[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83514-en":3,"doc-seo-83514-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83514,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Measuring Dead Directions Decomposing and Classifying Singular Structure off Canonical Alignment","A descent-free, alignment-free measurement framework extracts singular structure from trained networks using a directional-Fisher-based reading at a single frozen checkpoint. For each dead direction, the method recovers its order, yielding the exact per-direction learning coefficient 1/(2^d) independent of the optimizer’s basis. It classifies genuine singularities (fixed by architecture) versus flat gauge symmetries (distinguished by Fisher magnitude), and aggregates per-direction orders into the global Watanabe triple governing Bayesian free energy and WAIC. A pluggable detector supports transformer, convolutional, and normalization layers, enabling architecture-general order recovery.","arXiv :2607 .00603v 1 [ cs .LG] 1 Jul 2026  \nMeasuring Dead Directions: Decomposing and Classifying Singular Structure off Canonical Alignment  \nTejas Pradeep Shirodkar∗  \nIIIT, Hyderabad  \nAbstract  \nWe give a descent-free, alignment-free measurement of singular structure on trained networks. At a single frozen checkpoint the read recovers the order 􀀹 of each dead direction from the directional-Fisher rate, the master invariant from which the perdirection learning coefficient 1/(2􀀹) follows exactly, in whatever basis the optimizer left. The same read classifies each direction, separating a genuine singularity, whose order the architecture fixes, from a flat gauge symmetry; the directional-Fisher magnitude settles the cases the order cannot. A pluggable detector supplies the directions for transformer, convolutional, and normalisation layers. The read recovers the architecturepredicted order across constructed cells and trained networks, including a fine-tuned vision transformer whose dead structure is the LayerNorm-kernel gauge and a fromscratch one whose compressed MLP forms a node-death at its activation order. Where the singular structure enumerates, the per-direction orders assemble, through the typed intersection of the loci, into the global coefficient (􀁟, 􀀻) matching the closed form. The method removes the canonical-alignment and descent preconditions of the underlying rate result, turning order-recovery into a deterministic, architecture-general reading. We then map its reach into the Watanabe triple: the order determines the universal singular fluctuation 􀁡 (􀀹), though a trained network’s realized 􀁡 falls below it as the live structure absorbs the dead direction’s data fluctuation, and the multiplicity recovers from the dominant structure under a single-locus assumption.  \n1 Introduction  \nA dead direction is the object two traditions see at the same point. From Amari’s information geometry (Amari, 2016) it is a direction in which the Fisher metric loses non-degeneracy. From Watanabe’s singular learning theory (Watanabe, 2009) it is tangent to the analytic singular set, where the Kullback–Leibler divergence vanishes to an integer order that resolution of singularities recovers. The two readings name the same vector, and that order 􀀹 is the  \n∗ Correspondence: [tejas.shirodkar@research.iiit.ac.in](tejas.shirodkar@research.iiit.ac.in)  0009-0001-3034-0087  \n(a) the optimizer leaves u rotated off the axes  \nconstruct u from the K-FAC factors A  G  \n0 + tu ) u  \ndirectional Fisher u F(  \n(b) real gelu block: the read recovers k, the axis scan misses it  \n10 ~~ ~~ 3  \n10~~ ~~4  \n10 ~~ ~~ 5  \n\n|  | off-canonical (joint mode) per-coordinate scan |  |\n| --- | --- | --- |\n|  |  | \u003Cbr> |\n|  | d i | k = 1.95 (k = 2)\u003Cbr>1 26\u003Cbr> |\n|  |  ev ant | = . |\n\nk  \n10 ~~ ~~ 2 10 ~~ ~~ 1 scan distance t  \nFigure 1: Reading the order off canonical alignment. (a) A trained network leaves a dead direction 􀁃 rotated off the coordinate axes; we construct 􀁃 as the joint mode of the K-FAC factors 􀀖 ⊗ 􀀜 and scan the directional Fisher out from the frozen checkpoint 􀁜0 , with no descent and no alignment. (b) On a real gelu transformer block whose dead direction is rotated off the axes, the off-canonical joint-mode read recovers the activation order (ˆ􀀹 = 1.95,􀀹 = 2), while a per-coordinate scan along an axis follows the wrong direction and reads a deviant order (ˆ􀀹 = 1. 26) . The slope in the purity-matched window (shaded) returns 􀀹 .  \ninvariant that bridges them. The trajectory-rate result of Shirodkar (2026b) reads 􀀹 in original parameter coordinates, without resolution: move the parameters along a dead direction 􀁃 ,􀁜 (􀁂) = 􀁜0 + 􀁂􀁃, and the directional Fisher decays as 􀁃⊤ 􀀛(􀁜(􀁂)) 􀁃 = Θ (􀁂2 (􀀹−1)), so the log-log slope returns 􀀹 and the per-direction learning coefficient 􀁟 = 1/(2􀀹) .  \nSingular learning theory characterises a trained network by this learning coefficient together with the multiplicity 􀀻 and the singular fluctuation 􀁡 , the Watana","cbCaigQERg0gD61b","https://ap.wps.com/l/cbCaigQERg0gD61b","pdf",959188,1,45,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What does the directional-Fisher readout recover for each dead direction in a trained network?\",\"answer\":\"At a single frozen checkpoint, the read recovers the dead direction’s order from the directional-Fisher rate, which directly determines the per-direction learning coefficient 1/(2^d) exactly in any optimizer basis.\"},{\"question\":\"How does the method distinguish a genuine singularity from a gauge symmetry direction?\",\"answer\":\"A finite order indicates a genuine singularity (e.g., node-death or depth-induced degeneracy), while a Fisher magnitude that stays at the floor indicates a gauge direction. The magnitude separates cases where order alone would be insufficient.\"},{\"question\":\"Which network components can the approach handle, and how is order recovered across architectures?\",\"answer\":\"A pluggable detector supplies dead directions for transformer, convolutional, and normalization layers. The reading then recovers the architecture-predicted order across constructed cells and trained networks, including fine-tuned vision transformers and from-scratch models.\"}]",1784188552,113,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"measuring-dead-directions-decomposing-and-classifying-singular-structure-off-canonical-alignment","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/measuring-dead-directions-decomposing-and-classifying-singular-structure-off-canonical-alignment/83514/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What does the directional-Fisher readout recover for each dead direction in a trained network?","Question",{"text":74,"@type":75},"At a single frozen checkpoint, the read recovers the dead direction’s order from the directional-Fisher rate, which directly determines the per-direction learning coefficient 1/(2^d) exactly in any optimizer basis.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the method distinguish a genuine singularity from a gauge symmetry direction?",{"text":79,"@type":75},"A finite order indicates a genuine singularity (e.g., node-death or depth-induced degeneracy), while a Fisher magnitude that stays at the floor indicates a gauge direction. The magnitude separates cases where order alone would be insufficient.",{"name":81,"@type":72,"acceptedAnswer":82},"Which network components can the approach handle, and how is order recovered across architectures?",{"text":83,"@type":75},"A pluggable detector supplies dead directions for transformer, convolutional, and normalization layers. The reading then recovers the architecture-predicted order across constructed cells and trained networks, including fine-tuned vision transformers and from-scratch models.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]