[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85949-en":3,"doc-seo-85949-105":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},85949,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Conditional Optimal Bridge for Riemannian Activation Steering","Activation steering enables controlling large language models at inference time without fine-tuning. Many existing approaches heuristically optimize a log-density-ratio between desired and undesired activation distributions and produce query-independent steering directions that can harm both in-distribution and out-of-distribution performance. Cobras recasts activation steering as a Schrödinger Bridge on the residual-stream hypersphere. This yields a principled derivation of the log-density-ratio objective and query-adaptive directions via evaluation at the current activation, improving helpfulness, truthfulness, and detoxification.","arXiv :2607 . 105 17v 1 [ cs .LG] 12 Jul 2026  \nConditional Optimal Bridge for Riemannian Activation Steering  \nSeyed Arshan Dalili Ajay Narayanan Sridhar  \nVijaykrishnan Narayanan Mehrdad Mahdavi  \nThe Pennsylvania State University {sbd5760,afs6372,vxn9,[mzm616}@psu. edu](mzm616}@psu. edu)  \nAbstract  \nActivation steering offers a lightweight alternative to fine-tuning for controlling large language models at inference time. While many existing methods implicitly optimize a log-density-ratio objective between desired and undesired activation distributions, they do so heuristically rather than deriving it from a principled optimization problem. Moreover, these methods produce query-independent steering directions that can degrade performance on both in-distribution and out-of-distribution (OOD) inputs. We introduce Cobras (Conditional Optimal Bridge for Riemannian Activation Steering), which addresses both limitations by casting activation steering as a Schrödinger Bridge on the residual-stream hypersphere. This formulation yields, to our knowledge, the first principled derivation of the log-density-ratio steering objective from a well-posed optimization problem. Solving the bridge via entropic optimal transport and extracting the probability flow ODE recovers the widely used density-ratio gradient as a special case when the Sinkhorn potentials are uniform. Crucially, the Schrödinger potentials are evaluated at the current activation, making the resulting steering direction inherently query-adaptive. Empirically, across four models and three alignment axes (helpfulness, truthfulness, and detoxification), Cobras consistently outperforms prior activation steering baselines while avoiding the OOD degradation commonly observed in existing methods. The code can be found here.  \n1 Introduction  \nActivation steering, also termed representation engineering, has emerged as a compelling alternative to fine-tuning for shaping the behavior of large language models (LLMs) at inference time [Rimsky et al. , 2024 , Li et al. , 2023 , Wang et al. , 2025] . Its empirical success is often motivated by the linear representation hypothesis [Park et al. , 2023 , Alain and Bengio, 2016] where many high-level properties encoded by an LLM, such as sentiment, truthfulness, or harmfulness, appear to correspond to approximately linear directions in activation space. Under this view, changing a model’s behavior need not require modifying its weights; it can instead be achieved by moving its internal activations along directions associated with the target property at inference time. A standard approach is activation addition, which modifies the original activations by adding either a fixed steering vector or one that depends on the current activation.  \nThis perspective has inspired a growing body of work on inference-time control through activation interventions. Early methods typically construct a single steering direction from contrastive examples and then apply that direction uniformly across queries [Rimsky et al. , 2024], while more recent approaches refine the dynamics of the intervention by integrating a continuous steering field over multiple steps [Zhao et al. , 2026 , Wang et al. , 2025 , Rodriguez et al. , 2025] or working on a more faithful manifold by rotating activations on the hypersphere You et al. [2026], Pham and Nguyen [2024] . These advances show that steering can be both effective and computationally lightweight.  \nDespite this progress, existing methods share two limitations. First, most of the methods and their implicit objective are heuristic rather than derived from a well-posed optimization problem without formal justification for the objective itself. Second, these steering signals are query-independent. A steering vector that helps in-distribution prompts can make out-of-distribution (OOD) inputs wrong or gibberish, and our experiments confirm that several strong baselines improve TruthfulQA Lin et al. [2021] at the cost of se","cbCaigsUKzU4MNht","https://ap.wps.com/l/cbCaigsUKzU4MNht","pdf",2389496,2,1,24,"English","en",105,"# Introduction\n# Related Work\n## Theory of activation steering\n# Background and Setup\n# Method: Cobras and Schrödinger Bridge Formulation\n# Experiments\n## In-distribution and OOD Results\n# Conclusion","[{\"question\":\"What problem does Cobras address in existing activation steering methods?\",\"answer\":\"It addresses two gaps: most methods use heuristic objectives rather than a principled optimization formulation, and their steering directions are query-independent, which can degrade out-of-distribution performance.\"},{\"question\":\"How does Cobras formulate activation steering?\",\"answer\":\"Cobras casts activation steering as a Schrödinger Bridge on the residual-stream hypersphere, then solves it using entropic optimal transport and derives the steering objective from the resulting probability flow ODE.\"},{\"question\":\"Why is the steering direction in Cobras query-adaptive?\",\"answer\":\"The Schrödinger potentials are evaluated at the current activation, making the extracted steering direction change with the query rather than staying fixed.\"}]",1784207328,60,{"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},"conditional-optimal-bridge-for-riemannian-activation-steering","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/conditional-optimal-bridge-for-riemannian-activation-steering/85949/",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","2026-07-26","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 problem does Cobras address in existing activation steering methods?","Question",{"text":75,"@type":76},"It addresses two gaps: most methods use heuristic objectives rather than a principled optimization formulation, and their steering directions are query-independent, which can degrade out-of-distribution performance.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does Cobras formulate activation steering?",{"text":80,"@type":76},"Cobras casts activation steering as a Schrödinger Bridge on the residual-stream hypersphere, then solves it using entropic optimal transport and derives the steering objective from the resulting probability flow ODE.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is the steering direction in Cobras query-adaptive?",{"text":84,"@type":76},"The Schrödinger potentials are evaluated at the current activation, making the extracted steering direction change with the query rather than staying 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