)is an intermediate order statistic and generally induces a nonconvex optimization problem\. Second, once an observation lies beyond the active quantile, its additional score magnitude does not affect the VaR constraint\. Consequently, geometrically distinct upper\-tail residuals may have similar influence even when their severities differ substantially\.

### 3\.2CVaR Geometry Learning

In this study, we learn the ellipsoidal geometry using the empirical CVaR at tail fractionτ\\tau, which is defined as

Cτ​\(Ω\):\\displaystyle C\_\{\\tau\}\(\\Omega\):=CVaR^1−τ​\(s1​\(Ω\),…,sM​\(Ω\)\)\\displaystyle=\\widehat\{\\text\{CVaR\}\}\_\{1\-\\tau\}\\bigl\(s\_\{1\}\(\\Omega\),\\ldots,s\_\{M\}\(\\Omega\)\\bigr\)\(7\)=1τ​M​\(∑i=kτ\+1Ms\(i\)​\(Ω\)\+\(kτ−\(1−τ\)​M\)​s\(kτ\)​\(Ω\)\)\.\\displaystyle=\{\\frac\{1\}\{\\tau M\}\\left\(\\sum\_\{i=k\_\{\\tau\}\+1\}^\{M\}s\_\{\(i\)\}\(\\Omega\)\+\\big\(k\_\{\\tau\}\-\(1\-\\tau\)M\\big\)s\_\{\(k\_\{\\tau\}\)\}\(\\Omega\)\\right\)\}\.It is observed from \([7](https://arxiv.org/html/2609.27221#S3.E7)\) that CVaR quantifies the average magnitude in the upper tail of the score distribution, instead of only locating the tail boundary like VaR\. This avoids relying only on the single order statistic like \([6](https://arxiv.org/html/2609.27221#S3.E6)\)\. We then estimate the geometry through

minΩ−12logdetΩ,s\.t\.Cτ\(Ω\)≤1,Ω≻0\.\\displaystyle\\min\_\{\\Omega\}\\quad\-\\frac\{1\}\{2\}\\log\\det\\Omega,\\quad\\mathrm\{s\.t\.\}\\;C\_\{\\tau\}\(\\Omega\)\\leq 1,\\;\\Omega\\succ 0\.\(8\)Using the Rockafellar–Uryasev representation\[[23](https://arxiv.org/html/2609.27221#bib.bib11)\], the empirical CVaR \([7](https://arxiv.org/html/2609.27221#S3.E7)\) is equivalent to

Cτ​\(Ω\)=\\displaystyle C\_\{\\tau\}\(\\Omega\)=mint,\{ξm\}m=1Mt\+1τ​M​∑m=1Mξm\\displaystyle\\min\_\{t,\\\{\\xi\_\{m\}\\\}\_\{m=1\}^\{M\}\}\\quad t\+\\frac\{1\}\{\\tau M\}\\sum\_\{m=1\}^\{M\}\\xi\_\{m\}s\.t\.ξm≥rm⊤​Ω​rm−t,ξm≥0,m∈\[M\]\.\\displaystyle\\mathrm\{s\.t\.\}\\quad\\xi\_\{m\}\\geq r\_\{m\}^\{\\top\}\\Omega r\_\{m\}\-t,\\quad\\xi\_\{m\}\\geq 0,\\quad m\\in\[M\]\.\(9\)Equivalently, the problem \([8](https://arxiv.org/html/2609.27221#S3.E8)\) can be rewritten as

minΩ,t,ξ\\displaystyle\\min\_\{\\Omega,t,\\xi\}\\quad−12logdetΩ\\displaystyle\-\\frac\{1\}\{2\}\\log\\det\\Omegas\.t\.\\displaystyle\\mathrm\{s\.t\.\}\\quadt\+1τ​M​∑m=1Mξm≤1,Ω≻0\\displaystyle t\+\\frac\{1\}\{\\tau M\}\\sum\_\{m=1\}^\{M\}\\xi\_\{m\}\\leq 1,\\;\\;\\Omega\\succ 0ξm≥rm⊤​Ω​rm−t,ξm≥0,m∈\[M\]\\displaystyle\\xi\_\{m\}\\geq r\_\{m\}^\{\\top\}\\Omega r\_\{m\}\-t,\\;\\;\\xi\_\{m\}\\geq 0,\\;\\;m\\in\[M\]\(10\)Sincerm⊤​Ω​rmr\_\{m\}^\{\\top\}\\Omega r\_\{m\}is affine inΩ\\Omega, the objective is convex and all constraints define convex sets\. Problem \([10](https://arxiv.org/html/2609.27221#S3.E10)\) can therefore be solved to global optimality using standard convex optimization methods\[[24](https://arxiv.org/html/2609.27221#bib.bib2)\]\.

Here, the CVaR ellipsoid is not used directly as the final uncertainty region\. Instead,Ω^τ\\widehat\{\\Omega\}\_\{\\tau\}defines the geometry of the nonconformity score, while the final conformal threshold is determined independently on the calibration split at levelα\\alpha\. Thus,τ\\taucontrols which upper\-tail residuals shape the learned geometry, whereasα\\alphacontrols the final conformal miscoverage level\.

Bounded\-reweighting interpretation\.The empirical CVaR admits the dual representation\[[25](https://arxiv.org/html/2609.27221#bib.bib25)\]

Cτ​\(Ω\)=maxη∈𝒫τ⁡⟨Ω,Ση⟩\.\\displaystyle C\_\{\\tau\}\(\\Omega\)=\\max\_\{\\eta\\in\\mathcal\{P\}\_\{\\tau\}\}\\langle\\Omega,\\Sigma\_\{\\eta\}\\rangle\.\(11\)where

𝒫τ=\{η≥0:𝟏⊤η=1,ηm≤1τ​M\},Ση=∑m=1Mηmrmrm⊤\.\\displaystyle\\mathcal\{P\}\_\{\\tau\}=\\left\\\{\\eta\\geq 0:\\mathbf\{1\}^\{\\top\}\\eta=1,\\;\\eta\_\{m\}\\leq\\frac\{1\}\{\\tau M\}\\right\\\},\\;\\Sigma\_\{\\eta\}=\\sum\_\{m=1\}^\{M\}\\eta\_\{m\}r\_\{m\}r\_\{m\}^\{\\top\}\.\(12\)The corresponding problem \([8](https://arxiv.org/html/2609.27221#S3.E8)\) can be rewritten as

Ω^τ∈\\displaystyle\\widehat\{\\Omega\}\_\{\\tau\}\\in\\;arg⁡minΩ≻0−12​log​detΩ,\\displaystyle\\arg\\min\_\{\\Omega\\succ 0\}\\;\-\\frac\{1\}\{2\}\\log\\det\\Omega,s\.t\.Cτ​\(Ω\)=maxη∈𝒫τ⁡⟨Ω,Ση⟩≤1\.\\displaystyle\\mathrm\{s\.t\.\}\\quad C\_\{\\tau\}\(\\Omega\)=\\max\_\{\\eta\\in\\mathcal\{P\}\_\{\\tau\}\}\\langle\\Omega,\\Sigma\_\{\\eta\}\\rangle\\leq 1\.\(13\)Then, for problem \([13](https://arxiv.org/html/2609.27221#S3.E13)\), we have the following proposition\.

###### Proposition 2\.

Suppose that \([13](https://arxiv.org/html/2609.27221#S3.E13)\) admits an optimum and that the empirical residuals spanℝd\\mathbb\{R\}^\{d\}\. Then, there exists an optimal weightη⋆∈arg⁡maxη∈𝒫τ​⟨Ω^τ,Ση⟩\\eta^\{\\star\}\\in\\arg\\max\_\{\\eta\\in\\mathcal\{P\}\_\{\\tau\}\}\\left\\langle\\widehat\{\\Omega\}\_\{\\tau\},\\Sigma\_\{\\eta\}\\right\\ranglesuch that

Ω^τ−1=d​Ση⋆=d​∑m=1Mηm⋆​rm​rm⊤\.\\displaystyle\\widehat\{\\Omega\}\_\{\\tau\}^\{\-1\}=d\\Sigma\_\{\\eta^\{\\star\}\}=d\\sum\_\{m=1\}^\{M\}\\eta\_\{m\}^\{\\star\}r\_\{m\}r\_\{m\}^\{\\top\}\.\(14\)

###### Proof\.

See Appendix in the full paper\. ∎

Proposition[2](https://arxiv.org/html/2609.27221#Thmproposition2)provides an explicit link between tail\-aware geometry learning and minimum\-volume ellipsoidal geometry\. Specifically, CVaR identifies a tail\-focused residual scatter matrix through the optimal weightsη⋆\\eta^\{\\star\}, where the maximizing weights place greater emphasis on observations with large current scores\. The parameterτ\\taucontrols the concentration of this reweighting: a smallerτ\\tauincreases the allowable per\-sample weight1/\(τ​M\)1/\(\\tau M\)and permits greater concentration on a smaller subset of upper\-tail residuals\. The minimum\-volume objective then converts the resulting tail\-weighted scatter matrix into the optimal precision metric viaΩ^τ−1=d​Ση⋆\\widehat\{\\Omega\}\_\{\\tau\}^\{\-1\}=d\\Sigma\_\{\\eta^\{\\star\}\}\. Thus, the orientation and relative axis lengths of the ellipsoid are determined by the active tail\-weighted residual geometry rather than the unweighted covariance\.

