[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86076-en":3,"doc-seo-86076-105":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},86076,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","The Spectral Structure of Latent Treatment Effects","Identifying heterogeneous treatment effects under unobserved confounding is central to observational causal inference. In proxy models with a discrete latent confounder, prior Synthetic Potential Outcomes (SPO) methods recover a mixture of treatment effects via recursively constructed scalar moments. This work shows the moment sequence is only a projection of a more fundamental compressed observable operator. After projecting onto the shared proxy signal subspace, eigenvalues reveal latent effects and lifted eigenvectors recover proxy features and mixture proportions, enabling spectral estimation, overcomplete proxy handling, and high-probability perturbation bounds.","arXiv :2607 . 10926v 1 [ cs .LG] 12 Jul 2026  \nThe Spectral Structure of Latent Treatment Effects  \nHamza Virk 1 ∗ Bijan Mazaheri 1 ,2 Yihren Wu3  \n1 Dartmouth College  \n2 Broad Institute of MIT and Harvard  \n3 Hofstra University  \n{[hamza.a.virk.th](hamza.a.virk.th) , [bijan.h.mazaheri}@dartmouth.edu](bijan.h.mazaheri}@dartmouth.edu) , [yihren.wu@hofstra.edu](yihren.wu@hofstra.edu)  \nAbstract  \nIdentifying heterogeneous treatment effects under unobserved confounding is central in observational causal inference. In proxy models with a discrete latent confounder, prior Synthetic Potential Outcomes (SPO) [Mazaheri, Squires, and Uhler, 2025] recover the mixture of treatment effects through recursively constructed scalar moments. We show that this sequence is one projection of a more fundamental object. Under the same population factorization assumptions, there is an exact compressed observable operator: after projecting onto the shared proxy signal subspace, the difference of two treatment-arm quotient operators is similar to the diagonal matrix of latent treatment effects. Its eigenvalues are the latent effects; its lifted left eigenvectors, after anchor normalization, recover the target-proxy feature matrix and then the latent mixture proportions. Every scalar SPO moment is a bilinear functional of a power of this operator. The resulting estimator handles overcomplete proxy systems, replaces high-order scalar inversion with finite-dimensional spectral analysis, and admits high-probability first-order perturbation bounds for treatment effects, feature rows, and simplex-projected mixture weights.  \n1 Introduction  \nA ˜b˘r˚i`e¨f `g¨lˇi‹m¯p¯sfi`e ˚i‹n˚t´o ˚t‚h`e ¯p`a¯sfi˚t. The problem of drawing causal conclusions from non-experimental data is as old as empirical science itself. In the 1840s, John Snow traced the Broad Street cholera outbreak to a contaminated water pump in London, a landmark feat of causal reasoning conducted entirely through observational data and without a randomized trial in sight [Snow, 1855] . Decades later, Karl Pearson formalized the correlation coefficient and yet famously conflated correlation with causation, a confusion that would haunt empirical research for generations [Pearson, 1896] . The mid-twentieth century placed the distinction between association and causation on rigorous mathematical footing. Jerzy Neyman’s potential-outcomes framework [Spława-Neyman, 1990] and, independently, Donald Rubin’s later formalization [Rubin, 1974] gave statisticians a language for asking what would have happened had the treatment been different. The simultaneous-equations tradition in econometrics, crystallized at the Cowles Commission [Haavelmo, 1944], attacked the same problem from the angle of structural identification and produced the instrumental-variable methods still central to applied economics today. Pearl’s graphical reformulation [Pearl, 2009]  \n∗ Corresponding Author.  \nFigure 1: Causal triptych for the latent proxy mixture model. (a) depicts the proxy causal structure: a latent confounder U drives treatment T , the target proxy X, the reference proxy Z, and the observed outcome Y. The dashed arrow from T to Z indicates that the reference-proxy feature law may be treatment-arm-specific, as encoded by At(:, u) = E [Z | U = u, T = t] . (b) unrolls the latent classes uj, each of which generates shared target-proxy features, treatment-arm-specific reference-proxy features, and conditional potential-outcome means µ t(uj) = E[Y(t) | U = uj] . (c) shows the classwise counterfactual pairing recovered by the compressed spectral operator: each latent treatment effect is the paired contrast τj = µ 1 (uj) − µ0 (uj) .  \nunified the potential-outcomes and structural-equations traditions into the modern theory of causal inference. One adversary survived all of these frameworks. The unobserved confounder, a hidden variable that simultaneously influences who receives treatment and what outcome they experience, biases es","cbCairenrxKL0AbH","https://ap.wps.com/l/cbCairenrxKL0AbH","pdf",948763,4,1,51,"English","en",105,"# Abstract\n# Introduction\n## Background: causal inference under observational data\n## Proximal causal inference and recoverable heterogeneity\n## From SPO moments to a compressed spectral operator","[{\"question\":\"What problem does the document address in observational causal inference?\",\"answer\":\"It addresses identifying heterogeneous treatment effects when the confounder is unobserved, a key challenge in observational settings.\"},{\"question\":\"How does the proposed approach relate to Synthetic Potential Outcomes (SPO)?\",\"answer\":\"SPO recovers treatment-effect mixtures from recursively constructed scalar moments, but the document shows those moments are a projection of a more fundamental compressed observable operator.\"},{\"question\":\"What does the compressed observable operator reveal and how is it used?\",\"answer\":\"After projection onto the shared proxy signal subspace, the operator’s eigenvalues equal the latent treatment effects, and its lifted left eigenvectors (with anchor normalization) recover target-proxy features and mixture proportions; scalar SPO moments then become bilinear functionals of powers of this operator.\"}]",1784208372,129,{"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},"the-spectral-structure-of-latent-treatment-effects","",{"@graph":36,"@context":85},[37,53,68],{"@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":20},"https://docshare.wps.com/document/the-spectral-structure-of-latent-treatment-effects/86076/",{"url":52,"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","2026-07-27","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 the document address in observational causal inference?","Question",{"text":75,"@type":76},"It addresses identifying heterogeneous treatment effects when the confounder is unobserved, a key challenge in observational settings.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed approach relate to Synthetic Potential Outcomes (SPO)?",{"text":80,"@type":76},"SPO recovers treatment-effect mixtures from recursively constructed scalar moments, but the document shows those moments are a projection of a more fundamental compressed observable operator.",{"name":82,"@type":73,"acceptedAnswer":83},"What does the compressed observable operator reveal and how is it used?",{"text":84,"@type":76},"After projection onto the shared proxy signal subspace, the operator’s eigenvalues equal the latent treatment effects, and its lifted left eigenvectors (with anchor normalization) recover target-proxy features and mixture proportions; scalar SPO moments then become bilinear functionals of powers of this operator.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"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":20,"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"]