text stringlengths 55 9.15k | synonym_substitution stringlengths 55 9.22k | butter_fingers stringlengths 55 9.15k | random_deletion stringlengths 47 4.48k | change_char_case stringlengths 55 9.15k | whitespace_perturbation stringlengths 60 7k | underscore_trick stringlengths 55 9.15k |
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$h_{5[1,2]}\left( x^{i}\right) $ stated by boundary conditions;
b\) or, inversely, to compute $h_{4}$ for a given $h_{5}\left( x^{i},v\right)
,h_{5}^{\ast }\neq 0,$$$\sqrt{|h_{4}|}=h_{[0]}\left( x^{i}\right) (\sqrt{|h_{5}\left( x^{i},v\right) |})^{\ast }, \label{p1}$$with $h_{[0]}\left( x^{i}\right) $ given b... | $ h_{5[1,2]}\left (x^{i}\right) $ stated by boundary conditions;
b\) or, inversely, to compute $ h_{4}$ for a given $ h_{5}\left (x^{i},v\right)
, h_{5}^{\ast } \neq 0,$$$\sqrt{|h_{4}|}=h_{[0]}\left (x^{i}\right) (\sqrt{|h_{5}\left (x^{i},v\right) |})^{\ast }, \label{p1}$$with $ h_{[0]}\left (x^{i}\right) ... | $h_{5[1,2]}\lfft( x^{i}\right) $ stated by buundary conditions;
u\) or, inbersely, go compute $h_{4}$ for a given $h_{5}\leht( x^{u},v\rigyt)
,h_{5}^{\ast }\neq 0,$$$\sqrt{|h_{4}|}=h_{[0]}\lewt( x^{i}\righn) (\sqrt{|h_{5}\ledt( x^{m},v\right) |})^{\ast }, \lausl{p1}$$with $h_{[0]}\left( w^{i}\rigkt) $ given b... | $h_{5[1,2]}\left( x^{i}\right) $ stated by boundary conditions; inversely, compute $h_{4}$ a given $h_{5}\left( (\sqrt{|h_{5}\left( |})^{\ast }, \label{p1}$$with x^{i}\right) $ given boundary conditions. - The exact solutions (\[ep3a\]) for $\beta \neq 0$ are defined from an algebraic equation, $w_{i}\beta +\alpha wher... | $h_{5[1,2]}\left( x^{i}\right) $ stated by boundAry conditiOns;
b\) oR, inVerSeLy, to CompUte $h_{4}$ for a given $h_{5}\LEft( x^{I},v\right)
,h_{5}^{\ast }\neq 0,$$$\sqrt{|h_{4}|}=h_{[0]}\leFt( x^{i}\rIgHT) (\sqrT{|H_{5}\lEft( x^{i},V\right) |})^{\aST }, \lABEl{p1}$$WiTh $H_{[0]}\leFt( X^{I}\rIght) $ gIveN b... | $h_{5[1,2]}\left( x^{i}\r ight) $ st atedbybou nd arycond itions;
b \ ) or , inversely, to comput e $h_ {4 } $ fo r a give n $h_{5 } \l e f t(x^ {i },v \r i gh t)
,h _{5}^{\ ast }\neq0,$ $$ \sqrt{|h_{4} | }= h_{[0]}\le ft( x^{i}\right ) ( \sqrt{ |h _{5 } \left ( x ^{i}, v\righ t ) |})^ {\ast }, \ l abel{p 1 ... | $h_{5[1,2]}\left(_x^{i}\right) $_stated by boundary conditions;
_ __b\) or,_inversely,_to compute $h_{4}$_for a given_$h_{5}\left( x^{i},v\right)
_,h_{5}^{\ast }\neq 0,$$$\sqrt{|h_{4}|}=h_{[0]}\left(_x^{i}\right)_(\sqrt{|h_{5}\left( x^{i},v\right) |})^{\ast }, \label{p1}$$with $h_{[0]}\left( x^{i}\right) $ given b... |
mathsf{C}}}$, $deg(u_i) \geq 2\delta_s+1$. Therefore we can use the simple coding scheme described in SectionΒ \[sec:simple\_coding\] on $(m_{{\mathsf{C}}},n_{{\mathsf{C}}},{\mathcal{X}}_{{\mathsf{C}}},\delta_s)$ BNSI problem to save one transmission compared to uncoded transmission. Therefore the length of this code to... | mathsf{C}}}$, $ deg(u_i) \geq 2\delta_s+1$. Therefore we can use the simple coding schema trace in Section Β \[sec: simple\_coding\ ] on $ (m_{{\mathsf{C}}},n_{{\mathsf{C}}},{\mathcal{X}}_{{\mathsf{C}}},\delta_s)$ BNSI problem to save one transmission compare to uncoded transmission. Therefore the distance of this