### 3\.3Conformal Recalibration

After estimatingΩ^τ\\widehat\{\\Omega\}\_\{\\tau\}from𝒟est\\mathcal\{D\}\_\{\\mathrm\{est\}\}, we keep it fixed and computes^n=rn⊤​Ω^τ​rn\\widehat\{s\}\_\{n\}=r\_\{n\}^\{\\top\}\\widehat\{\\Omega\}\_\{\\tau\}r\_\{n\}on the independent calibration set𝒟cal=\{\(Xn,Yn\)\}n=1N\\mathcal\{D\}\_\{\\mathrm\{cal\}\}=\\\{\(X\_\{n\},Y\_\{n\}\)\\\}\_\{n=1\}^\{N\}\. The finite\-sample corrected conformal quantileq^α\\widehat\{q\}\_\{\\alpha\}is then computed at miscoverage levelα\\alpha, and the final prediction region is

𝒬α,τ​\(X\)=\{Y:\(Y−f^​\(X\)\)⊤​Ω^τ​\(Y−f^​\(X\)\)≤q^α\}\.\\displaystyle\\mathcal\{Q\}\_\{\\alpha,\\tau\}\(X\)=\\left\\\{Y:\(Y\-\\widehat\{f\}\(X\)\)^\{\\top\}\\widehat\{\\Omega\}\_\{\\tau\}\(Y\-\\widehat\{f\}\(X\)\)\\leq\\widehat\{q\}\_\{\\alpha\}\\right\\\}\.\(15\)By Proposition[1](https://arxiv.org/html/2609.27221#Thmproposition1), for a new test sample\(XN\+1,YN\+1\)\(X\_\{N\+1\},Y\_\{N\+1\}\),

ℙ⁡\(YN\+1∈𝒬α,τ​\(XN\+1\)\)≥1−α,\\displaystyle\\mathbb\{P\}\\left\(Y\_\{N\+1\}\\in\\mathcal\{Q\}\_\{\\alpha,\\tau\}\(X\_\{N\+1\}\)\\right\)\\geq 1\-\\alpha,\(16\)provided thatΩ^τ\\widehat\{\\Omega\}\_\{\\tau\}is fixed before the calibration data are used\. Therefore, changingτ\\taualters the learned geometry without changing the nominal conformal coverage level, while changingα\\alpharescales the final prediction region without requiring the geometry to be relearned\.

Algorithm 1CVaR\-based Ellipsoidal Conformal Prediction0:Trained predictor

f^\\widehat\{f\}, estimation set

𝒟est\\mathcal\{D\}\_\{\\mathrm\{est\}\}, calibration set

𝒟cal\\mathcal\{D\}\_\{\\mathrm\{cal\}\}, tail parameter

τ\\tau, and miscoverage level

α\\alpha\.

1:Estimate

Ω^τ\\widehat\{\\Omega\}\_\{\\tau\}using

𝒟est\\mathcal\{D\}\_\{\\mathrm\{est\}\}by solving \([10](https://arxiv.org/html/2609.27221#S3.E10)\)

2:Compute

s^n​\(Ω^τ\)=rn⊤​Ω^τ​rn\\widehat\{s\}\_\{n\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)=r\_\{n\}^\{\\top\}\\widehat\{\\Omega\}\_\{\\tau\}r\_\{n\}for samples in

𝒟cal\\mathcal\{D\}\_\{\\mathrm\{cal\}\}
3:Compute the conformal quantile

q^α\\widehat\{q\}\_\{\\alpha\}based on

\{s^n\}n=1N\\\{\\widehat\{s\}\_\{n\}\\\}\_\{n=1\}^\{N\}\.

4:Construct the prediction set for

XN\+1X\_\{N\+1\}based on \([15](https://arxiv.org/html/2609.27221#S3.E15)\)

4:The prediction set

Qα,τ​\(XN\+1\)Q\_\{\\alpha,\\tau\}\(X\_\{N\+1\}\)for

XN\+1X\_\{N\+1\}

## 4Geometry–Tail Characterization

In this section, we theoretically characterize the trade\-off between ellipsoidal volume and tail risk severity for the proposed method\. First, for anyΩ\\OmegawithVτ​\(Ω\)\>0V\_\{\\tau\}\(\\Omega\)\>0, define the tail\-severity ratio

Γτ​\(Ω\)=Cτ​\(Ω\)Vτ​\(Ω\)≥1\.\\displaystyle\\Gamma\_\{\\tau\}\(\\Omega\)=\\frac\{C\_\{\\tau\}\(\\Omega\)\}\{V\_\{\\tau\}\(\\Omega\)\}\\geq 1\.\(17\)This ratio measures the relative severity of scores beyond the VaR boundary: a value close to one indicates that upper\-tail scores remain near the boundary, whereas a larger value indicates more severe tail excursions\. LetΩV⋆\\Omega\_\{V\}^\{\\star\}andΩC⋆\\Omega\_\{C\}^\{\\star\}denote optimal solutions of the VaR\-based \([6](https://arxiv.org/html/2609.27221#S3.E6)\) and CVaR\-based \([8](https://arxiv.org/html/2609.27221#S3.E8)\) geometry\-learning problems at the same tail fractionτ\\tau, respectively, and define

Γτ,V=Γτ​\(ΩV⋆\),Γτ,C=Γτ​\(ΩC⋆\),\\displaystyle\\Gamma\_\{\\tau,V\}=\\Gamma\_\{\\tau\}\(\\Omega\_\{V\}^\{\\star\}\),\\qquad\\Gamma\_\{\\tau,C\}=\\Gamma\_\{\\tau\}\(\\Omega\_\{C\}^\{\\star\}\),\(18\)as the tail\-severity ratio ofΩV⋆\\Omega\_\{V\}^\{\\star\}andΩC⋆\\Omega\_\{C\}^\{\\star\}, respectively\. We further introduce the VaR\-normalized geometry ellipsoids

ℰτ,j=\{r:r⊤​Ωj⋆​r≤Vτ​\(Ωj⋆\)\},j∈\{V,C\}\.\\displaystyle\\mathcal\{E\}\_\{\\tau,j\}=\\left\\\{r:r^\{\\top\}\\Omega\_\{j\}^\{\\star\}r\\leq V\_\{\\tau\}\(\\Omega\_\{j\}^\{\\star\}\)\\right\\\},\\qquad j\\in\\\{V,C\\\}\.\(19\)The setℰτ,j\\mathcal\{E\}\_\{\\tau,j\}defines an ellipsoidal region induced by the metric matrixΩj⋆\\Omega\_\{j\}^\{\\star\}, with its overall scale determined by the thresholdVτ​\(Ωj⋆\)V\_\{\\tau\}\(\\Omega\_\{j\}^\{\\star\}\)\. Hence,Ωj⋆\\Omega\_\{j\}^\{\\star\}characterizes the shape and orientation of the ellipsoid, whileVτ​\(Ωj⋆\)V\_\{\\tau\}\(\\Omega\_\{j\}^\{\\star\}\)controls its size at risk levelτ\\tau\. Then, we have the following proposition\.

###### Proposition 3\.

Suppose that problems \([6](https://arxiv.org/html/2609.27221#S3.E6)\) and \([8](https://arxiv.org/html/2609.27221#S3.E8)\) admit optimal solutions, and that the empirical residuals spanℝd\\mathbb\{R\}^\{d\}\. Then the ellipsoids associated withΩV⋆\\Omega\_\{V\}^\{\\star\}andΩC⋆\\Omega\_\{C\}^\{\\star\}satisfy

1≤Vol⁡\(ℰτ,C\)Vol⁡\(ℰτ,V\)≤\(Γτ,VΓτ,C\)d/2\.\\displaystyle 1\\leq\\frac\{\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,C\}\)\}\{\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,V\}\)\}\\leq\\left\(\\frac\{\\Gamma\_\{\\tau,V\}\}\{\\Gamma\_\{\\tau,C\}\}\\right\)^\{d/2\}\.\(20\)IfVol⁡\(ℰτ,C\)=ρ​Vol⁡\(ℰτ,V\)\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,C\}\)=\\rho\\,\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,V\}\)forρ≥1\\rho\\geq 1, then

Γτ,C≤ρ−2/dΓτ,V\.\\displaystyle\\Gamma\_\{\\tau,C\}\\leq\\rho^\{\-2/d\}\\Gamma\_\{\\tau,V\}\.\(21\)

###### Proof\.

See Appendix in the full paper\. ∎

Proposition[3](https://arxiv.org/html/2609.27221#Thmproposition3)quantifies the trade\-off between geometric compactness and tail sensitivity\. The VaR solution provides the smallest VaR\-normalized volume, while the CVaR solution achieves a no\-worse tail\-severity ratio,Γτ,C≤Γτ,V\\Gamma\_\{\\tau,C\}\\leq\\Gamma\_\{\\tau,V\}\. Moreover, any increase in the CVaR\-based normalized volume must be compensated by a corresponding reduction in tail severity according to \([21](https://arxiv.org/html/2609.27221#S4.E21)\)\. Thus, CVaR\-based learning may allocate more geometric capacity to directions containing severe residual excursions rather than merely optimizing the VaR boundary\. This result provides a theoretical rationale for using CVaR to learn a more tail\-aware residual geometry\. Importantly, it concerns the geometry\-estimation stage and does not directly imply a smaller final conformal region, whose scale is determined by the miscoverage levelα\\alphaduring calibration\. Accordingly, Section[5](https://arxiv.org/html/2609.27221#S5)evaluates whether this geometry\-level tail improvement translates into favorable calibrated volume behavior at coverage level1−α1\-\\alpha\.

