[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573026-105":53,"doc-detail-573026-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","ai-neural-networks-neural-networks-backpropagation-and-core-concepts","AI-Neural-Networks - Neural Networks - Backpropagation and Core Concepts","","Neural networks overview covering architectures from deep feed forward to recurrent models including LSTM, GRU, and various memory and probabilistic cell types. Presents neural network graphs and connections between input/output cells and recurrent/memory mechanisms. Includes mathematical foundations for linear vector spaces and inner products, spanning, orthogonality, Gram-Schmidt, reciprocal bases, perceptron architecture and learning rule, Hebbian postulates, optimization conditions, transformations, eigenvalues, and common minimization algorithms. Concludes with heuristic backpropagation variations such as batching and gradient estimation across training sets.",{"@graph":63,"@context":122},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/ai-neural-networks-neural-networks-backpropagation-and-core-concepts/573026/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/ai-neural-networks-neural-networks-backpropagation-and-core-concepts/573026.png","ImageObject",442,249,{"name":88,"@type":89},"BAd apple","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-18","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":76},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114,118],{"name":105,"@type":106,"acceptedAnswer":107},"Which neural network cell and architecture types are covered?","Question",{"text":108,"@type":109},"The content lists deep feed forward (DFF), noisy and probabilistic hidden cells, spiking hidden cells, and recurrent architectures. It also enumerates LSTM, GRU, and multiple memory cell variants, plus several network families such as GAN and residual networks.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What mathematical topics are included to support neural network reasoning?",{"text":113,"@type":109},"It covers linear vector spaces, linear independence, spanning, inner products, norms, angles, orthogonality, Gram-Schmidt orthogonalization, reciprocal basis vectors, transformations, and eigenvalues/eigenvectors with diagonalization.",{"name":115,"@type":106,"acceptedAnswer":116},"How are learning and optimization ideas described?",{"text":117,"@type":109},"It includes perceptron learning and decision boundary concepts, Hebb’s postulate and related learning rules, and optimization conditions (stationary points, Hessian positivity). It also presents general minimization and steepest descent algorithms.",{"name":119,"@type":106,"acceptedAnswer":120},"What heuristic variation of backpropagation is discussed?",{"text":121,"@type":109},"Batching is described, where parameter updates occur only after the full training set is presented. Gradients from each training example are averaged to improve the estimate of the gradient.","https://schema.org",{"og:url":78,"og:type":124,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":126,"canonical":78},"index,follow",{"doc_id":128,"site_id":56},573026,1792241794,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":88,"user_avatar":133,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":76,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":45,"language":139,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":61,"update_tm":143,"read_time":144},19241458193282,"https://us-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","A mostly complete chart of  \n# Neural NetworksDeep Feed Forward(DFF)Input Cell\n\n◎Backfed Input Cell  \n△Noisy Input CellHidden Cell◎Probablistic Hidden Cell  \n△Spiking Hidden Cell  \nRecurrent Neural Network(RNN)Long/Short Term Memory (LSTM)Gated Recurrent Unit(GRU)Output Cell◎Match Input Output CellRecurrent Cell◎Memory Cell  \nAuto Encoder(AE)Variational AE(VAE)Denoising AE(DAE)Sparse AE(SAE)  \n△  \nDifferent Memory CellKernelO Convolution or PoolMarkov Chain(MC)Hopfield Network(HN)Boltzmann Machine(BM)Restricted BM(RBM)Deep Belief Network(DBN)Deep Convolutional Network(DCN)Deconvolutional Network(DN)Deep Convolutional Inverse Graphics Network(DCIGN)Generative Adversarial Network(GAN)Liquid State Machine(LSM)Extreme Learning Machine(ELM)Echo State Network(ESN)Deep Residual Network(DRN)Kohonen Network(KN)Support Vector Machine(SVM)  NeuralTuring Machine(NTM)  \n# Neural Network Graphs\n\n◎2016 Fjodor van Veen -asimovinstitute.org  \nDeep Feed Forward Example  \nDeep Recurrent Example  \n(previous iteration)  \nDeep Recurrent Example  \nDeep GRU Example  \n(previous iteration)  \nDeep GRU Example  \nDeep LSTM Example  \n(previous iteration)  \nDeep LSTM Example  \n\n| Linear Vector Spaces:  \u003Cbr>Definition:A linear vector space,X is aset of elements(vectors)defined over a  \u003Cbr>scalar field,F,that stisfies the following conditions  \u003Cbr>1)ifxeXandy∈Xthen eyEX.2)+y-y+x 3)(+y+z=x+(+)  \u003Cbr>4)There is a unique vector o∈X,such that x+0=x for all x∈X  \u003Cbr>5)For each vector xEXthere is aunique vector in X,to be called(x),such that  \u003Cbr>x+(x-0.   