[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86089-en":3,"doc-seo-86089-105":30,"detail-sidebar-cat-0-en-105":92},{"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},86089,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Sticky Jump Diffusions: A Unifying View of Masked, Continuous, and Hybrid Diffusion","Sticky Jump Diffusions (SJDs) define continuous-time Markov processes on Rd where discrete anchors correspond to token embeddings. Forward dynamics release anchor mass at a hazard rate and diffuse in the continuous space; time reversal couples a score-driven SDE with a sticky jump kernel determined by flux balance. Denoising Hazard Matching estimates the score and per-anchor reverse hazards from a single denoising classifier using simulation-free cross-entropy training. SJDs unify masked, continuous, and hybrid diffusion as limiting cases and extend to a design space via cross-position blending.","Sticky Jump Diffusions: A Unifying View of Masked, Continuous, and Hybrid Diffusion  \nPascal Jutras-Dubé  \nDepartment of Computer Science Purdue University [pjutrasd@purdue.edu](pjutrasd@purdue.edu)  \nPatrick Pynadath  \nDepartment of Computer Science Purdue University [ppynadat@purdue.edu](ppynadat@purdue.edu)  \narXiv :2607 . 1095 1v 1 [ cs .LG] 12 Jul 2026  \nJeremy Lu  \nDepartment of Computer Science Purdue University [lu1008@purdue.edu](lu1008@purdue.edu)  \nYuan Gao  \nDepartment of Mathematics Purdue University [gao662@purdue.edu](gao662@purdue.edu)  \nRuqi Zhang  \nDepartment of Computer Science  \nPurdue University  \n[ruqiz@purdue.edu](ruqiz@purdue.edu)  \nAbstract  \nWe introduce Sticky Jump Diffusions (SJDs), continuous-time Markov processes on Rd whose discrete anchors are token embeddings. In forward time, anchors release their mass at a hazard rate and the released mass diffuses in the continuous ambient space; time reversal couples a score-driven SDE with a sticky jump kernel whose rate and destination are fixed by flux balance with the forward law. We estimate the score and the per-anchor reverse hazards from a single denoising classifier via Denoising Hazard Matching, the hazard analogue of denoising score matching, with simulation-free cross-entropy training. SJD recovers masked diffusion, continuous diffusion, and hybrid diffusion as limits. Its reversal explains features that each family treats as given: the mask of masked diffusion carries no evidence about the source token because the unsticking kernel of every anchor collapses to the same absorbing point; the terminal projection of continuous diffusion is required due to the absence of atoms in its forward marginal, without which flux balance yields no reverse jumps; and the update rules of hybrid diffusion (commit rate, destination, and drift) all follow from flux balance rather than from separate design.  \nBeyond these limits, the unsticking kernel becomes a design space: a cross-position blending corrupts each position toward a blend of its neighbors’ clean values or embeddings, turning dependency structure such as spatial locality or a constraint graph into an inductive bias of the corruption itself, and improves over the identitykernel hybrid on CIFAR-10, Text8, and Sudoku. Our code is available at [https:](https:)//[github.com/PascalJD/sticky-jump-diffusions](github.com/PascalJD/sticky-jump-diffusions).  \n1 Introduction  \nDiffusion-based generative modeling has been extended from continuous data [Sohl-Dickstein et al., 2015, Song and Ermon, 2019, Ho et al., 2020, Song et al., 2021, Karras et al., 2022] to discrete sequences such as text. Two families dominate the discrete setting. Masked diffusion operates on the token space and outputs valid tokens by construction, but each position is mapped to a single  \nPreprint.  \nMasked diffusion  \n0.0 0.2 0.4 0.6 0.8 1.0  \nreverse time ¿✗ Uses Continuous Gradients ✓ Outputs Discrete Tokens  \nContinuous diffusion  \n0.0 0.2 0.4 0.6 0.8 1.0  \nreverse time ¿✓ Uses Continuous Gradients ✗ Outputs Discrete Tokens  \n2  \n1  \n0  \n-1  \n-2  \nSticky Jump Diffusion  \n0.0 0.2 0.4 0.6 0.8 1.0  \nreverse time ¿✓ Uses Continuous Gradients  \n✓ Outputs Discrete Tokens  \nData PMF  \n\n| \u003Cbr>\u003Cbr> |  |\n| --- | --- |\n\n0.0 0.5  \nFigure 1: Reverse-time trajectories over each model’s path space. Masked diffusion is discrete but gradient-free; continuous diffusion is gradient-based but requires a terminal projection; SJD couples ambient gradients with sticky jumps onto anchors.  \nabsorbing token, and the per-position factorization erases graded information about how close the corruption is to a candidate token [Austin et al., 2021, Sahoo et al., 2024, Shi et al., 2024, Lou et al., 2024] . Continuous diffusion operates on token embeddings and uses standard score-based learning, but its terminal state lies off the discrete support and must be mapped back by a projection step external to the dynamics [Li et al., 2022, Dieleman et al., 2022, Chen et al","cbCaiv8dRKWVUxLA","https://ap.wps.com/l/cbCaiv8dRKWVUxLA","pdf",813609,5,1,32,"English","en",105,"# Introduction\n## Sticky Jump Diffusions (SJDs)\n## Forward process with hazards and diffusion\n## Time reversal and sticky jump kernel\n## Denoising Hazard Matching (DHM)","[{\"question\":\"What are Sticky Jump Diffusions (SJDs)?\",\"answer\":\"SJDs are continuous-time Markov processes on Rd with a finite or countable set of anchors identified with token embeddings.\"},{\"question\":\"How does time reversal work in SJDs?\",\"answer\":\"Time reversal couples a score-driven SDE on the continuous region with a sticky jump kernel that commits mass to anchors, with rates fixed by flux balance.\"},{\"question\":\"How are the model’s score and reverse hazards learned?\",\"answer\":\"Denoising Hazard Matching estimates both from a single denoising classifier using simulation-free cross-entropy training.\"}]",1784208437,81,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"sticky-jump-diffusions-a-unifying-view-of-masked-continuous-and-hybrid-diffusion","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/sticky-jump-diffusions-a-unifying-view-of-masked-continuous-and-hybrid-diffusion/86089/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What 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