## 5Experiments

Table 1:Experimental results on synthetic data\.α\\alphaMethodCoverageEfficiency↓\\downarrowSeverity↓\\downarrow0\.1Naive0\.9002±0\.00650\.9002\\pm\\\!\\mbox\{\\scriptsize$0\.0065$\}3\.9105±0\.02473\.9105\\pm\\\!\\mbox\{\\scriptsize$0\.0247$\}1\.2434±0\.02191\.2434\\pm\\\!\\mbox\{\\scriptsize$0\.0219$\}Cov0\.9002±0\.00550\.9002\\pm\\\!\\mbox\{\\scriptsize$0\.0055$\}3\.6929±0\.01593\.6929\\pm\\\!\\mbox\{\\scriptsize$0\.0159$\}1\.1854±0\.01391\.1854\\pm\\\!\\mbox\{\\scriptsize$0\.0139$\}MVCS0\.8998±0\.00600\.8998\\pm\\\!\\mbox\{\\scriptsize$0\.0060$\}3\.6081±0\.0166\\textbf\{3\.6081\}\\pm\\\!\\mbox\{\\scriptsize$0\.0166$\}1\.1893±0\.01511\.1893\\pm\\\!\\mbox\{\\scriptsize$0\.0151$\}Ours:0\.10\.10\.9000±0\.00590\.9000\\pm\\\!\\mbox\{\\scriptsize$0\.0059$\}3\.6128±0\.01603\.6128\\pm\\\!\\mbox\{\\scriptsize$0\.0160$\}1\.1691±0\.01441\.1691\\pm\\\!\\mbox\{\\scriptsize$0\.0144$\}Ours:0\.080\.080\.9001±0\.00590\.9001\\pm\\\!\\mbox\{\\scriptsize$0\.0059$\}3\.6082¯±0\.0156\\underline\{3\.6082\}\\pm\\\!\\mbox\{\\scriptsize$0\.0156$\}1\.1665¯±0\.0163\\underline\{1\.1665\}\\pm\\\!\\mbox\{\\scriptsize$0\.0163$\}Ours:0\.050\.050\.9001±0\.00610\.9001\\pm\\\!\\mbox\{\\scriptsize$0\.0061$\}3\.6250±0\.01603\.6250\\pm\\\!\\mbox\{\\scriptsize$0\.0160$\}1\.1652±0\.0177\\textbf\{ 1\.1652\}\\pm\\\!\\mbox\{\\scriptsize$0\.0177$\}0\.05Naive0\.9493±0\.00500\.9493\\pm\\\!\\mbox\{\\scriptsize$0\.0050$\}4\.1995±0\.03054\.1995\\pm\\\!\\mbox\{\\scriptsize$0\.0305$\}1\.1940±0\.02441\.1940\\pm\\\!\\mbox\{\\scriptsize$0\.0244$\}Cov0\.9501±0\.00400\.9501\\pm\\\!\\mbox\{\\scriptsize$0\.0040$\}3\.8849±0\.01833\.8849\\pm\\\!\\mbox\{\\scriptsize$0\.0183$\}1\.1562±0\.01731\.1562\\pm\\\!\\mbox\{\\scriptsize$0\.0173$\}MVCS0\.9499±0\.00460\.9499\\pm\\\!\\mbox\{\\scriptsize$0\.0046$\}3\.8430¯±0\.0185\\underline\{3\.8430\}\\pm\\\!\\mbox\{\\scriptsize$0\.0185$\}1\.1664±0\.01931\.1664\\pm\\\!\\mbox\{\\scriptsize$0\.0193$\}Ours:0\.10\.10\.9501±0\.00430\.9501\\pm\\\!\\mbox\{\\scriptsize$0\.0043$\}3\.8455±0\.01803\.8455\\pm\\\!\\mbox\{\\scriptsize$0\.0180$\}1\.1551±0\.01791\.1551\\pm\\\!\\mbox\{\\scriptsize$0\.0179$\}Ours:0\.080\.080\.9501±0\.00430\.9501\\pm\\\!\\mbox\{\\scriptsize$0\.0043$\}3\.8430¯±0\.0183\\underline\{3\.8430\}\\pm\\\!\\mbox\{\\scriptsize$0\.0183$\}1\.1541¯±0\.0185\\underline\{1\.1541\}\\pm\\\!\\mbox\{\\scriptsize$0\.0185$\}Ours:0\.050\.050\.9501±0\.00450\.9501\\pm\\\!\\mbox\{\\scriptsize$0\.0045$\}3\.8403±0\.0192\\textbf\{ 3\.8403\}\\pm\\\!\\mbox\{\\scriptsize$0\.0192$\}1\.1534±0\.0190\\textbf\{1\.1534\}\\pm\\\!\\mbox\{\\scriptsize$0\.0190$\}↓\\downarrow: lower is better; bold: best result; underlined: second\-best result\.In this section, we evaluate the proposed method on synthetic and real\-world data\.

### 5\.1Synthetic Data\.

We consider a bivariate regression problem withX∼Uniform⁡\(0,1\)X\\sim\\mathrm\{Uniform\}\(0,1\)andY=f∗​\(X\)\+A⁡\(X\)​ϵ∈ℝ2Y=f^\{\*\}\(X\)\+A\(X\)\\epsilon\\in\\mathbb\{R\}^\{2\}, wheref1∗​\(X\)=X\+sin⁡\(2​π​X\)f\_\{1\}^\{\*\}\(X\)=X\+\\sin\(2\\pi X\)andf2∗​\(X\)=X2\+cos⁡\(2​π​X\)f\_\{2\}^\{\*\}\(X\)=X^\{2\}\+\\cos\(2\\pi X\)\. The noise isϵ=\(z1,z2\+0\.4​z12−0\.4​σ12\)\\epsilon=\(z\_\{1\},z\_\{2\}\+0\.4z\_\{1\}^\{2\}\-0\.4\\sigma\_\{1\}^\{2\}\), wherez1∼𝒩⁡\(0,0\.1\)z\_\{1\}\\sim\\mathcal\{N\}\(0,0\.1\),z2∼𝒩⁡\(0,0\.02\)z\_\{2\}\\sim\\mathcal\{N\}\(0,0\.02\), andσ12\\sigma\_\{1\}^\{2\}is the variance ofz1z\_\{1\}\. To introduce a rare anisotropic high\-variance regime, we setA⁡\(X\)=IA\(X\)=IforX≤0\.92X\\leq 0\.92andA⁡\(X\)=Rπ/4​\(1005\)​Rπ/4⊤A\(X\)=R\_\{\\pi/4\}\\begin\{pmatrix\}1&0\\\\ 0&5\\end\{pmatrix\}R\_\{\\pi/4\}^\{\\top\}otherwise, whereRπ/4R\_\{\\pi/4\}is the rotation matrix with angleπ/4\\pi/4\. Thus, the amplified regime occurs with probability0\.080\.08\. We generate 20,000 samples and split them into training, estimation, calibration, and test sets with proportions0\.40\.4,0\.20\.2,0\.20\.2, and0\.20\.2\. A two\-layer MLP is fitted on the training set, while the estimation and calibration sets are used for score construction and split conformal calibration, respectively\. We compareOurswithNaive, which uses theℓ2\\ell\_\{2\}norm;Cov, which uses the Mahalanobis distance based on empirical covariance\[[3](https://arxiv.org/html/2609.27221#bib.bib3)\]; andMVCS, which learns a minimum\-volume ellipsoid under a VaR constraint\[[9](https://arxiv.org/html/2609.27221#bib.bib10)\]\. For our method, we letτ=0\.1,0\.08\\tau=0\.1,0\.08and0\.050\.05, respectively\. The evaluation metrics are Coverage and Efficiency, where Efficiency is defined as thedd\-th root of the conformal ellipsoid volume\. Moreover, we employ a new metric termed Severity:

Severity=CVaR^1−α​\(s¯1,⋯,s¯J\)\\displaystyle\\mathrm\{Severity\}=\\widehat\{\\operatorname\{CVaR\}\}\_\{1\-\\alpha\}\(\\overline\{s\}\_\{1\},\\cdots,\\overline\{s\}\_\{J\}\)\(22\)whereJJis the number of test samples\. Severity measures the average magnitude of the worstα\\alpha\-tail of normalized scoress¯j=sj/q^α\\overline\{s\}\_\{j\}=s\_\{j\}/\\widehat\{q\}\_\{\\alpha\}and therefore quantifies how severe extreme prediction outcomes are relative to the calibrated boundary\. It can also be interpreted as an empirically calibrated counterpart to the tail\-severity ratioΓα\\Gamma\_\{\\alpha\}in Proposition[3](https://arxiv.org/html/2609.27221#Thmproposition3)across calibration and test stages\.