6)multiplication,for all scalarsaEF,and all vectors xEX  \u003Cbr>7)For any x∈X,1x=x(for scalar 1).  \u003Cbr>8)For any two scalars aEF amdbEF and any xEX,a(60=(a6)x,  \u003Cbr>9)(a+b)x=ax+bx.10)a(x+y=ax+ay  \u003Cbr>Linear Independence:Consider n vectors {x,X2x△}.If there  \u003Cbr>existsn scalars aj,a.,…an,at least one of which is nonzero,such that  \u003Cbr>ax₁+aX+…+aXa=0,then the {x₁)are linearly dependent.  \u003Cbr>Spanning a Space:  \u003Cbr>Let Xbe a linear vector space and let(u₂,u₂…vn)be a subset of vectors inX.  \u003Cbr>This subset spansX if and only if for every vector x EXthere exist scalars xj,  \u003Cbr>x.…xsuch thatx=x;Mz+xuz+..+xmMs  \u003Cbr>Inner Product:(x y)for any scalar function of xandy.  \u003Cbr>1.(x,y)=(y,x)2.(x,ay+by₂=a(x,y₁)+b(x,y₂)  \u003Cbr>3.(x,x)≥0,where equality holds iffx is the zerovector.  \u003Cbr>Norm:Ascalar function xll is called a norm ifit satisfies:  \u003Cbr>1.||xll≥0  \u003Cbr>2.lx|l=0 if and only ifx=0.  \u003Cbr>3.llax|l=|ax| 4.llx+yll≤||x|l+llyll  \u003Cbr>Angle:The angle e bet.2 vectors x andy is defined by co  \u003Cbr>Orthogonality:2 vectorsx,ye Xare said to be orthogonal if (xy)=0.  \u003Cbr>Gram Schmidt Orthogonalization:  \u003Cbr>Assume that we have n independent vectors yny2…,y..From  \u003Cbr>these vectors we will obt  \u003Cbr>,V₂…,V₃.  \u003Cbr>r₂=Yu  \u003Cbr>Vector E  \u003Cbr>Reciprocal Basis Vectors:  \u003Cbr>,x=(ry,x)  \u003Cbr>To compute the reciprocal basis vectors:set B=[v₁V2…Vn],  \u003Cbr>R-[r₁r₂…rm],R⁷=B-¹In matrix form:x”=B-¹x⁵   | Perceptron Architecture:  \u003Cbr>a=hardlim(Wp+b),W=[₁w⁷₂w⁷….sw],  \u003Cbr>a₁=hardtim(n;)=hardlim(w⁷p+b)  \u003Cbr>Decision Boundary:w⁷p+b₁=0  \u003Cbr>The decision boundary is always orthogonal to the weight vector.  \u003Cbr>Single-layer perceptrons can only classify linearly separable vectors.  \u003Cbr>Perceptron Learning Rule  \u003Cbr>wnew=wd+ep⁷,bnew=boud+e,  \u003Cbr>where e =t-a   |\n| --- | --- |\n|  | Hebb's Postulate:\"When an axon of cell A is near enough to excite a  \u003Cbr>cell B and repeatedly or persistently takes part in firing it,some growth  \u003Cbr>process or metabolic change takes place in one or both cells such that A's  \u003Cbr>efficiency,as one of the cells firing B,is increased\"  \u003Cbr>Linear Associator:a=purelin(Wp)  \u003Cbr>Pseudoinverse Rule:W=TP+  \u003Cbr>When the number,R,of rows of P is greater than the num ber of  \u003Cbr>columns,Q,of P and the columns of P are independent,then the  \u003Cbr>pseudoinverse can be computed by P+=(PTP)-1PT  \u003Cbr>Variations of Hebbian Learning:  \u003Cbr>Filtered Learningch14:  \u003Cbr>Delta Rule(Ch.10):  \u003Cbr>Unsupervised Hebb(   |\n|  | (x-x\")+  \u003Cbr>Directional Derivatives:","cbCaiuc4QLzfVljz","https://ap.wps.com/l/cbCaiuc4QLzfVljz","pdf",8067403,"English","# Neural NetworksDeep Feed Forward (DFF) Input Cell\n## Neural Network Graphs\n## Linear Vector Spaces and Geometry\n## Perceptron Architecture and Learning Rules\n## Hebbian Learning and Associative Models\n## Optimization, Transformations, and Eigen-Related Concepts\n## General Minimization Algorithms and Steepest Descent\n## Backpropagation Variations (Batching)","[{\"question\":\"Which neural network cell and architecture types are covered?\",\"answer\":\"The content lists deep feed forward (DFF), noisy and probabilistic hidden cells, spiking hidden cells, and recurrent architectures. It also enumerates LSTM, GRU, and multiple memory cell variants, plus several network families such as GAN and residual networks.\"},{\"question\":\"What mathematical topics are included to support neural network reasoning?\",\"answer\":\"It covers linear vector spaces, linear independence, spanning, inner products, norms, angles, orthogonality, Gram-Schmidt orthogonalization, reciprocal basis vectors, transformations, and eigenvalues/eigenvectors with diagonalization.\"},{\"question\":\"How are learning and optimization ideas described?\",\"answer\":\"It includes perceptron learning and decision boundary concepts, Hebb’s postulate and related learning rules, and optimization conditions (stationary points, Hessian positivity). It also presents general minimization and steepest descent algorithms.\"},{\"question\":\"What heuristic variation of backpropagation is discussed?\",\"answer\":\"Batching is described, where parameter updates occur only after the full training set is presented. Gradients from each training example are averaged to improve the estimate of the gradient.\"}]","AI-Neural-Networks - Neural Networks - Backpropagation and Core Concepts | PDF",1791930791,7]