code... | matjsf{C}}}$, $deg(u_i) \geq 2\delta_s+1$. Thtrefore we can usg rhe sikple ckding scfeme described in SectionΒ \[sec:dinple\_ciding\] on $(m_{{\mathsf{C}}},n_{{\mathrf{C}}},{\mathcap{X}}_{{\mathsf{X}}},\delua_s)$ BNSI problem vk save one trzksmisvmon compared to uncoded tsansmission. Thardflre the length of this code to... | mathsf{C}}}$, $deg(u_i) \geq 2\delta_s+1$. Therefore we can simple scheme described Section \[sec:simple\_coding\] on one compared to uncoded Therefore the length this code to transmit all the symbols indexed by ${\mathsf{C}}\subseteq [n]$ over ${\mathds{F}}_q$ is $N_{{\mathsf{C}}}=|{\mathsf{C}}|-1$. For some integer $... | mathsf{C}}}$, $deg(u_i) \geq 2\delta_s+1$. TherEfore we can Use thE siMplE cOdinG schEme described in sEctiOnΒ \[sec:simple\_coding\] on $(m_{{\maThsf{C}}},N_{{\mAThsf{c}}},{\MaThcal{x}}_{{\mathsf{c}}},\DeLTA_s)$ BnSi pRobLeM To Save oNe tRansmisSion comparEd tO uNcoded transmISsIon. TherefoRe tHe length of thIs cOde to... | mathsf{C}}}$, $deg(u_i) \g eq 2\delta _s+1$ . T her ef orewe c an use the sim p le c oding scheme described in S ec t ion\ [s ec:si mple\_c o di n g \]on $ (m_ {{ \ ma thsf{ C}} },n_{{\ mathsf{C}} },{ \m athcal{X}}_{ { \m athsf{C}}} ,\d elta_s)$ BNS I p roblem t o s a ve on e t ransm ission compar ed to unc od e d t... | mathsf{C}}}$, $deg(u_i)_\geq 2\delta_s+1$._Therefore we can use_the simple_coding_scheme described_in_SectionΒ \[sec:simple\_coding\] on $(m_{{\mathsf{C}}},n_{{\mathsf{C}}},{\mathcal{X}}_{{\mathsf{C}}},\delta_s)$_BNSI problem to_save one transmission compared_to uncoded transmission._Therefore_the length of this code to... |
, parallelizable, compact Riemannian $n$-manifold can be embedded isometrically as a special Lagrangian submanifold in a manifold with holonomy ${\mathrm{SU}}(n)$.
Notice that the assumption of real analyticity refers not only to the manifold, but to the structure as well.
$\alpha$-Einstein-Sasaki geometry and hypers... | , parallelizable, compact Riemannian $ n$-manifold can be embedded isometrically as a special Lagrangian submanifold in a manifold with holonomy $ { \mathrm{SU}}(n)$.
Notice that the premise of veridical analyticity refers not only to the manifold, but to the structure equally well.
$ \alpha$-Einstein - Sasaki ge... | , pagallelizable, compact Rieoannian $n$-manifold can ue embesded isooetrically as a special Lagrengiqn suvmanifold in a manifola with hoponomy ${\mqthrn{WU}}(n)$.
Notice vgat the assumlbion mh real analyticlty refers tot only to tha oauifold, but to the structure as well.
$\ajpha$-Einxtfin-Sasaki geomgtry sgd hgievs... | , parallelizable, compact Riemannian $n$-manifold can be as special Lagrangian in a manifold the of real analyticity not only to manifold, but to the structure as $\alpha$-Einstein-Sasaki geometry and hypersurfaces {#sec:go} =================================================== In this section we classify the constant to... | , parallelizable, compact RiemAnnian $n$-manIfold Can Be eMbEddeD isoMetrically as a sPEciaL Lagrangian submanifold In a maNiFOld wITh HolonOmy ${\mathRM{Su}}(N)$.
notIcE tHat ThE AsSumptIon Of real aNalyticity RefErS not only to thE MaNifold, but tO thE structure as WelL.
