[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-151837-en":3,"doc-seo-151837-105":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":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},151837,962085570644,"Melati","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","A. Adversarial Dynamics Embedding - Derivation Details","Appendix sections provide rigorous derivations for Adversarial Dynamics Embedding (ADE) within an energy-based modeling framework. It explains the augmented model formulation, the equivalence of its maximum-likelihood estimate to the original model, and a primal-dual objective using a dual sampler q(x,v). To make the dual sampler flexible and tractable, it details Hamiltonian and Langevin dynamics embedding parameterizations, including concrete RNN-based initialization and iterative dynamics updates. Further sections derive special cases of energy-based processes for Gaussian processes, Student-t processes, and topic-model families such as LDA.","Appendix  \nA. Adversarial Dynamics Embedding  \nThe details of ADE derivations were originally provided in (Dai et al., 2019) . We include relevant details here for completeness, since we make explicit use of these techniques in our training methods.  \nGiven the EBM with pf (x) / exp(f (x)), ADE considers the augmented model  \npf (x, v ) / exp ✓f (x) − λ2v>v◆ , (A.1)  \nwhere v is the auxiliary momentum variable. It has been proved in Dai et al. (2019) that the MLE of (A.1) is the same as the original model, i.e.,  \nargmfax D 􀀤log Zp (x, v ) dv& = argmfax D [log pf (x)] . (A.2)  \nWe then apply the primal-dual view of the MLE to the augmented model, leading to  \nmafx qm(xi,nv)  [f (x)] − Eq (x,v ) 􀀤f (x) − λ2v>v − log q (x, v )& , (A.3)  \nwhere q (x, v ) is the dual sampler in the exponential family of distributions.  \nTo ensure that the dual sampler q (x, v ) is flexible and tractable, ADE utilizes the dynamics embedding parametrization. Specifically, we consider the Hamiltonian dynamics embedding as an example. In this setting, the sample first comes from an initial distribution 'x0 , v0 ( ⇠ q0! (x, v ), and then moves according to  \n(x0 , v0 ) = Lf,⌘ (x, v ) := 0 v 12 v0 +x fv  1(x) 1 (A.4)  \n@ v0 = v + ⌘2 ⌘rx 2f (x0 ) A ,  \nwhere ⌘ is defined as the leapfrog stepsize. After T iterations, we obtain  \n'xT, vT ( = Lf,⌘ ◦ Lf,⌘ ◦ ... ◦ Lf,⌘ 'x0 , v0 (, (A.5)  \nwhere Lf,⌘ can be one layer of the neural network. Together with the initial distribution q0!, we obtain the parametrization of q (x, v ) with learnable parameters (⌘ , !) . As justified in Dai et al. (2019), this parametrization is flexible and has a tractable density,  \nqT 'xT, vT ( = q0! 'x0 , v0 ( . (A.6)  \nBy plugging this parametrization into (A.3), we obtain the final objective,  \nmaxf2F  ` (f,!) := D [f] − E(x0 ,v0 )⇠q0!(x,v ) 􀀤f 'xT ( − λ2--vT--22& − H 'q0!( . (A.7)  \nDai et al. (2019) also introduces the Langevin and generalized Hamiltonian dynamics embedding for dual density parametrization in ADE. For more details, please refer to Dai et al. (2019) .  \nA.1. Dynamics Embedding Distribution Parametrization  \nWe now present the concrete implementation of the dual sampler q (x1:n, v |✓) used in our paper. Following the ADE technique introduced above, parametrization of q (x1:n, v |✓) is separated into parameterizing the initial distribution and the dynamics embedding. In our real-data experiment, we use the block RNN to parametrize q0! (x1:n, v |✓), as introduced in Appendix F.1 . Block RNN is simply a design choice; other alterantives such as normalizing flows can also parametrize q0! (x1:n, v |✓), as shown in our synthetic-data experiment.  \nThe output ofthe RNN is then treated as the starting sample to which T Hamiltonian/Langevin dynamics updates are applied, i.e.,  \n'xT1:n, vT ( = L ◦ L ◦ ... ◦ L 'x01:n, v0 (, (A.8)  \nwhere L can be either a Hamiltonian layer or a Langevin layer as specalized below,  \n0 vi12 = vi + ⌘2 rxf (i ; ✓) / 1  \nHamiltonian layer: (x01:n, v0 ) = Lf,⌘ (x1:n, v ) := B x0 = xi + ⌘v 2 / i = 1 , . . . , n C , (A.9)  \n@ v0i = vii~~1~~2 + ⌘2 rx f i(x0i; ✓) / A  \nwith v0 = {v0i} .  \nLangevin layer: (x01:n, v0 ) = L⇠f,⌘ (x1:n) :=  \nv0i = ⇠i + ⌘2 rxf (xi ; ✓) x0i = xi + v0i  \n/ i = 1,..., n! , (A.10)  \nwith ⇠ = {⇠i } , ⇠i ⇠ q! (⇠) and v0 = {v0i} .  \nFinally, following Theorem 4 in Dai et al. (2019), we obtain the parametrized dual sampler with tractable density as  \nHamiltonian embedding: qT 'xT1:n, vT |✓( = q0 'x01:n, v0 |✓(, (A.11)  \nT −1  \nLangevin embedding: qT ⇣xT1:n , 4vt  |✓⌘ = q0 'x01:n , ⇠0 |✓( Y q!i '⇠t ( . (A.12)  \nt=1  \nBy plugging this into the primal-dual view objective (21), we are able to learn the paramters in EBP s and dual samplers.  \nB. Derivation of Special Cases of EBPs  \nIn this section, we provide the details for instantiating (un)conditional EBPs to other specific models.  \nB.1. Latent Variable Representation of Gaussian Processes  \nWe consider the latent variable model specified in (8), i.e.,  \n✓ ⇠ N (0, Id) , (B.1)  \nfw (x, t; ✓) =","cbCaitIijJbEb7Et","https://ap.wps.com/l/cbCaitIijJbEb7Et","pdf",7256917,1,14,"English","en",105,"# Appendix A. Adversarial Dynamics Embedding\n## Dynamics Embedding Distribution Parametrization\n# Appendix B. Derivation of Special Cases of EBPs\n## Latent Variable Representation of Gaussian Processes\n## Latent Variable Representation of Student-t Processes\n## Topic Models","[{\"question\":\"What is the augmented model used in Adversarial Dynamics Embedding (ADE)?\",\"answer\":\"ADE augments the original energy-based model by introducing an auxiliary momentum variable v, yielding a joint distribution pf(x,v) normalized by an intractable partition term. The augmented formulation enables a primal-dual learning objective for the dual sampler.\"},{\"question\":\"Why does ADE use a dynamics embedding for the dual sampler q(x,v)?\",\"answer\":\"The dynamics embedding parameterization makes the dual sampler q(x,v) flexible while keeping its density tractable. Hamiltonian dynamics and Langevin dynamics are used to construct q through iterative updates starting from an initial distribution q0.\"},{\"question\":\"Which models are shown as special cases of unconditional EBPs in the appendix?\",\"answer\":\"The appendix derives special cases including Gaussian processes, Student-t processes, and topic-model families such as Bayesian sets, probabilistic latent semantic indexing, LDA, and replicated softmax. It presents LDA and replicated softmax as directed and undirected examples.\"}]","A. Adversarial Dynamics Embedding - Derivation Details | PDF",1787851138,35,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"a-adversarial-dynamics-embedding-derivation-details","",{"@graph":36,"@context":86},[37,54,69],{"@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/a-adversarial-dynamics-embedding-derivation-details/151837/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"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","2026-09-05","2026-08-27",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 is the augmented model used in Adversarial Dynamics Embedding (ADE)?","Question",{"text":76,"@type":77},"ADE augments the original energy-based model by introducing an auxiliary momentum variable v, yielding a joint distribution pf(x,v) normalized by an intractable partition term. The augmented formulation enables a primal-dual learning objective for the dual sampler.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why does ADE use a dynamics embedding for the dual sampler q(x,v)?",{"text":81,"@type":77},"The dynamics embedding parameterization makes the dual sampler q(x,v) flexible while keeping its density tractable. Hamiltonian dynamics and Langevin dynamics are used to construct q through iterative updates starting from an initial distribution q0.",{"name":83,"@type":74,"acceptedAnswer":84},"Which models are shown as special cases of unconditional EBPs in the appendix?",{"text":85,"@type":77},"The appendix derives special cases including Gaussian processes, Student-t processes, and topic-model families such as Bayesian sets, probabilistic latent semantic indexing, LDA, and replicated softmax. 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