Table[1](https://arxiv.org/html/2609.27221#S5.T1)reports results averaged over 100 simulations\. All methods achieve empirical coverage close to the nominal1−α1\-\\alpha, consistent with Proposition 1 that the independent calibration step ensures coverage guarantees for any fixedΩ^\\widehat\{\\Omega\}\. Atα=0\.1\\alpha=0\.1,MVCSand our method have nearly identical efficiency, while ours achieves lower tail severity\. Notably,τ=0\.08\\tau=0\.08yields the best efficiency, matching the 8% high\-variance population\. Since this proportion is below the allowed miscoverage level, a VaR\-based objective can favor compact sets while underweighting severe tail losses, whereas CVaR remains sensitive beyond the target quantile\. Atα=0\.05\\alpha=0\.05, the allowed miscoverage falls below the rare\-regime probability, requiring the geometry to accommodate part of the high\-variance population\. Our method then achieves both the smallest sets and the lowest severity, withτ=0\.05\\tau=0\.05performing best on average\. This highlights the benefit of tail\-aware geometry learning when the coverage constraint extends into the rare regime\. The gap may also partly reflect the non\-convex optimization of the VaR\-based formulation\. Overall, CVaR maintains competitive efficiency while providing better tail control\.

Fig\.[1](https://arxiv.org/html/2609.27221#S5.F1)further provides the visualization of the obtained residues and the coverage area of MVCS and our method\. We letα=0\.1\\alpha=0\.1andτ=0\.1\\tau=0\.1\. We can observe that, compared with MVCS, our method produces a conformal region that is better aligned with the residual distribution\. By explicitly emphasizing high\-loss samples in the tail, the CVaR objective improves sensitivity to extreme residuals and encourages a more risk\-aware region\.

\(a\)\(b\)
Figure 1:Visualization of the residuals and coverage: \(a\) MVCS; \(b\) Our method\.
### 5\.2Real Data\.

We further test the proposed method on three signal processing benchmark datasets: Gas\[[26](https://arxiv.org/html/2609.27221#bib.bib16)\], Protein\[[27](https://arxiv.org/html/2609.27221#bib.bib17)\], and House\[[28](https://arxiv.org/html/2609.27221#bib.bib18)\]\. These datasets are used for multivariate regression, with a two\-layer MLP serving as the base predictor\. Further details regarding the datasets and the base predictor are provided in the Appendix of the full paper due to space constraints\. We letα=0\.1\\alpha=0\.1, and the results are reported in Table[2](https://arxiv.org/html/2609.27221#S5.T2)\. Across all datasets and choices ofτ\\tau, the empirical coverage remains close to the nominal level of0\.90\.9, consistent with the marginal coverage guarantee of Proposition[1](https://arxiv.org/html/2609.27221#Thmproposition1)\. Compared withMVCS, our method incurs only a modest increase in prediction\-set size while substantially reducing miscoverage severity\. Moreover, decreasingτ\\taufrom0\.10\.1to0\.050\.05further reduces severity on all three datasets, at the cost of a decrease in efficiency, while leaving coverage essentially unchanged\. This behavior illustrates the role ofτ\\tauin controlling the efficiency–severity trade\-off independently of the target coverage levelα\\alpha\. Overall, the results show that the proposed CVaR\-based construction preserves conformal coverage and competitive set efficiency while providing substantially improved control over severe miscoverage\.

Table 2:Results on real data:α=0\.1\\alpha=0\.1\.MethodCoverageEfficiency↓\\downarrowSeverity↓\\downarrowGasNaive0\.8980±0\.00470\.8980\\pm\\\!\\mbox\{\\scriptsize$0\.0047$\}12\.4092±0\.196712\.4092\\pm\\\!\\mbox\{\\scriptsize$0\.1967$\}3\.9242±0\.17703\.9242\\pm\\\!\\mbox\{\\scriptsize$0\.1770$\}Cov0\.8982±0\.00580\.8982\\pm\\\!\\mbox\{\\scriptsize$0\.0058$\}8\.5057±0\.12178\.5057\\pm\\\!\\mbox\{\\scriptsize$0\.1217$\}4\.0354±0\.45024\.0354\\pm\\\!\\mbox\{\\scriptsize$0\.4502$\}MVCS0\.8992±0\.00580\.8992\\pm\\\!\\mbox\{\\scriptsize$0\.0058$\}7\.5024±0\.1229\\textbf\{ 7\.5024\}\\pm\\\!\\mbox\{\\scriptsize$0\.1229$\}4\.0410±0\.45374\.0410\\pm\\\!\\mbox\{\\scriptsize$0\.4537$\}Ours:0\.10\.10\.8990±0\.00530\.8990\\pm\\\!\\mbox\{\\scriptsize$0\.0053$\}7\.7740¯±0\.1287\\underline\{7\.7740\}\\pm\\\!\\mbox\{\\scriptsize$0\.1287$\}2\.2924¯±0\.1595\\underline\{2\.2924\}\\pm\\\!\\mbox\{\\scriptsize$0\.1595$\}Ours:0\.050\.050\.8994±0\.00560\.8994\\pm\\\!\\mbox\{\\scriptsize$0\.0056$\}7\.8556±0\.12987\.8556\\pm\\\!\\mbox\{\\scriptsize$0\.1298$\}1\.9917±0\.1848\\textbf\{1\.9917\}\\pm\\\!\\mbox\{\\scriptsize$0\.1848$\}ProteinNaive0\.9000±0\.00630\.9000\\pm\\\!\\mbox\{\\scriptsize$0\.0063$\}2\.9395±0\.24062\.9395\\pm\\\!\\mbox\{\\scriptsize$0\.2406$\}7\.7390±2\.67247\.7390\\pm\\\!\\mbox\{\\scriptsize$2\.6724$\}Cov0\.9006±0\.00640\.9006\\pm\\\!\\mbox\{\\scriptsize$0\.0064$\}2\.6816±0\.19732\.6816\\pm\\\!\\mbox\{\\scriptsize$0\.1973$\}5\.4533±2\.49815\.4533\\pm\\\!\\mbox\{\\scriptsize$2\.4981$\}MVCS0\.9007±0\.00610\.9007\\pm\\\!\\mbox\{\\scriptsize$0\.0061$\}2\.5095±0\.1924\\textbf\{2\.5095\}\\pm\\\!\\mbox\{\\scriptsize$0\.1924$\}5\.5840±2\.56485\.5840\\pm\\\!\\mbox\{\\scriptsize$2\.5648$\}Ours:0\.10\.10\.9007±0\.00560\.9007\\pm\\\!\\mbox\{\\scriptsize$0\.0056$\}2\.5316¯±0\.1976\\underline\{2\.5316\}\\pm\\\!\\mbox\{\\scriptsize$0\.1976$\}3\.7681¯±1\.3876\\underline\{3\.7681\}\\pm\\\!\\mbox\{\\scriptsize$1\.3876$\}Ours:0\.050\.050\.9002±0\.00630\.9002\\pm\\\!\\mbox\{\\scriptsize$0\.0063$\}2\.5771±0\.19922\.5771\\pm\\\!\\mbox\{\\scriptsize$0\.1992$\}3\.3327±1\.5972\\textbf\{3\.3327\}\\pm\\\!\\mbox\{\\scriptsize$1\.5972$\}HouseNaive0\.9019±0\.00600\.9019\\pm\\\!\\mbox\{\\scriptsize$0\.0060$\}2\.3647±0\.02802\.3647\\pm\\\!\\mbox\{\\scriptsize$0\.0280$\}2\.4940±0\.18492\.4940\\pm\\\!\\mbox\{\\scriptsize$0\.1849$\}Cov0\.9018±0\.00730\.9018\\pm\\\!\\mbox\{\\scriptsize$0\.0073$\}2\.1926±0\.03562\.1926\\pm\\\!\\mbox\{\\scriptsize$0\.0356$\}2\.5127±0\.31222\.5127\\pm\\\!\\mbox\{\\scriptsize$0\.3122$\}MVCS0\.9018±0\.00720\.9018\\pm\\\!\\mbox\{\\scriptsize$0\.0072$\}2\.1212±0\.0354\\textbf\{2\.1212\}\\pm\\\!\\mbox\{\\scriptsize$0\.0354$\}2\.5179±0\.31412\.5179\\pm\\\!\\mbox\{\\scriptsize$0\.3141$\}Ours:0\.10\.10\.9012±0\.00620\.9012\\pm\\\!\\mbox\{\\scriptsize$0\.0062$\}2\.1458¯±0\.0364\\underline\{2\.1458\}\\pm\\\!\\mbox\{\\scriptsize$0\.0364$\}2\.3191¯±0\.2716\\underline\{2\.3191\}\\pm\\\!\\mbox\{\\scriptsize$0\.2716$\}Ours:0\.050\.050\.9009±0\.00680\.9009\\pm\\\!\\mbox\{\\scriptsize$0\.0068$\}2\.1677±0\.037432\.1677\\pm\\\!\\mbox\{\\scriptsize$0\.03743$\}2\.2214±0\.2889\\textbf\{2\.2214\}\\pm\\\!\\mbox\{\\scriptsize$0\.2889$\}

## 6Conclusion

We propose a tail\-aware geometry learning framework for multivariate conformal prediction with ellipsoidal scores\. Our approach decouples tail sensitivity from the target miscoverage level, using an independent parameter to control the shape of the learned geometry via a CVaR criterion\. The resulting formulation is convex and admits a bounded\-reweighting interpretation, revealing how upper\-tail residuals shape the learned metric\. We further characterize the trade\-off between tail severity and ellipsoidal volume\. Extensive experiments validate the effectiveness of the proposed method\.