$\alpha$-eiNstEIn-SasAki GeomeTry and HYpers... | , parallelizable, compactRiemannian $n$- man ifo ld can beembedded isome t rica lly as a special Lagra ngian s u bman i fo ld in a mani f ol d wit hho lon om y $ {\mat hrm {SU}}(n )$.
Notic e t ha t the assump t io n of realana lyticity ref ers not o nl y t o theman ifold , butt o thestructure a s well.
$\a... | , parallelizable,_compact Riemannian_$n$-manifold can be embedded_isometrically as_a_special Lagrangian_submanifold_in a manifold_with holonomy ${\mathrm{SU}}(n)$.
Notice_that the assumption of_real analyticity refers_not_only to the manifold, but to the structure as well.
$\alpha$-Einstein-Sasaki geometry and hypers... |
}(X^-)\ar[d] \\
\mathrm{res}_{\pi}(\pi_n^{-}( \Sigma^{\infty}(X^n/ X^{n-1}))) \ar[r]^{d_n \ \ \ } & \mathrm{res}_{\pi}(\pi_{n-1}^{-}( \Sigma^{\infty}(X^{n-1}/ X^{n-2}))). }$$ Using the adjointness of $\mathrm{res}_{\pi}$ and $\mathrm{ind}_{\pi}$ and $\mathrm{ind}_{\pi}(\mathbb{Z}[-,G/K])=\mathbb{Z}^{G}[-,K]$, we conclu... | } (X^-)\ar[d ] \\
\mathrm{res}_{\pi}(\pi_n^{- } (\Sigma^{\infty}(X^n/ X^{n-1 }) )) \ar[r]^{d_n \ \ \ } & \mathrm{res}_{\pi}(\pi_{n-1}^{- } (\Sigma^{\infty}(X^{n-1}/ X^{n-2 }) )). } $ $ Using the adjointness of $ \mathrm{res}_{\pi}$ and $ \mathrm{ind}_{\pi}$ and $ \mathrm{ind}_{\pi}(\mathbb{Z}[-,G / K])=\mathbb{Z}^{G}... | }(X^-)\ar[f] \\
\mathrm{res}_{\pi}(\pi_n^{-}( \Sigma^{\innty}(X^n/ X^{n-1}))) \ar[r]^{d_n \ \ \ } & \matirm{res}_{\pj}(\pi_{n-1}^{-}( \Sigoa^{\infty}(X^{n-1}/ X^{n-2}))). }$$ Using the adjomntnwss od $\mathrm{res}_{\pi}$ and $\mathfm{ind}_{\pi}$ ajd $\mathrn{ind}_{\ki}(\mathbb{Z}[-,G/K])=\mathbb{V}^{F}[-,K]$, we conclu... | }(X^-)\ar[d] \\ \mathrm{res}_{\pi}(\pi_n^{-}( \Sigma^{\infty}(X^n/ X^{n-1}))) \ar[r]^{d_n \ } \mathrm{res}_{\pi}(\pi_{n-1}^{-}( \Sigma^{\infty}(X^{n-1}/ }$$ Using the and we conclude that chain complex obtained $ \Sigma^{\infty}X_+$ by applying the methods Subsection 4.1 to the stable cofiber sequences obtained by appl... | }(X^-)\ar[d] \\
\mathrm{res}_{\pi}(\pi_n^{-}( \Sigma^{\inFty}(X^n/ X^{n-1}))) \ar[r]^{D_n \ \ \ } & \matHrm{Res}_{\Pi}(\Pi_{n-1}^{-}( \SIgma^{\Infty}(X^{n-1}/ X^{n-2}))). }$$ Using THe adJointness of $\mathrm{res}_{\pi}$ And $\maThRM{ind}_{\PI}$ aNd $\matHrm{ind}_{\pI}(\MaTHBb{Z}[-,g/K])=\MaThbB{Z}^{g}[-,k]$, wE concLu... | }(X^-)\ar[d] \\
\mathrm{re s}_{\pi}(\ pi_n^ {-} ( \ Si gma^ {\in fty}(X^n/ X^{n - 1})) ) \ar[r]^{d_n \ \ \ }& \ma th r m{re s }_ {\pi} (\pi_{n - 1} ^ { -}( \ Si gma ^{ \ in fty}( X^{ n-1}/ X ^{n-2}))). }$ $Using the ad j oi ntness of$\m athrm{res}_{ \pi }$ and $ \ma t hrm{i nd} _{\pi }$ and $\math rm{ind}_{ \p i }(\... | }(X^-)\ar[d] \\
\mathrm{res}_{\pi}(\pi_n^{-}(_\Sigma^{\infty}(X^n/ X^{n-1})))_\ar[r]^{d_n \ \ \_} &_\mathrm{res}_{\pi}(\pi_{n-1}^{-}(_\Sigma^{\infty}(X^{n-1}/ X^{n-2})))._}$$_Using the adjointness_of $\mathrm{res}_{\pi}$ and_$\mathrm{ind}_{\pi}$ and $\mathrm{ind}_{\pi}(\mathbb{Z}[-,G/K])=\mathbb{Z}^{G}[-,K]$, we_conclu... |
Hodge, P.W. 1961,, 66, 83
Ibata, R., Gilmore, G., &Β Irwin, M. 1994,, 370, 194
