[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82459-en":3,"doc-seo-82459-105":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},82459,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Dead Directions: Geometric Singular Learning","Singular learning theory and information geometry analyze the same parameter spaces using largely separate languages. The former computes Bayesian invariants in resolved coordinates, while the latter works in original coordinates under non-degeneracy assumptions that overparameterised models often violate. The paper introduces a unifying primitive, the dead direction, linking Fisher metric degeneracy to a KL-order tangent of the analytic singular set. It recovers KL order from directional Fisher curvature decay, extends to multi-component crossings and RLCT contributions, and lifts the rate to deep networks via K-FAC factorisation and quotient results.","Dead Directions: Geometric Singular Learning  \nTejas Pradeep Shirodkar∗ IIIT, Hyderabad  \narXiv :2606 .05957v 1 [ cs .LG] 4 Jun 2026  \nAbstract  \nSingular learning theory and information geometry have studied the same parameter spaces in mostly separate vocabularies: the former computes Bayesian invariants in resolved coordinates, the latter works in original coordinates under a non-degeneracy assumption that overparameterised models routinely violate. We bridge them through one primitive, the dead direction: a unit vector along which the Fisher metric degenerates, equivalently a tangent to the analytic singular set with a definite KL order, set by how fast the KL divergence vanishes. The two readings name the same vector; our central move shows its KL order is recoverable as the decay rate of the directional Fisher curvature approaching the singularity, in original parameter coordinates and without a Hironaka resolution. A selection rule on smooth fibres translates this rate into Watanabe’s singledirection contribution to the real log canonical threshold, and we extend the recovery to multi-component crossings, multiplicity 􀀻 , the singular fluctuation 􀁡 (universal in the KL order for 1D directions), prior-RLCT shifts, and tempered posteriors. We then lift this rate to a deep network: a multi-layer K-FAC factorisation writes each Fisher block as a product of activation-and gradient-side rates with a duality between them, instantiated at modern-network primitives (residual streams, layer normalisation, attention) . A quotient theorem carries the rate to the gauge quotient Θ/􀀜 under gradient flow on a 􀀜-invariant metric; SGD qualifies, standard Adam does not, and we construct a 􀀜-equivariant Adamfamily preconditioner (DDCAdam) that does. The bridge yields a parameter-coordinate handle on singular geometry, closed-form per-architecture predictions, and a trajectoryrate readout of Watanabe’s triple (􀁟, 􀀻, 􀁡) from one checkpoint’s forward and backward passes, without posterior sampling.  \n∗ Correspondence: [tejas.shirodkar@research.iiit.ac.in](tejas.shirodkar@research.iiit.ac.in)  0009-0001-3034-0087  \nPart I  \nFoundations and the bridging primitive  \n1 Introduction  \nA trained neural network is a single point in a high-dimensional parameter space: one coordinate per weight, with training tracing a path to a setting that fits the data. In small classical models this endpoint is isolated, and perturbing the weights in any direction degrades the fit. Overparameterised networks behave differently. With far more parameters than the data constrains, the settings that fit equally well form continuous families rather than isolated points. Along some directions the loss does not move at all; along others it changes only at high order. The local geometry at the solution is degenerate, a singularity of the parameter space, and that degeneracy is informative: it reflects how much of the network’s capacity the task uses and which directions the learnt function ignores.  \nThe natural instrument for this geometry is the Fisher information metric 􀀛 (􀁜) = E􀁆 ∼􀀾 ∗ [􀁭 􀁜 log 􀀾 􀁜 (􀁆 ) 􀁭 􀁜 log 􀀾 􀁜 (􀁆 ) ⊤ ], which measures how sharply a model’s predictions respond as the parameters move. A direction in which 􀀛 is large is tightly constrained by the data; a direction in which 􀀛 degenerates to zero is left free. The exactly-fitting parameters Σ 􀀩 = {􀁜 : 􀀾 􀁜 = 􀀾 ∗ } form the singular set, and the Fisher metric loses rank along it.  \nTwo communities have spent two decades studying this parameter space, in mostly separate vocabularies. Information geometry, following Amari (2016), treats a parametric family {􀀾 􀁜 : 􀁜 ∈ Θ} as a Riemannian manifold under the Fisher metric. Natural gradient, the dual (∇, ∇∗ ) connection structure, and the exponential / mixture flatness duality are its central constructions, and all require the metric to be non-singular. Singular learning theory, following Watanabe (2009), addresses the opposite case: a non-identifiable model wh","cbCaieDELkggtoqy","https://ap.wps.com/l/cbCaieDELkggtoqy","pdf",6017195,3,1,139,"English","en",105,"# Introduction\n## Part I: Foundations and the bridging primitive\n### Dead directions bridge two traditions","[{\"question\":\"What is the “dead direction” introduced in the work?\",\"answer\":\"A dead direction is a unit parameter-space vector along which the Fisher information metric degenerates. It is also a tangent to the analytic singular set with a definite KL order determined by how the KL divergence vanishes.\"},{\"question\":\"How does the paper connect Fisher-metric behavior to Watanabe’s invariants?\",\"answer\":\"It shows the KL order can be recovered as the decay rate of directional Fisher curvature approaching the singularity. A selection rule then translates this rate into the singledirection contribution to Watanabe’s real log canonical threshold.\"},{\"question\":\"Does the approach rely on Hironaka resolution?\",\"answer\":\"No. The central recovery of the relevant order is performed in original parameter coordinates without performing a Hironaka resolution.\"}]",1784180592,350,{"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},"dead-directions-geometric-singular-learning","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/dead-directions-geometric-singular-learning/82459/",4,{"url":51,"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","2026-07-23","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},"What is the “dead direction” introduced in the work?","Question",{"text":75,"@type":76},"A dead direction is a unit parameter-space vector along which the Fisher information metric degenerates. It is also a tangent to the analytic singular set with a definite KL order determined by how the KL divergence vanishes.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper connect Fisher-metric behavior to Watanabe’s invariants?",{"text":80,"@type":76},"It shows the KL order can be recovered as the decay rate of directional Fisher curvature approaching the singularity. A selection rule then translates this rate into the singledirection contribution to Watanabe’s real log canonical threshold.",{"name":82,"@type":73,"acceptedAnswer":83},"Does the approach rely on Hironaka resolution?",{"text":84,"@type":76},"No. The central recovery of the relevant order is performed in original parameter coordinates without performing a Hironaka resolution.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"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":52,"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"]