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- \[24\]S\. Boyd, S\. P\. Boyd, and L\. Vandenberghe\(2004\)Convex optimization\.Cambridge university press\.Cited by:[§3\.2](https://arxiv.org/html/2609.27221#S3.SS2.p1.5)\.
- \[25\]R\. T\. Rockafellar and S\. Uryasev\(2002\)Conditional value\-at\-risk for general loss distributions\.Journal of banking & finance26\(7\),pp\. 1443–1471\.Cited by:[§3\.2](https://arxiv.org/html/2609.27221#S3.SS2.p3.1)\.
- \[26\]\(2019\)Gas Turbine CO and NOx Emission Data Set\.Note:UCI Machine Learning RepositoryDOI: https://doi\.org/10\.24432/C5WC95Cited by:[§5\.2](https://arxiv.org/html/2609.27221#S5.SS2.p1.1)\.
- \[27\]P\. Rana\(2013\)Physicochemical Properties of Protein Tertiary Structure\.Note:UCI Machine Learning RepositoryDOI: https://doi\.org/10\.24432/C5QW3HCited by:[§5\.2](https://arxiv.org/html/2609.27221#S5.SS2.p1.1)\.
- \[28\]S\. Feldman, S\. Bates, and Y\. Romano\(2023\)Calibrated multiple\-output quantile regression with representation learning\.Journal of Machine Learning Research24\(24\),pp\. 1–48\.Cited by:[§5\.2](https://arxiv.org/html/2609.27221#S5.SS2.p1.1),[§7\.3](https://arxiv.org/html/2609.27221#S7.SS3.p1.1)\.

## 7Appendix

### 7\.1Proof of Proposition[2](https://arxiv.org/html/2609.27221#Thmproposition2)

Recall that

Cτ​\(Ω\)=maxη∈𝒫τ⁡⟨Ω,Ση⟩,Ση=∑m=1Mηm​rm​rm⊤,C\_\{\\tau\}\(\\Omega\)=\\max\_\{\\eta\\in\\mathcal\{P\}\_\{\\tau\}\}\\left\\langle\\Omega,\\Sigma\_\{\\eta\}\\right\\rangle,\\qquad\\Sigma\_\{\\eta\}=\\sum\_\{m=1\}^\{M\}\\eta\_\{m\}r\_\{m\}r\_\{m\}^\{\\top\},\(23\)where

𝒫τ=\{η∈ℝM:η≥0,1⊤η=1,ηm≤1τ​M,m∈\[M\]\}\.\\mathcal\{P\}\_\{\\tau\}=\\left\\\{\\eta\\in\\mathbb\{R\}^\{M\}:\\eta\\geq 0,\\;\\mathbf\{1\}^\{\\top\}\\eta=1,\\;\\eta\_\{m\}\\leq\\frac\{1\}\{\\tau M\},\\;m\\in\[M\]\\right\\\}\.Throughout the proof, we assumeτ∈\(0,1\]\\tau\\in\(0,1\]and that\{rm\}m=1M\\\{r\_\{m\}\\\}\_\{m=1\}^\{M\}spansℝd\\mathbb\{R\}^\{d\}\. We consider the convex optimization problem

minΩ≻0−12​log​detΩ,s\.t\.Cτ​\(Ω\)≤1\.\\displaystyle\\min\_\{\\Omega\\succ 0\}\\;\-\\frac\{1\}\{2\}\\log\\det\\Omega,\\quad\\mathrm\{s\.t\.\}\\;C\_\{\\tau\}\(\\Omega\)\\leq 1\.\(24\)Define the empirical residual scatter matrix asS≜1M​∑m=1Mrm​rm⊤\.S\\triangleq\\frac\{1\}\{M\}\\sum\_\{m=1\}^\{M\}r\_\{m\}r\_\{m\}^\{\\top\}\.Since\{rm\}m=1M\\\{r\_\{m\}\\\}\_\{m=1\}^\{M\}spansℝd\\mathbb\{R\}^\{d\}, we haveS≻0\.S\\succ 0\.Consider the uniform weight vectorη¯=\(1M,…,1M\)\.\\bar\{\\eta\}=\\left\(\\frac\{1\}\{M\},\\ldots,\\frac\{1\}\{M\}\\right\)\.Sinceτ∈\(0,1\]\\tau\\in\(0,1\], we have1M≤1τ​M,\\frac\{1\}\{M\}\\leq\\frac\{1\}\{\\tau M\},and henceη¯∈𝒫τ\.\\bar\{\\eta\}\\in\\mathcal\{P\}\_\{\\tau\}\.Therefore, by the dual representation \([23](https://arxiv.org/html/2609.27221#S7.E23)\), we haveCτ​\(Ω\)≥⟨Ω,S⟩\.C\_\{\\tau\}\(\\Omega\)\\geq\\left\\langle\\Omega,S\\right\\rangle\.

Letμ≜λmin​\(S\)\>0\.\\mu\\triangleq\\lambda\_\{\\min\}\(S\)\>0\.For everyΩ⪰0\\Omega\\succeq 0, we obtain⟨Ω,S⟩≥μ​tr⁡\(Ω\)\.\\left\\langle\\Omega,S\\right\\rangle\\geq\\mu\\,\\operatorname\{tr\}\(\\Omega\)\.Consequently, every feasibleΩ\\Omegasatisfies

tr⁡\(Ω\)≤1μ\.\\operatorname\{tr\}\(\\Omega\)\\leq\\frac\{1\}\{\\mu\}\.\(25\)Thus the feasible set is bounded in the positive\-semidefinite cone\. The problem is also strictly feasible\. Indeed, for anyε\>0\\varepsilon\>0,Cτ​\(ε​I\)=ε​Cτ​\(I\),C\_\{\\tau\}\(\\varepsilon I\)=\\varepsilon C\_\{\\tau\}\(I\),by positive homogeneity ofCτC\_\{\\tau\}\. Hence, for sufficiently smallCτ​\(ε​I\)<1\.C\_\{\\tau\}\(\\varepsilon I\)<1\.Therefore Slater’s condition holds\.

To establish attainment of the optimum, letΩ0≻0\\Omega\_\{0\}\\succ 0be any strictly feasible point and let\{Ωn\}n≥1\\\{\\Omega\_\{n\}\\\}\_\{n\\geq 1\}be a minimizing sequence such that, without loss of generality, we have

−12logdetΩn≤−12logdetΩ0\.\-\\frac\{1\}\{2\}\\log\\det\\Omega\_\{n\}\\leq\-\\frac\{1\}\{2\}\\log\\det\\Omega\_\{0\}\.\(26\)Equivalently, this leads todet\(Ωn\)≥det\(Ω0\)\>0\.\\det\(\\Omega\_\{n\}\)\\geq\\det\(\\Omega\_\{0\}\)\>0\.Together with the trace bound \([25](https://arxiv.org/html/2609.27221#S7.E25)\), this implies that the eigenvalues ofΩn\\Omega\_\{n\}are bounded above and bounded away from zero\. Consequently, the minimizing sequence lies in a compact subset of the positive\-definite cone\. Since bothCτ​\(Ω\)C\_\{\\tau\}\(\\Omega\)and−logdetΩ\-\\log\\det\\Omegaare continuous on this set, an optimal solutionΩ^τ≻0\\widehat\{\\Omega\}\_\{\\tau\}\\succ 0exists\. Moreover, the functionΩ↦−logdetΩ\\Omega\\mapsto\-\\log\\det\\Omegais strictly convex on the positive\-definite cone, while the feasible set is convex becauseCτC\_\{\\tau\}is convex\. Hence the optimizerΩ^τ\\widehat\{\\Omega\}\_\{\\tau\}is unique\.

We next explore the activity of the CVaR constraint\. For every scalarc\>0c\>0, the dual representation \([23](https://arxiv.org/html/2609.27221#S7.E23)\) gives

Cτ​\(c​Ω\)=maxη∈𝒫τ⁡⟨c​Ω,Ση⟩=c​maxη∈𝒫τ​⟨Ω,Ση⟩=c​Cτ​\(Ω\)\.\\displaystyle C\_\{\\tau\}\(c\\Omega\)=\\max\_\{\\eta\\in\\mathcal\{P\}\_\{\\tau\}\}\\left\\langle c\\Omega,\\Sigma\_\{\\eta\}\\right\\rangle=c\\max\_\{\\eta\\in\\mathcal\{P\}\_\{\\tau\}\}\\left\\langle\\Omega,\\Sigma\_\{\\eta\}\\right\\rangle=cC\_\{\\tau\}\(\\Omega\)\.\(27\)ThusCτC\_\{\\tau\}is positively homogeneous\. Suppose, for contradiction, thatCτ​\(Ω^τ\)<1\.C\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)<1\.Then one can choosec\>1c\>1sufficiently close to one such thatc​Cτ​\(Ω^τ\)≤1\.cC\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)\\leq 1\.By \([27](https://arxiv.org/html/2609.27221#S7.E27)\),c​Ω^τc\\widehat\{\\Omega\}\_\{\\tau\}remains feasible\.

