[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83671-en":3,"doc-seo-83671-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83671,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Coordinate Singularities Break Conformal Coverage for Gaze and Head Pose","Conformal prediction offers distribution-free reliability for vision tasks, yet its guarantees depend on how prediction errors are measured in the output space. For gaze on the sphere and 3D head pose on SO(3), common conformal scoring relies on flat chart coordinates (e.g., yaw–pitch or Euler angles), which distort geometry near coordinate singularities. Across ETH-XGaze, Gaze360, BIWI, and AFLW2000-3D, 90% slice-conditional coverage collapses by 30–50 points (e.g., to 38.9%/42.0% and 57.5%/55.2%). Structural proofs show scalar thresholding cannot correct the induced axis-shape distortion, while geodesic, coordinate-free scoring removes it without retraining or meaningful added cost.","arXiv :2607 .02565v1 [ cs .CV] 29 Jun 2026  \nCoordinate Singularities Break Conformal Coverage for Gaze and Head Pose  \nMohammadreza Jamalifard, Yaxiong Lei∗ , Parastoo Azizinezhad, and  \nJavier Andreu-Perez∗†  \nSmart Health Technologies Group, Centre for Computational Intelligence, School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK {m.jamalifard,yaxiong.lei,p.azizinezhad,[j.andreu-perez}@essex.ac.uk](j.andreu-perez}@essex.ac.uk)  \n∗ Equal contribution. †Corresponding author.  \nAbstract. Conformal prediction provides distribution-free reliability guarantees for vision systems, but these guarantees depend on how prediction errors are measured in the output space. Many vision tasks produce outputs on curved spaces (e.g . gaze directions on the sphere or 3D head rotations), yet intermediate prediction heads, residuals, uncertainty estimates, or conformal scores are often defined in flat coordinate charts such as yaw–pitch or Euler angles. We show that this scoring choice introduces systematic geometric distortion near coordinate singularities (large pitch angles on the sphere and poses approaching gimbal lock in 3D rotations) . Across four datasets (ETH-XGaze, Gaze360, BIWI, AFLW2000- 3D), slice-conditional coverage at a nominal 90% target drops by 30–50 percentage points in these regions, falling to 38.9% on ETH-XGaze and 42.0% on Gaze360 at gaze pitch above 70◦ , and to 57 .5% on BIWI and 55.2% on AFLW2000-3D at head pose pitch above 60 ◦ near gimbal lock, despite marginal coverage remaining near 90% . We prove that this is structural. Scalar thresholding changes the size of chart-coordinate prediction sets but leaves their distorted axis ratios unchanged. To diagnose this hidden failure mode, we show that a simple geometric quantity, the Riemannian volume density, strongly correlates with where coverage collapse occurs. Finally, we show that coordinate-free geodesic scoring removes this distortion. It requires no retraining and adds negligible computational cost.  \nKeywords: Conformal prediction · Gaze estimation · Head pose estimation · Riemannian geometry  \n1 Introduction  \nConsider a driver monitoring system using conformal prediction to guarantee 90% coverage of a driver’s gaze direction. Marginal conformal evaluation appears nominal: overall coverage is 90 .8% . Yet at extreme pitch angles associated with distraction and drowsiness [29], coverage drops to roughly 39%, so fewer than half of the prediction sets contain the true gaze direction. Such viewing angles  \n2 M. Jamalifard et al.  \nare not rare: combined eye and cervical rotation regularly exceeds 70 ◦ [7,21] during mirror checks, instrument-panel glances, and drowsiness-related head drops, consistent with natural driving variation [10] . The failure persists across architectures, training procedures, and calibration set sizes. Its source is more fundamental: the coordinate system used for the conformal nonconformity score.  \nGaze directions lie on the sphere S2 [40], and head poses are rotations in SO(3) [13, 26] . Vision pipelines often represent intermediate predictions, labels, or downstream calibration residuals using yaw–pitch or Euler angles, even when final benchmark accuracy is reported with angular error. Our analysis concerns the nonconformity score used inside conformal calibration. If this score is computed from chart-coordinate residuals, the resulting prediction sets inherit the chart geometry. These charts distort distances: they compress regions near coordinate singularities (the poles of S2 or gimbal lock in SO(3)) and stretch others [15, 45] . A fixed threshold corresponds to a large region near the equator but only a narrow sliver near the pole (Fig. 1) . Because marginal coverage is enforced by construction, coverage is redistributed across the manifold. Wellconditioned regions overcover while singular regions undercover [9, 37] . Global evaluation metrics therefore do not reveal this failure.  \nA natural ","cbCairQRcZGnopG3","https://ap.wps.com/l/cbCairQRcZGnopG3","pdf",4080662,4,1,18,"English","en",105,"# Introduction\n## Problem setup: conformal scoring on chart coordinates\n## Geometric distortion near singularities\n# Contributions\n## Coverage audit across datasets\n## Impossibility of scalar chart correction\n## Riemannian diagnostic and geodesic remedy","[{\"question\":\"Why does coverage collapse occur for gaze and head pose conformal prediction?\",\"answer\":\"Coverage collapses when nonconformity scores are computed in flat chart coordinates (yaw–pitch or Euler angles), because these charts distort distances near coordinate singularities like sphere poles or gimbal lock. The marginal calibration remains correct, but slice-conditional coverage is redistributed unfavorably.\"},{\"question\":\"How large is the reported coverage drop in the singular regions?\",\"answer\":\"Slice-conditional 90% target coverage drops by about 30–50 percentage points near problematic pitch angles. The abstract reports specific cases such as 38.9% (ETH-XGaze) and 42.0% (Gaze360) for gaze pitch above 70°, and 57.5% (BIWI) and 55.2% (AFLW2000-3D) for head pose pitch above 60° near gimbal lock.\"},{\"question\":\"Why can scalar thresholding not fix the problem?\",\"answer\":\"Scalar adaptive conformal methods can change the size of prediction sets but not their local shape. The axis ratios are determined by the chart’s metric tensor, so threshold rescaling cannot correct the geometric distortion.\"}]",1784189646,45,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"coordinate-singularities-break-conformal-coverage-for-gaze-and-head-pose","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/coordinate-singularities-break-conformal-coverage-for-gaze-and-head-pose/83671/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why does coverage collapse occur for gaze and head pose conformal prediction?","Question",{"text":75,"@type":76},"Coverage collapses when nonconformity scores are computed in flat chart coordinates (yaw–pitch or Euler angles), because these charts distort distances near coordinate singularities like sphere poles or gimbal lock. The marginal calibration remains correct, but slice-conditional coverage is redistributed unfavorably.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How large is the reported coverage drop in the singular regions?",{"text":80,"@type":76},"Slice-conditional 90% target coverage drops by about 30–50 percentage points near problematic pitch angles. The abstract reports specific cases such as 38.9% (ETH-XGaze) and 42.0% (Gaze360) for gaze pitch above 70°, and 57.5% (BIWI) and 55.2% (AFLW2000-3D) for head pose pitch above 60° near gimbal lock.",{"name":82,"@type":73,"acceptedAnswer":83},"Why can scalar thresholding not fix the problem?",{"text":84,"@type":76},"Scalar adaptive conformal methods can change the size of prediction sets but not their local shape. The axis ratios are determined by the chart’s metric tensor, so threshold rescaling cannot correct the geometric distortion.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]