Kodama, T. &Β Bower, R.G. 2001,, 321, 18
Karachentsev, [[*etΒ al.*]{}]{}Β 2003,, 398, 479
McLaughlin, D.E. 1999,, 117, 2398
Meurer, G.R., Mackie, G., &Β Carignan, C. 1994,, 107, 2021 (MMC94)
Meurer, G.R., Carignan, C., Beaulieu, S., &Β Fre... | Hodge, P.W. 1961, , 66, 83
Ibata, R., Gilmore, G., & Β Irwin, M. 1994, , 370, 194
Kodama, T. & Β Bower, R.G. 2001, , 321, 18
Karachentsev, [ [ * et Β al. * ] { } ] { } Β 2003, , 398, 479
McLaughlin, D.E. 1999, , 117, 2398
Meurer, G.R., Mackie, G., & Β Carignan, C. 1994, , 107, 2021 (MMC94)
Meurer, G.R... |
Hodhe, P.W. 1961,, 66, 83
Ibata, R., Gilmore, N., &Β Irwin, M. 1994,, 370, 194
Kodama, T. &Β Boxer, R.G. 2001,, 321, 18
Karachdntsev, [[*etΒ al.*]{}]{}Β 2003,, 398, 479
McLaughlin, D.E. 1999,, 117, 2398
Mwurer, G.R., Mackie, G., &Β Carignan, Z. 1994,, 107, 2021 (MMC94)
Mvurer, G.R., Xarijnan, C., Beaulieu, S., &Β Fre... | Hodge, P.W. 1961,, 66, 83 Ibata, R., & M. 1994,, 194 Kodama, T. 18 [[*et al.*]{}]{} 2003,, 479 McLaughlin, D.E. 117, 2398 Meurer, G.R., Mackie, G., Carignan, C. 1994,, 107, 2021 (MMC94) Meurer, G.R., Carignan, C., Beaulieu, S., & K.C. 1996,, 111, 1551 (MCBF96) Meylan, G., Sarajedeni, A., Jablonka, P., Djorgovski, S.G.,... |
Hodge, P.W. 1961,, 66, 83
Ibata, R., Gilmore, G., &Β IrwiN, M. 1994,, 370, 194
Kodama, T. &Β BOwer, R.g. 2001,, 321, 18
KaRacHeNtseV, [[*etΒ aL.*]{}]{}Β 2003,, 398, 479
McLaughlin, D.E. 1999,, 117, 2398
MEUrer, g.R., Mackie, G., &Β Carignan, C. 1994,, 107, 2021 (MMC94)
MEurer, g.R., cArigNAn, c., BeauLieu, S., &Β FrE... |
Hodge, P.W. 1961,, 66, 8 3
Ibata,R., G ilm ore ,G.,&Β Ir win, M. 1994,, 370, 194
Kodama, T. &Β Bow er, R .G . 200 1 ,, 321, 18
Ka r ac h e nts ev ,[[* et al .*]{} ]{} Β 2003,, 398, 479
Mc La ughlin, D.E. 19 99,, 117,239 8
Meurer, G .R. , Mack ie , G . , &Β C ari gnan, C. 19 9 4,, 10 7, 2021 ( MM C 94)
M e urer... |
Hodge, P.W._1961,, 66,_83
Ibata, R., Gilmore, G.,_&Β Irwin, M._1994,,_370, 194
Kodama,_T._&Β Bower, R.G. 2001,,_321, 18
Karachentsev, [[*etΒ al.*]{}]{}Β 2003,,_398, 479
McLaughlin, D.E. 1999,,_117, 2398
Meurer, G.R.,_Mackie,_G., &Β Carignan, C. 1994,, 107, 2021 (MMC94)
Meurer, G.R., Carignan, C., Beaulieu, S., &Β Fre... |
gamma=-\kappa.$$ Making use of the continuity method, one can easily prove that the solvability of this equation is equivalent to the one of $$\int_{X}\langle\kappa,\vartheta\rangle_{H}\frac{\omega^{n}}{n!}=0,$$ for any $\vartheta\in\Gamma(X,E)$ satisfying $D^{''}_{E}\vartheta=D_{H}^{'}\vartheta=0$. By the assumption $... | gamma=-\kappa.$$ Making use of the continuity method, one can easily rise that the solvability of this equality is equivalent to the one of $ $ \int_{X}\langle\kappa,\vartheta\rangle_{H}\frac{\omega^{n}}{n!}=0,$$ for any $ \vartheta\in\Gamma(X, E)$ satisfying $ D^{''}_{E}\vartheta = D_{H}^{'}\vartheta=0$. By the assump... | gamla=-\kappa.$$ Making use of tht continuity method, one ran easjly provd that the solvability of thms ewuatiin is equivalent to thd one of $$\pnt_{X}\langlw\kapka,\vartheta\rangle_{H}\hdac{\omega^{n}}{n!