[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-122481-en":3,"doc-seo-122481-105":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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},122481,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Relational Computation for Very Large-Scale Machine Learning","Relational Computation for Very Large-Scale Machine Learning develops a tensor-program view where Einstein summation can be interpreted as relational operations over primary keys. By mapping tensor calculus to sequences of joins and aggregations, the thesis enables execution within relational database systems, leveraging parallelization, distribution, scalability, and efficient handling of sparsity. It proposes Upper-Case-Lower-Case Einstein Notation and the SparseEinSum compiler to generate optimized tensor-relational algebra with sparsity estimation and cost-based schema selection, yielding strong benchmark performance and competitive scaling.","RICE UNIVERSITY  \nRelational Computation for Very Large-Scale Machine Learning  \nBy Yuxin Tang  \nA THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE  \nDoctor of Philosophy  \nAPPROVED, THESIS COMMITTEE  \n Christopher Jermaine   \nChristopher Jermaine (Apr 25, 2025 00:15 CDT)  \nChristopher M. Jermaine (Chair)  \nJ.S. Abercrombie Professor of Engineering Professor and Chair, Department of Computer Science, Rice University  \nArlei Lopes da Silva (Apr 25, 2025 01:41 CDT)  \nArlei Lopes da Silva  \nAssistant Professor, Department of Computer Science, Rice University  \nHOUSTON, TEXAS December 2024  \nABSTRACT  \nRelational Computation for Very Large-Scale Machine Learning  \nby  \nYuxin Tang  \nIn mathematics, a tensor is an algebraic object that describes multilinear relationships among sets of algebraic entities associated with a vector space. From a computational perspective, tensors are commonly represented as multi-dimensional arrays—a format that plays a central role in machine learning. A widely used convention for expressing tensor operations is Einstein summation notation (EinSum), which compactly encodes summation over indexed terms. This notational framework not only streamlines the expression of complex tensor computations but also lends itself to an alternative interpretation: a multi-dimensional array can be viewed as a mapping from a vector of integers (i.e., a primary key) to a real number. This perspective aligns closely with the classical definition of a relational database relation. As a result, many numerical and machine learning computations in tensor calculus can be reformulated as sequences of joins and aggregations over relational data. Executing these computations within a relational database system offers several key advantages, including automatic parallelization, distribution, and scalability. Moreover, relational databases are particularly effective at handling sparsity, as they are designed to efficiently represent and process cases where only a small subset of the possible primary keys actually occur in the relation.  \nIn this thesis, I propose an extension to Einstein notation called Upper-Case-  \nLower-Case Einstein Notation—a simple yet expressive framework for describing tensor programs that interleave operations over sparse (relational) data with efficient kernel calls over dense tensors. This notation enables the concise representation of computations optimized for complex sparsity patterns. To support this notation, I develop a compiler, SparseEinSum, which takes standard EinSum expressions as input, transforms them into extended Upper-Case-Lower-Case Einstein Notation as intermediate representation, and compiles them into tensor-relational algebra. The compiler incorporates sparsity estimation and cost-based schema selection to guide the transformation. The resulting programs can be executed on virtually any relational database system, leveraging arrays to manage dense tensors within a relational execution model. Experiments across tensor computation benchmarks demonstrate that the generated tensor-relational computations offer significant performance improvements.  \nTo support automatic differentiation of relational computation compiled from EinSum, I derive key rules that enable automatic differentiation for relational algebra. I introduce functional relational algebra to build functions in the relational domain and define relational analogs of partial derivatives, Jacobians, gradients, and a set of relation-Jacobian product rules for core relational operators, including table scan, selection, aggregation, and join. This functional framework builds the foundation for differentiation in relational algebra. Then, I propose a relational algebra automatic differentiation algorithm using an efficient, correctness-preserving implementation of the relation-Jacobian product. Through extensive experiments, I show that executing machine learning computations on top of a relational engine","cbCaive84wXLkvYk","https://ap.wps.com/l/cbCaive84wXLkvYk","pdf",1990424,1,147,"English","en",105,"# Abstract\n## Relational view of tensor operations\n## Upper-Case-Lower-Case Einstein Notation and SparseEinSum\n## Tensor-relational execution and benchmarks\n## Relational algebra automatic differentiation","[{\"question\":\"How does the thesis connect tensor computation with relational databases?\",\"answer\":\"It interprets Einstein summation as a mapping from a multi-index (primary key) to values, so tensor calculus can be reformulated as joins and aggregations over relational data.\"},{\"question\":\"What are Upper-Case-Lower-Case Einstein Notation and SparseEinSum?\",\"answer\":\"Upper-Case-Lower-Case Einstein Notation extends Einstein notation to express programs that interleave sparse relational operations with dense tensor kernel calls. SparseEinSum compiles standard EinSum into this extended notation and then into tensor-relational algebra using sparsity estimation and cost-based schema selection.\"},{\"question\":\"How is automatic differentiation handled for relational computation?\",\"answer\":\"The thesis derives differentiation rules for relational algebra by introducing functional relational algebra and relation-Jacobian product rules for key operators, then builds an efficient, correctness-preserving relational algebra automatic differentiation algorithm.\"}]","Relational Computation for Very Large-Scale Machine Learning | PDF",1785810881,370,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"relational-computation-for-very-large-scale-machine-learning","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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/relational-computation-for-very-large-scale-machine-learning/122481/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How does the thesis connect tensor computation with relational databases?","Question",{"text":75,"@type":76},"It interprets Einstein summation as a mapping from a multi-index (primary key) to values, so tensor calculus can be reformulated as joins and aggregations over relational data.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What are Upper-Case-Lower-Case Einstein Notation and SparseEinSum?",{"text":80,"@type":76},"Upper-Case-Lower-Case Einstein Notation extends Einstein notation to express programs that interleave sparse relational operations with dense tensor kernel calls. SparseEinSum compiles standard EinSum into this extended notation and then into tensor-relational algebra using sparsity estimation and cost-based schema selection.",{"name":82,"@type":73,"acceptedAnswer":83},"How is automatic differentiation handled for relational computation?",{"text":84,"@type":76},"The thesis derives differentiation rules for relational algebra by introducing functional relational algebra and relation-Jacobian product rules for key operators, then builds an efficient, correctness-preserving relational algebra automatic differentiation algorithm.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"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":53,"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"]