[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86079-en":3,"doc-seo-86079-105":30,"detail-sidebar-cat-0-en-105":84},{"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},86079,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","The Singularity Space: A Generative Diffusion Framework for Signal Representation","Generative models often encode signals as dense amplitude grids, which can blur sharp transients essential for physical correctness. Singularity Space introduces a generative diffusion framework that represents signals via complex-plane singularities, grounded in classical pole-residue descriptions of meromorphic functions. A latent space of physically constrained per-signal singular configurations solves inverse problems from degraded or partial observations, delivering interpretability, structural stability against Gibbs artifacts, and resolution-free reconstruction on arbitrary grids. Evaluated on 1D Burgers shocks, the method reduces representation size by 8×, lowers reconstruction error 4.2× in zero-shot sub-resolution generalization, and recovers physical parameters with ~1e-4 absolute error, suggesting broader applicability to transient-driven signals like speech and biomedical data.","arXiv :2607 . 10930v 1 [ cs .LG] 12 Jul 2026  \nThe Singularity Space: A Generative Diffusion Framework for Signal  \nRepresentation  \nEli Bar-Yosef ∗1, Amir Averbuch †2, and Eli Turkel ‡1  \n1 Department of Applied Mathematics, Tel Aviv University, Israel  \n2 School of Computer Science and AI, Tel Aviv University, Israel  \nAbstract  \nGenerative models often represent signals as dense grids of amplitudes, blurring sharp transients that are crucial for the correctness of physical signals. We introduce Singularity Space, a generative framework that represents signals through complex-plane singularities, rooted in the classical pole-residue representation of meromorphic functions. We learn a latent space of physically constrained, per-signal singularity configurations to solve an inverse problem from degraded or partial observations. The framework has three key properties: interpretability, in which each generated singularity configuration corresponds to a set of physical parameters; structural stability, which mitigates Gibbs artifacts at discontinuities; and resolution-free output reconstruction on arbitrary grids without retraining or interpolation. Our framework employs a transformer-based diffusion model that directly predicts samples at complex-plane singularity coordinates, subject to geometric constraints during sampling. As a controlled test case for sharp-feature recovery, we evaluate our framework on 1D Burgers shocks, where each shock is represented by 32 predicted singularities (an 8 × reduction versus a 1024-point grid signal) . Our framework preserves signal structure (TV ratio ≈ 1) under unseen test-time observation noise, achieves a 4.2 × lower reconstruction error in zero-shot sub-resolution generalization thana grid-based baseline, and recovers physical parameters to 10 −4 absolute error in-distribution. These results suggest that singularity-based representations may provide a practical foundation for other transient-dominated signals such as speech and biomedical signals, with potential extension to higher-dimensional domains.  \nKeywords: Diffusion Models, Inverse Problems, Meromorphic Representations, Zero-Shot Generalization, Interpretability, Rational Approximation, Scientific Machine Learning, Signal Reconstruction, Singularity Space  \n1 Introduction  \nReal-world multiscale signals, whether sharp acoustic transients, physiological spikes (e.g., QRS complexes, neural spikes), or discontinuities in fluid dynamics, are often characterized by a localized transient structure. Generative models, such as diffusion models and autoencoders, have achieved significant success in natural image and audio synthesis by treating data as values on fixed grid sampled arrays. However, these models are optimized for perceptual fidelity rather than physical  \n∗ Email: [elibaryosef@mail.tau.ac.il](elibaryosef@mail.tau.ac.il)  \n†Email: [amir@math.tau.ac.il](amir@math.tau.ac.il)  \n‡Email: [turkel@tauex.tau.ac.il](turkel@tauex.tau.ac.il)  \naccuracy, where the amplitude, location and sharpness of a discontinuity or transient are crucial for physical correctness. Moreover, the grid formulation creates a representational mismatch: the physical phenomenon is localized and transient, yet the network is forced to represent it on a dense grid. To resolve these challenges, modern generative architectures must coordinate many independent amplitudes to approximate a single transient. A small transient misalignment can push grid-based models toward low-frequency, averaged reconstructions, suppressing the high-frequency components needed to accurately represent the signal. This results in non-physical smoothed or blurred transients. This phenomenon is related to spectral bias [25], where neural networks converge on low-frequency approximations early during training and are slow to learn high-frequency parts of the signal. These high-frequency parts are crucial for accurately representing sharp transients. Spectral representations are often used","cbCaitQ2sP9aYpAW","https://ap.wps.com/l/cbCaitQ2sP9aYpAW","pdf",764494,22,1,29,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What are the key properties and results reported for the framework?\",\"answer\":\"The framework provides interpretability, structural stability that mitigates Gibbs artifacts at discontinuities, and resolution-free reconstruction on arbitrary grids; on 1D Burgers shocks it achieves an 8× reduction versus a 1024-point grid, about 4.2× lower reconstruction error in zero-shot sub-resolution generalization, and physical-parameter recovery with ~1e-4 absolute error.\"}]",1784208380,73,{"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":79,"head_meta":81,"extra_data":83,"updated_unix":28},"the-singularity-space-a-generative-diffusion-framework-for-signal-representation","",{"@graph":36,"@context":78},[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/the-singularity-space-a-generative-diffusion-framework-for-signal-representation/86079/",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","2026-07-26","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72],{"name":73,"@type":74,"acceptedAnswer":75},"What are the key properties and results reported for the framework?","Question",{"text":76,"@type":77},"The framework provides interpretability, structural stability that mitigates Gibbs artifacts at discontinuities, and resolution-free reconstruction on arbitrary grids; on 1D Burgers shocks it achieves an 8× reduction versus a 1024-point grid, about 4.2× lower reconstruction error in zero-shot sub-resolution generalization, and physical-parameter recovery with ~1e-4 absolute error.","Answer","https://schema.org",{"og:url":52,"og:type":80,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":82,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":85},[86,90,94,98,103,108,113,116,121,124,128],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":87,"show_sort_weight":88,"slug":89},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":91,"show_sort_weight":92,"slug":93},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Exam",70,"exam",{"id":99,"doc_module":4,"doc_module_name":46,"category_name":100,"show_sort_weight":101,"slug":102},5,"Comic",60,"comic",{"id":104,"doc_module":4,"doc_module_name":46,"category_name":105,"show_sort_weight":106,"slug":107},6,"Technology",50,"technology",{"id":109,"doc_module":4,"doc_module_name":46,"category_name":110,"show_sort_weight":111,"slug":112},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":114,"slug":115},30,"research-report",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":118,"show_sort_weight":119,"slug":120},9,"Religion & Spirituality",20,"religion-spirituality",{"id":119,"doc_module":4,"doc_module_name":46,"category_name":122,"show_sort_weight":119,"slug":123},"World Cup","world-cup",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":126,"show_sort_weight":125,"slug":127},10,"Lifestyle","lifestyle",{"id":129,"doc_module":4,"doc_module_name":46,"category_name":130,"show_sort_weight":99,"slug":131},19,"General","general"]