}=0,$$ fod any $\tartheta\in\Gamma(W,E)$ satisfyitg $D^{''}_{E}\vartheta=D_{V}^{'}\vxrcheta=0$. By the assumption $... | gamma=-\kappa.$$ Making use of the continuity method, easily that the of this equation of for any $\vartheta\in\Gamma(X,E)$ $D^{''}_{E}\vartheta=D_{H}^{'}\vartheta=0$. By the $\int_{X}\partial [\eta]\wedge\frac{\omega^{n-1}}{(n-1)!}=0$ for any Dolbeault class H^{0,1}(X)$, we know $$\int_{X}\langle\sqrt{-1}\Lambda_{\ome... | gamma=-\kappa.$$ Making use of the cOntinuity mEthod, One Can EaSily ProvE that the solvabILity Of this equation is equivaLent tO tHE one OF $$\iNt_{X}\laNgle\kapPA,\vARTheTa\RaNglE_{H}\FRaC{\omegA^{n}}{n!}=0,$$ For any $\vArtheta\in\GAmmA(X,e)$ satisfying $D^{''}_{e}\VaRtheta=D_{H}^{'}\vaRthEta=0$. By the assuMptIon $... | gamma=-\kappa.$$ Making us e of the c ontin uit y m et hod, one can easily pr o ve t hat the solvability of this e q uati o nis eq uivalen t t o the o ne of $ $ \i nt_{X }\l angle\k appa,\vart het a\ rangle_{H}\f r ac {\omega^{n }}{ n!}=0,$$ for an y $\va rt het a \in\G amm a(X,E )$ sat i sfying $D^{''}_ {E }... | gamma=-\kappa.$$ Making_use of_the continuity method, one_can easily_prove_that the_solvability_of this equation_is equivalent to_the one of $$\int_{X}\langle\kappa,\vartheta\rangle_{H}\frac{\omega^{n}}{n!}=0,$$_for any $\vartheta\in\Gamma(X,E)$_satisfying_$D^{''}_{E}\vartheta=D_{H}^{'}\vartheta=0$. By the assumption $... |
To get around these limitations, new, separate options must be defined, increasing the problemβs branching factor, and care must be taken to avoid loops (if so desired). An MMDP coarse action leaves the βdirectionβ of the action undecided: the same fine policy may be executed starting in several bottleneck states, and... | To get around these limitations, new, separate choice must be define, increasing the problem βs branching divisor, and care must be taken to debar loops (if so desire). An MMDP coarse action leaves the β commission β of the action undecided: the like fine policy may be executed starting in several bottleneck states, an... | To get around these limitauions, new, separatg iptionv must be defived, increasing the problemβs uranxhing factor, and care must ce taken no avoid ooopw (if so desmded). An MMDP ckwrse ection leaves tme βdirectiotβ of the actiot jnbecided: the same fine policy may be qxecutec dtarting in sederak botfleneck states, and... | To get around these limitations, new, separate be increasing the branching factor, and avoid (if so desired). MMDP coarse action the βdirectionβ of the action undecided: same fine policy may be executed starting in several bottleneck states, and may the agent in one of several directions until arriving at one of multip... | To get around these limitatioNs, new, separAte opTioNs mUsT be dEfinEd, increasing thE ProbLemβs branching factor, and Care mUsT Be taKEn To avoId loops (IF sO DEsiReD). AN MMdP COaRse acTioN leaves The βdirectiOnβ oF tHe action undeCIdEd: the same fIne Policy may be eXecUted stArTinG In sevEraL bottLeneck STates, aNd... | To get around these limit ations, ne w, se par ate o ptio ns m ust be defined , inc reasing the problemβsbranc hi n g fa c to r, an d carem us t beta ke n t oa vo id lo ops (if so desired). An M MDP coarse a c ti on leavesthe βdirectionβ of the a ct ion undec ide d: th e same fine p olicy may b e execu t ed... | To_get around_these limitations, new, separate_options must_be_defined, increasing_the_problemβs branching factor,_and care must_be taken to avoid_loops (if so_desired)._An MMDP coarse action leaves the βdirectionβ of the action undecided: the same fine_policy_may be_executed_starting_in several bottleneck states, and... |
$\Lambda^{2k-1}_n$ that is both cs and cs-$k$-neighborly. We then delete the cs-$(k-1)$-neighborly and $(k-1)$-stacked balls $\operatorname{\mathrm{lk}}\big(\{1,2\}, \pm B^{2k+1, k}_{n+2}\big)$ that are antipodal and share no common facets, and insert the cones over the boundary of these two balls. Thus, the resulting... | $ \Lambda^{2k-1}_n$ that is both cs and cs-$k$-neighborly. We then delete the cs-$(k-1)$-neighborly and $ (k-1)$-stacked balls $ \operatorname{\mathrm{lk}}\big(\{1,2\ }, \pm B^{2k+1, k}_{n+2}\big)$ that are antipodal and share no common aspect, and tuck the cones over the boundary of these two balls. therefore, the res... | $\Lalbda^{2k-1}_n$ that is both cs akd cs-$k$-neighborly. We thei delets the cs-$(y-1)$-neighborly and $(k-1)$-stacked balps $\operqtorname{\mathrm{lk}}\big(\{1,2\}, \pm B^{2k+1, k}_{n+2}\big)$ that arw anuipodal and share no common facsbs, anb mnsert the conex over the boundary of tvere two balls. Thus, the resulting... | $\Lambda^{2k-1}_n$ that is both cs and cs-$k$-neighborly. delete cs-$(k-1)$-neighborly and balls $\operatorname{\mathrm{lk}}\big(\{1,2\}, \pm and no common facets, insert the cones the boundary of these two balls. the resulting complex is also cs; furthermore, by Lemma \[lm: induction method\], it cs-$k$-neighborly. In... | $\Lambda^{2k-1}_n$ that is both cs and cs-$K$-neighborlY. We thEn dEleTe The cS-$(k-1)$-neIghborly and $(k-1)$-stACked Balls $\operatorname{\mathrM{lk}}\biG(\{1,2\}, \pM b^{2k+1, k}_{n+2}\BIg)$ That aRe antipODaL ANd sHaRe No cOmMOn FacetS, anD insert The cones ovEr tHe Boundary of thESe Two balls. ThUs, tHe resulting... | $\Lambda^{2k-1}_n$ that i s both csand c s-$ k$- ne ighb orly . We then dele t e th e cs-$(k-1)$-neighborl y and $ ( k-1) $ -s tacke d balls $\ o p era to rn ame {\ m at hrm{l k}} \big(\{ 1,2\}, \pm B^ {2 k+1, k}_{n+2 } \b ig)$ thatare antipodal a ndshareno co m mon f ace ts, a nd ins e rt the cones ov er the bo ... | $\Lambda^{2k-1}_n$_that is_both cs and cs-$k$-neighborly._We then_delete_the cs-$(k-1)$-neighborly_and_$(k-1)$-stacked balls $\operatorname{\mathrm{lk}}\big(\{1,2\},_\pm B^{2k+1, k}_{n+2}\big)$_that are antipodal and_share no common_facets,_and insert the cones over the boundary of these two balls. Thus, the resulting... |
1$, while modules are indicated by $\tau = 0$. Mixtures correspond to groups with $0 <\tau < 1$. For the rest of the paper, we refer to groups with $\tau\approx 1$ as community-like and groups with $\tau\approx 0$ as module-like.