On the other hand,

−12logdet\(cΩ^τ\)\\displaystyle\-\\frac\{1\}\{2\}\\log\\det\(c\\widehat\{\\Omega\}\_\{\\tau\}\)=−12​log⁡\(cd​detΩ^τ\)\\displaystyle=\-\\frac\{1\}\{2\}\\log\\left\(c^\{d\}\\det\\widehat\{\\Omega\}\_\{\\tau\}\\right\)=−12logdetΩ^τ−d2logc\\displaystyle=\-\\frac\{1\}\{2\}\\log\\det\\widehat\{\\Omega\}\_\{\\tau\}\-\\frac\{d\}\{2\}\\log c<−12logdetΩ^τ,\\displaystyle<\-\\frac\{1\}\{2\}\\log\\det\\widehat\{\\Omega\}\_\{\\tau\},\(28\)which contradicts the optimality ofΩ^τ\\widehat\{\\Omega\}\_\{\\tau\}\. Therefore, we have

Cτ​\(Ω^τ\)=1\.C\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)=1\.\(29\)
Then, we need to study the active CVaR weights and the subdifferential\. Define the active set of CVaR weights atΩ^τ\\widehat\{\\Omega\}\_\{\\tau\}by

𝒜τ​\(Ω^τ\)≜arg⁡maxη∈𝒫τ​⟨Ω^τ,Ση⟩\.\\mathcal\{A\}\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)\\triangleq\\arg\\max\_\{\\eta\\in\\mathcal\{P\}\_\{\\tau\}\}\\left\\langle\\widehat\{\\Omega\}\_\{\\tau\},\\Sigma\_\{\\eta\}\\right\\rangle\.\(30\)Since𝒫τ\\mathcal\{P\}\_\{\\tau\}is compact and the maximized objective is continuous inη\\eta,𝒜τ​\(Ω^τ\)\\mathcal\{A\}\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)is nonempty\. Moreover, because the objective is linear inη\\eta,𝒜τ​\(Ω^τ\)\\mathcal\{A\}\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)is a convex face of𝒫τ\\mathcal\{P\}\_\{\\tau\}\. BecauseCτ​\(Ω\)C\_\{\\tau\}\(\\Omega\)is the pointwise maximum of linear functions ofΩ\\Omega, Danskin’s theorem yields

∂Cτ​\(Ω^τ\)=conv⁡\{Ση:η∈𝒜τ​\(Ω^τ\)\}\.\\partial C\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)=\\operatorname\{conv\}\\left\\\{\\Sigma\_\{\\eta\}:\\eta\\in\\mathcal\{A\}\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)\\right\\\}\.\(31\)Since𝒜τ​\(Ω^τ\)\\mathcal\{A\}\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)is convex andη↦Ση\\eta\\mapsto\\Sigma\_\{\\eta\}is linear, the image set\{Ση:η∈𝒜τ​\(Ω^τ\)\}\\left\\\{\\Sigma\_\{\\eta\}:\\eta\\in\\mathcal\{A\}\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)\\right\\\}is itself convex\. Hence, it leads to

∂Cτ​\(Ω^τ\)=\{Ση:η∈𝒜τ​\(Ω^τ\)\}\.\\partial C\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)=\\left\\\{\\Sigma\_\{\\eta\}:\\eta\\in\\mathcal\{A\}\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)\\right\\\}\.\(32\)Therefore every subgradient ofCτC\_\{\\tau\}atΩ^τ\\widehat\{\\Omega\}\_\{\\tau\}can be represented by the scatter matrix associated with an active CVaR weight vector\.

Finally, we derive the KKT conditions\. Since Slater’s condition holds, the KKT conditions are necessary and sufficient for \([24](https://arxiv.org/html/2609.27221#S7.E24)\)\. Therefore there exist a multiplierλ⋆≥0\\lambda^\{\\star\}\\geq 0and a matrixG⋆∈∂Cτ​\(Ω^τ\)G^\{\\star\}\\in\\partial C\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)such that

−12​Ω^τ−1\+λ⋆​G⋆=0\.\-\\frac\{1\}\{2\}\\widehat\{\\Omega\}\_\{\\tau\}^\{\-1\}\+\\lambda^\{\\star\}G^\{\\star\}=0\.\(33\)The multiplier must satisfyλ⋆\>0,\\lambda^\{\\star\}\>0,becauseλ⋆=0\\lambda^\{\\star\}=0would implyΩ^τ−1=0,\\widehat\{\\Omega\}\_\{\\tau\}^\{\-1\}=0,which is impossible forΩ^τ≻0\\widehat\{\\Omega\}\_\{\\tau\}\\succ 0\. By \([32](https://arxiv.org/html/2609.27221#S7.E32)\), there existsη⋆∈𝒜τ​\(Ω^τ\)\\eta^\{\\star\}\\in\\mathcal\{A\}\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)such thatG⋆=Ση⋆\.G^\{\\star\}=\\Sigma\_\{\\eta^\{\\star\}\}\.Thus \([33](https://arxiv.org/html/2609.27221#S7.E33)\) becomes

Ω^τ−1=2​λ⋆​Ση⋆\.\\widehat\{\\Omega\}\_\{\\tau\}^\{\-1\}=2\\lambda^\{\\star\}\\Sigma\_\{\\eta^\{\\star\}\}\.\(34\)Sinceη⋆\\eta^\{\\star\}is active, we have⟨Ω^τ,Ση⋆⟩=Cτ​\(Ω^τ\)\.\\left\\langle\\widehat\{\\Omega\}\_\{\\tau\},\\Sigma\_\{\\eta^\{\\star\}\}\\right\\rangle=C\_\{\\tau\}\(\\widehat\{\\Omega\}\_\{\\tau\}\)\.Using \([29](https://arxiv.org/html/2609.27221#S7.E29)\), we have⟨Ω^τ,Ση⋆⟩=1\.\\left\\langle\\widehat\{\\Omega\}\_\{\\tau\},\\Sigma\_\{\\eta^\{\\star\}\}\\right\\rangle=1\.Multiplying \([34](https://arxiv.org/html/2609.27221#S7.E34)\) byΩ^τ\\widehat\{\\Omega\}\_\{\\tau\}and taking the trace gives

d\\displaystyle d=tr⁡\(Ω^τ​Ω^τ−1\)=2​λ⋆​tr⁡\(Ω^τ​Ση⋆\)\\displaystyle=\\operatorname\{tr\}\\left\(\\widehat\{\\Omega\}\_\{\\tau\}\\widehat\{\\Omega\}\_\{\\tau\}^\{\-1\}\\right\)=2\\lambda^\{\\star\}\\operatorname\{tr\}\\left\(\\widehat\{\\Omega\}\_\{\\tau\}\\Sigma\_\{\\eta^\{\\star\}\}\\right\)=2​λ⋆​⟨Ω^τ,Ση⋆⟩=2​λ⋆\.\\displaystyle=2\\lambda^\{\\star\}\\left\\langle\\widehat\{\\Omega\}\_\{\\tau\},\\Sigma\_\{\\eta^\{\\star\}\}\\right\\rangle=2\\lambda^\{\\star\}\.\(35\)
Substituting \([35](https://arxiv.org/html/2609.27221#S7.E35)\) into \([34](https://arxiv.org/html/2609.27221#S7.E34)\) yields

Ω^τ−1=d​Ση⋆=d​∑m=1Mηm⋆​rm​rm⊤\.\\widehat\{\\Omega\}\_\{\\tau\}^\{\-1\}=d\\,\\Sigma\_\{\\eta^\{\\star\}\}=d\\sum\_\{m=1\}^\{M\}\\eta\_\{m\}^\{\\star\}r\_\{m\}r\_\{m\}^\{\\top\}\.\(36\)SinceΩ^τ≻0\\widehat\{\\Omega\}\_\{\\tau\}\\succ 0, \([36](https://arxiv.org/html/2609.27221#S7.E36)\) impliesΣη⋆≻0\.\\Sigma\_\{\\eta^\{\\star\}\}\\succ 0\.Consequently, we obtain

Ω^τ=1d​Ση⋆−1=1d​\(∑m=1Mηm⋆​rm​rm⊤\)−1\.\\widehat\{\\Omega\}\_\{\\tau\}=\\frac\{1\}\{d\}\\Sigma\_\{\\eta^\{\\star\}\}^\{\-1\}=\\frac\{1\}\{d\}\\left\(\\sum\_\{m=1\}^\{M\}\\eta\_\{m\}^\{\\star\}r\_\{m\}r\_\{m\}^\{\\top\}\\right\)^\{\-1\}\.\(37\)
The active weight vectorη⋆\\eta^\{\\star\}need not be unique\. Nevertheless, every KKT\-compatible active weight vector satisfying \([36](https://arxiv.org/html/2609.27221#S7.E36)\) induces the same scatter matrix, namely

Ση⋆=1d​Ω^τ−1\.\\Sigma\_\{\\eta^\{\\star\}\}=\\frac\{1\}\{d\}\\widehat\{\\Omega\}\_\{\\tau\}^\{\-1\}\.\(38\)This proves the proposition\.

### 7\.2Proof of Proposition[3](https://arxiv.org/html/2609.27221#Thmproposition3)

For the fixed empirical residual sample, define the quadratic score associated withΩ≻0\\Omega\\succ 0assm​\(Ω\)=rm⊤​Ω​rm,m∈\[M\]\.s\_\{m\}\(\\Omega\)=r\_\{m\}^\{\\top\}\\Omega r\_\{m\},m\\in\[M\]\.For anyc\>0c\>0, we havesm​\(c​Ω\)=c​sm​\(Ω\)\.s\_\{m\}\(c\\Omega\)=c\\,s\_\{m\}\(\\Omega\)\.Since multiplication by a positive scalar preserves the ordering of the empirical scores, both the empirical VaR and CVaR are positively homogeneous:

Vτ​\(c​Ω\)=c​Vτ​\(Ω\),Cτ​\(c​Ω\)=c​Cτ​\(Ω\),c\>0\.V\_\{\\tau\}\(c\\Omega\)=cV\_\{\\tau\}\(\\Omega\),\\qquad C\_\{\\tau\}\(c\\Omega\)=cC\_\{\\tau\}\(\\Omega\),\\qquad c\>0\.\(39\)
We first verify that the tail\-severity ratios appearing in the proposition are well defined\. In particular, the assumed existence of an optimal solution to the VaR\-based problem \([6](https://arxiv.org/html/2609.27221#S3.E6)\) implies that

Vτ​\(Ω\)\>0for every​Ω≻0\.V\_\{\\tau\}\(\\Omega\)\>0\\qquad\\text\{for every \}\\Omega\\succ 0\.\(40\)Indeed, suppose that there existed someΩ~≻0\\widetilde\{\\Omega\}\\succ 0such thatVτ​\(Ω~\)=0\.V\_\{\\tau\}\(\\widetilde\{\\Omega\}\)=0\.Then, by \([39](https://arxiv.org/html/2609.27221#S7.E39)\), we haveVτ​\(c​Ω~\)=c​Vτ​\(Ω~\)=0≤1V\_\{\\tau\}\(c\\widetilde\{\\Omega\}\)=cV\_\{\\tau\}\(\\widetilde\{\\Omega\}\)=0\\leq 1for everyc\>0c\>0\. Hencec​Ω~c\\widetilde\{\\Omega\}would remain feasible for the VaR\-based problem for arbitrarily largecc\. However,

−12logdet\(cΩ~\)\\displaystyle\-\\frac\{1\}\{2\}\\log\\det\(c\\widetilde\{\\Omega\}\)=−12​log⁡\(cd​det\(Ω~\)\)\\displaystyle=\-\\frac\{1\}\{2\}\\log\\left\(c^\{d\}\\det\(\\widetilde\{\\Omega\}\)\\right\)=−12logdet\(Ω~\)−d2logc⟶−∞,\\displaystyle=\-\\frac\{1\}\{2\}\\log\\det\(\\widetilde\{\\Omega\}\)\-\\frac\{d\}\{2\}\\log c\\longrightarrow\-\\infty,\(41\)contradicting the assumed existence of an optimal solution to \([6](https://arxiv.org/html/2609.27221#S3.E6)\)\. Therefore, \([40](https://arxiv.org/html/2609.27221#S7.E40)\) holds\. In particular,Vτ​\(ΩV⋆\)\>0,Vτ​\(ΩC⋆\)\>0,V\_\{\\tau\}\(\\Omega\_\{V\}^\{\\star\}\)\>0,V\_\{\\tau\}\(\\Omega\_\{C\}^\{\\star\}\)\>0,and hence bothΓτ,V\\Gamma\_\{\\tau,V\}andΓτ,C\\Gamma\_\{\\tau,C\}are well defined\.

We next show that the corresponding risk constraints are active at the two optima\. Suppose, for contradiction, thatVτ​\(ΩV⋆\)<1\.V\_\{\\tau\}\(\\Omega\_\{V\}^\{\\star\}\)<1\.SinceVτ​\(ΩV⋆\)\>0V\_\{\\tau\}\(\\Omega\_\{V\}^\{\\star\}\)\>0, there exists somec\>1c\>1sufficiently close to one such thatc​Vτ​\(ΩV⋆\)≤1\.cV\_\{\\tau\}\(\\Omega\_\{V\}^\{\\star\}\)\\leq 1\.By \([39](https://arxiv.org/html/2609.27221#S7.E39)\), we haveVτ​\(c​ΩV⋆\)=c​Vτ​\(ΩV⋆\)≤1,V\_\{\\tau\}\(c\\Omega\_\{V\}^\{\\star\}\)=cV\_\{\\tau\}\(\\Omega\_\{V\}^\{\\star\}\)\\leq 1,soc​ΩV⋆c\\Omega\_\{V\}^\{\\star\}is feasible for the VaR\-based problem\. On the other hand, we can obtain

−12logdet\(cΩV⋆\)\\displaystyle\-\\frac\{1\}\{2\}\\log\\det\(c\\Omega\_\{V\}^\{\\star\}\)=−12logdet\(ΩV⋆\)−d2logc\\displaystyle=\-\\frac\{1\}\{2\}\\log\\det\(\\Omega\_\{V\}^\{\\star\}\)\-\\frac\{d\}\{2\}\\log c<−12logdet\(ΩV⋆\),\\displaystyle<\-\\frac\{1\}\{2\}\\log\\det\(\\Omega\_\{V\}^\{\\star\}\),\(42\)which contradicts the optimality ofΩV⋆\\Omega\_\{V\}^\{\\star\}\. Therefore, this leads to

Vτ​\(ΩV⋆\)=1\.V\_\{\\tau\}\(\\Omega\_\{V\}^\{\\star\}\)=1\.\(43\)The same scaling argument applied to the CVaR\-based problem gives

Cτ​\(ΩC⋆\)=1\.C\_\{\\tau\}\(\\Omega\_\{C\}^\{\\star\}\)=1\.\(44\)By the definition ofΓτ,C\\Gamma\_\{\\tau,C\}and \([44](https://arxiv.org/html/2609.27221#S7.E44)\), we have

Γτ,C=Cτ​\(ΩC⋆\)Vτ​\(ΩC⋆\)=1Vτ​\(ΩC⋆\),\\Gamma\_\{\\tau,C\}=\\frac\{C\_\{\\tau\}\(\\Omega\_\{C\}^\{\\star\}\)\}\{V\_\{\\tau\}\(\\Omega\_\{C\}^\{\\star\}\)\}=\\frac\{1\}\{V\_\{\\tau\}\(\\Omega\_\{C\}^\{\\star\}\)\},\(45\)Using positive homogeneity,

Vτ​\(Γτ,C​ΩC⋆\)=Γτ,C​Vτ​\(ΩC⋆\)=1\.\\displaystyle V\_\{\\tau\}\\left\(\\Gamma\_\{\\tau,C\}\\Omega\_\{C\}^\{\\star\}\\right\)=\\Gamma\_\{\\tau,C\}V\_\{\\tau\}\(\\Omega\_\{C\}^\{\\star\}\)=1\.\(46\)Therefore,Γτ,C​ΩC⋆\\Gamma\_\{\\tau,C\}\\Omega\_\{C\}^\{\\star\}is feasible for the VaR\-based problem \([6](https://arxiv.org/html/2609.27221#S3.E6)\)\. By the optimality ofΩV⋆\\Omega\_\{V\}^\{\\star\}, we have

−12logdet\(ΩV⋆\)\\displaystyle\-\\frac\{1\}\{2\}\\log\\det\(\\Omega\_\{V\}^\{\\star\}\)≤−12logdet\(Γτ,CΩC⋆\)\\displaystyle\\leq\-\\frac\{1\}\{2\}\\log\\det\\left\(\\Gamma\_\{\\tau,C\}\\Omega\_\{C\}^\{\\star\}\\right\)⟹det\(ΩV⋆\)\\displaystyle\\implies\\det\(\\Omega\_\{V\}^\{\\star\}\)≥det\(Γτ,C​ΩC⋆\)\.\\displaystyle\\geq\\det\\left\(\\Gamma\_\{\\tau,C\}\\Omega\_\{C\}^\{\\star\}\\right\)\.\(47\)
We now express the two VaR\-normalized ellipsoids in unit\-threshold form\. By \([43](https://arxiv.org/html/2609.27221#S7.E43)\),ℰτ,V=\{r:r⊤​ΩV⋆​r≤1\}\\mathcal\{E\}\_\{\\tau,V\}=\\left\\\{r:r^\{\\top\}\\Omega\_\{V\}^\{\\star\}r\\leq 1\\right\\\}\. Similarly, by \([45](https://arxiv.org/html/2609.27221#S7.E45)\), we haveℰτ,C=\{r:r⊤​\(Γτ,C​ΩC⋆\)​r≤1\}\.\\mathcal\{E\}\_\{\\tau,C\}=\\left\\\{r:r^\{\\top\}\\left\(\\Gamma\_\{\\tau,C\}\\Omega\_\{C\}^\{\\star\}\\right\)r\\leq 1\\right\\\}\.For anyA≻0A\\succ 0, the ellipsoidℰ⁡\(A\)=\{r∈ℝd:r⊤​A​r≤1\}\\mathcal\{E\}\(A\)=\\left\\\{r\\in\\mathbb\{R\}^\{d\}:r^\{\\top\}Ar\\leq 1\\right\\\}has volumeVol\(ℰ\(A\)\)=λ\(𝔹d\)det\(A\)−1/2,\\operatorname\{Vol\}\(\\mathcal\{E\}\(A\)\)=\\lambda\(\\mathbb\{B\}\_\{d\}\)\\det\(A\)^\{\-1/2\},whereλ⁡\(𝔹d\)\\lambda\(\\mathbb\{B\}\_\{d\}\)denotes the volume of thedd\-dimensional Euclidean unit ball\. Consequently,