Groups in networks are revealed by a sequential extraction procedure proposed inΒ [@ZLZ11... | 1 $, while modules are indicated by $ \tau = 0$. Mixtures correspond to group with $ 0 < \tau < 1$. For the remainder of the paper, we refer to groups with $ \tau\approx 1 $ as residential district - like and groups with $ \tau\approx 0 $ as module - like.
group in networks are revealed by a consecutive origin proce... | 1$, wjile modules are indicattd by $\tau = 0$. Mixtutew corrxspond fo groupr with $0 <\tau < 1$. For the rest oh thw paptg, we refer to groups dith $\tau\aiprox 1$ as comnynity-like ehd grouif wifm $\tau\cp'rox 0$ as module-kike.
Groups in networks ase rzvealed by a sequential extraction pwocedurr oroposed inΒ [@ZLZ11... | 1$, while modules are indicated by $\tau Mixtures to groups $0 <\tau < the we refer to with $\tau\approx 1$ community-like and groups with $\tau\approx 0$ module-like. Groups in networks are revealed by a sequential extraction procedure proposed in @SBB13; @Weiss]. One first finds the group $S$ and its linking pattern ... | 1$, while modules are indicated bY $\tau = 0$. MixturEs corResPonD tO groUps wIth $0 <\tau < 1$. For the reST of tHe paper, we refer to groups With $\tAu\APproX 1$ As CommuNity-likE AnD GRouPs WiTh $\tAu\APpRox 0$ as ModUle-like.
groups in neTwoRkS are revealed BY a Sequential ExtRaction proceDurE propoSeD inΒ [@zlZ11... | 1$, while modules are ind icated by$\tau =0$. M ixtu rescorrespond tog roup s with $0 <\tau < 1$.For t he rest of thepaper,w er e fer t ogro up s w ith $ \ta u\appro x 1$ as co mmu ni ty-like andg ro ups with $ \ta u\approx 0$asmodule -l ike .
Gro ups in n etwork s are r evealed b ya seque n tial ex t r ac t... | 1$,_while modules_are indicated by $\tau_= 0$._Mixtures_correspond to_groups_with $0 <\tau_< 1$. For_the rest of the_paper, we refer_to_groups with $\tau\approx 1$ as community-like and groups with $\tau\approx 0$ as module-like.
Groups in_networks_are revealed_by_a_sequential extraction procedure proposed inΒ [@ZLZ11... |
βdataβ at $x_0$ (i.e., derivatives $f^{(i)}(x_0)$).
Our paper proceeds as follows. In SectionΒ \[sec:terl\], we start with a general result of applying Taylor expansions to Q-functions. When we apply the same technique to the RL objective, we reuse the general result and derive a higher-order policy optimization objec... | β data β at $ x_0 $ (i.e., derivatives $ f^{(i)}(x_0)$).
Our paper proceeds as follows. In Section Β \[sec: terl\ ], we depart with a cosmopolitan result of applying Taylor expansion to Q - function. When we apply the same proficiency to the RL objective, we reuse the general consequence and derive a higher - club p... | βdahaβ at $x_0$ (i.e., derivatives $f^{(l)}(x_0)$).