Vol⁡\(ℰτ,C\)Vol⁡\(ℰτ,V\)=\(det\(ΩV⋆\)det\(Γτ,C​ΩC⋆\)\)1/2\.\\frac\{\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,C\}\)\}\{\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,V\}\)\}=\\left\(\\frac\{\\det\(\\Omega\_\{V\}^\{\\star\}\)\}\{\\det\(\\Gamma\_\{\\tau,C\}\\Omega\_\{C\}^\{\\star\}\)\}\\right\)^\{1/2\}\.\(48\)Combining \([48](https://arxiv.org/html/2609.27221#S7.E48)\) with \([47](https://arxiv.org/html/2609.27221#S7.E47)\) gives

Vol⁡\(ℰτ,C\)Vol⁡\(ℰτ,V\)≥1\.\\frac\{\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,C\}\)\}\{\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,V\}\)\}\\geq 1\.\(49\)
We next establish the upper bound\. From \([43](https://arxiv.org/html/2609.27221#S7.E43)\) and the definition ofΓτ,V\\Gamma\_\{\\tau,V\}, we haveCτ​\(ΩV⋆\)=Γτ,V​Vτ​\(ΩV⋆\)=Γτ,V\.C\_\{\\tau\}\(\\Omega\_\{V\}^\{\\star\}\)=\\Gamma\_\{\\tau,V\}V\_\{\\tau\}\(\\Omega\_\{V\}^\{\\star\}\)=\\Gamma\_\{\\tau,V\}\.Therefore, by positive homogeneity,Cτ​\(ΩV⋆Γτ,V\)=Cτ​\(ΩV⋆\)Γτ,V=1\.C\_\{\\tau\}\\left\(\\frac\{\\Omega\_\{V\}^\{\\star\}\}\{\\Gamma\_\{\\tau,V\}\}\\right\)=\\frac\{C\_\{\\tau\}\(\\Omega\_\{V\}^\{\\star\}\)\}\{\\Gamma\_\{\\tau,V\}\}=1\.Hence,ΩV⋆/Γτ,V\\Omega\_\{V\}^\{\\star\}/\\Gamma\_\{\\tau,V\}is feasible for the CVaR\-based problem \([8](https://arxiv.org/html/2609.27221#S3.E8)\)\. By the optimality ofΩC⋆\\Omega\_\{C\}^\{\\star\}, we can obtain

−12logdet\(ΩC⋆\)≤−12logdet\(ΩV⋆Γτ,V\),\-\\frac\{1\}\{2\}\\log\\det\(\\Omega\_\{C\}^\{\\star\}\)\\leq\-\\frac\{1\}\{2\}\\log\\det\\left\(\\frac\{\\Omega\_\{V\}^\{\\star\}\}\{\\Gamma\_\{\\tau,V\}\}\\right\),\(50\)which equivalently leads to

det\(ΩC⋆\)≥det\(ΩV⋆Γτ,V\)=Γτ,V−d​det\(ΩV⋆\)\.\\displaystyle\\det\(\\Omega\_\{C\}^\{\\star\}\)\\geq\\det\\left\(\\frac\{\\Omega\_\{V\}^\{\\star\}\}\{\\Gamma\_\{\\tau,V\}\}\\right\)=\\Gamma\_\{\\tau,V\}^\{\-d\}\\det\(\\Omega\_\{V\}^\{\\star\}\)\.\(51\)Multiplying both sides byΓτ,Cd\\Gamma\_\{\\tau,C\}^\{d\}yields

det\(Γτ,C​ΩC⋆\)≥\(Γτ,CΓτ,V\)d​det\(ΩV⋆\)\.\\det\\left\(\\Gamma\_\{\\tau,C\}\\Omega\_\{C\}^\{\\star\}\\right\)\\geq\\left\(\\frac\{\\Gamma\_\{\\tau,C\}\}\{\\Gamma\_\{\\tau,V\}\}\\right\)^\{d\}\\det\(\\Omega\_\{V\}^\{\\star\}\)\.\(52\)Substituting \([52](https://arxiv.org/html/2609.27221#S7.E52)\) into \([48](https://arxiv.org/html/2609.27221#S7.E48)\), we obtain

Vol⁡\(ℰτ,C\)Vol⁡\(ℰτ,V\)\\displaystyle\\frac\{\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,C\}\)\}\{\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,V\}\)\}≤\[det\(ΩV⋆\)\(Γτ,C/Γτ,V\)d​det\(ΩV⋆\)\]1/2\\displaystyle\\leq\\left\[\\frac\{\\det\(\\Omega\_\{V\}^\{\\star\}\)\}\{\\left\(\\Gamma\_\{\\tau,C\}/\\Gamma\_\{\\tau,V\}\\right\)^\{d\}\\det\(\\Omega\_\{V\}^\{\\star\}\)\}\\right\]^\{1/2\}=\(Γτ,VΓτ,C\)d/2\.\\displaystyle=\\left\(\\frac\{\\Gamma\_\{\\tau,V\}\}\{\\Gamma\_\{\\tau,C\}\}\\right\)^\{d/2\}\.\(53\)Combining \([49](https://arxiv.org/html/2609.27221#S7.E49)\) and \([53](https://arxiv.org/html/2609.27221#S7.E53)\) gives

1≤Vol⁡\(ℰτ,C\)Vol⁡\(ℰτ,V\)≤\(Γτ,VΓτ,C\)d/2\.1\\leq\\frac\{\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,C\}\)\}\{\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,V\}\)\}\\leq\\left\(\\frac\{\\Gamma\_\{\\tau,V\}\}\{\\Gamma\_\{\\tau,C\}\}\\right\)^\{d/2\}\.\(54\)This proves \([20](https://arxiv.org/html/2609.27221#S4.E20)\)\.

Moreover, since the volume ratio in \([54](https://arxiv.org/html/2609.27221#S7.E54)\) is at least one, the upper bound immediately implies1≤\(Γτ,VΓτ,C\)d/2,1\\leq\\left\(\\frac\{\\Gamma\_\{\\tau,V\}\}\{\\Gamma\_\{\\tau,C\}\}\\right\)^\{d/2\},and thereforeΓτ,C≤Γτ,V\\Gamma\_\{\\tau,C\}\\leq\\Gamma\_\{\\tau,V\}\. Thus, the CVaR\-optimal geometry has a tail\-severity ratio no larger than that of the VaR\-optimal geometry\.

Finally, suppose thatVol⁡\(ℰτ,C\)=ρ​Vol⁡\(ℰτ,V\),ρ≥1\.\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,C\}\)=\\rho\\,\\operatorname\{Vol\}\(\\mathcal\{E\}\_\{\\tau,V\}\),\\rho\\geq 1\.Then \([53](https://arxiv.org/html/2609.27221#S7.E53)\) givesρ≤\(Γτ,VΓτ,C\)d/2\.\\rho\\leq\\left\(\\frac\{\\Gamma\_\{\\tau,V\}\}\{\\Gamma\_\{\\tau,C\}\}\\right\)^\{d/2\}\.Raising both sides to the power2/d2/dgivesρ2/d≤Γτ,VΓτ,C\.\\rho^\{2/d\}\\leq\\frac\{\\Gamma\_\{\\tau,V\}\}\{\\Gamma\_\{\\tau,C\}\}\.Rearranging yieldsΓτ,C≤ρ−2/dΓτ,V\\Gamma\_\{\\tau,C\}\\leq\\rho^\{\-2/d\}\\Gamma\_\{\\tau,V\}\. This proves \([21](https://arxiv.org/html/2609.27221#S4.E21)\)\.

### 7\.3Details of Real\-data Experiments

For the Protein and House datasets, we construct response variables following\[[28](https://arxiv.org/html/2609.27221#bib.bib18)\]\. For all datasets, we first reserve half of the data to train the base predictor\. The remaining data are further partitioned into training, calibration and test subsets with a ratio of 0\.4, 0\.3 and 0\.3 for each trial\. The training subset serves to estimateΩ\\Omega, while the calibration set is utilized for conformal prediction\.

#### 7\.3\.1Dataset details

The details of the used dataset are summarized as Table\.[3](https://arxiv.org/html/2609.27221#S7.T3)

Table 3:Experimental results on real data\.Dataset\# SamplesDimension of covariateDimension of responsesGas3673392Protein4573082House21613182
#### 7\.3\.2Architecture and training of base predictor

For all dataset, we simply train a 2\-layer MLP and the size of the two layer are both 32\. The activation function isReLU, solver isadam, and learning rate is 0\.001\.