Our paper procgees as hollows. In SectkonΒ \[sec:terl\], we start with a gxnerql rewult of applying Taylof expansilns to Q-duncuions. When we apply the same tedmniquz vo the RL objecjive, we reusa the general serupt and derive a higher-order policy jptimizstlon objec... | βdataβ at $x_0$ (i.e., derivatives $f^{(i)}(x_0)$). Our as In Section we start with Taylor to Q-functions. When apply the same to the RL objective, we reuse general result and derive a higher-order policy optimization objective. This leads to Section where we formally present the *Taylor Expansion Policy Optimization* ... | βdataβ at $x_0$ (i.e., derivatives $f^{(i)}(x_0)$).
OuR paper procEeds aS foLloWs. in SeCtioNΒ \[sec:terl\], we starT With A general result of applyiNg TayLoR ExpaNSiOns to q-functiONs. wHEn wE aPpLy tHe SAmE techNiqUe to the rL objectivE, we ReUse the generaL ReSult and derIve A higher-order PolIcy optImIzaTIon obJec... | βdataβ at $x_0$ (i.e., de rivatives$f^{( i)} (x_ 0) $).
Our paper proceed s asfollows. In SectionΒ \[ sec:t er l \],w estart with a ge n e ral r es ult o f a pplyi ngTaylorexpansions to Q -functions.W he n we apply th e same techn iqu e to t he RL objec tiv e, we reuse the ge neral res ul t and d e rive ah ... | βdataβ_at $x_0$_(i.e., derivatives $f^{(i)}(x_0)$).
Our paper_proceeds as_follows._In SectionΒ \[sec:terl\],_we_start with a_general result of_applying Taylor expansions to_Q-functions. When we_apply_the same technique to the RL objective, we reuse the general result and derive_a_higher-order policy_optimization_objec... |
+1} \binom{j}{r}\right\}
\frac{z^{j+1}}{k}
+ (\mbox{polynomial of $k$})\\[8pt]
& \qquad = \frac{z^{j+1}}{j+1}\frac{1}{k}
+ (\mbox{a polynomial of $k$}).
\end{aligned}$$]{} Hence, if we put [ $$\begin{aligned}
& B(k,z) := (I)_{k}+(II)_{k}+(III)_{k}+(IV)_{k}\\[4pt]
& \qquad - \fra... | +1 } \binom{j}{r}\right\ }
\frac{z^{j+1}}{k }
+ (\mbox{polynomial of $ k$})\\[8pt ]
& \qquad = \frac{z^{j+1}}{j+1}\frac{1}{k }
+ (\mbox{a polynomial of $ k$ }).
\end{aligned}$$ ] { } Hence, if we put [ $ $ \begin{aligned }
& B(k, z): = (I)_{k}+(II)_{k}+(III)_{k}+(IV)_{k}... | +1} \bijom{j}{r}\right\}
\frac{z^{j+1}}{y}
+ (\mbox{kooynomiel of $k$})\\[8lt]
& \qduad = \frac{z^{j+1}}{j+1}\frac{1}{k}
+ (\mvox{a kjlynomial of $k$}).
\ena{aligned}$$]{} Jence, if we kut [ $$\begin{aligned}
& B(k,z) := (I)_{k}+(II)_{i}+(LII)_{k}+(IR)_{k}\\[4't]
& \qquad - \fta... | +1} \binom{j}{r}\right\} \frac{z^{j+1}}{k} + (\mbox{polynomial of $k$})\\[8pt] = + (\mbox{a of $k$}). \end{aligned}$$]{} $$\begin{aligned} B(k,z) := (I)_{k}+(II)_{k}+(III)_{k}+(IV)_{k}\\[4pt] \qquad - \frac{1}{j+1} \sum_{r=0}^{j+1}\binom{j+1}{r} \frac{(-1)^{r}z^{r+1}}{r+1}B_{j+1-r}(z)\right\} \left(\frac{1}{k}-\frac{1}... | +1} \binom{j}{r}\right\}
\frac{z^{j+1}}{k}
+ (\mbox{poLynomial of $K$})\\[8pt]
& \qqUad = \FraC{z^{J+1}}{j+1}\frAc{1}{k}
+ (\mBox{a polynomial OF $k$}).
\enD{aligned}$$]{} Hence, if we put [ $$\beGin{alIgNEd}
& B(k,Z) := (i)_{k}+(iI)_{k}+(IIi)_{k}+(IV)_{k}\\[4pt]
& \QQuAD - \Fra... | +1} \binom{j}{r}\right\}
\fr ac{z^ {j+ 1}} {k }
+ (\mbox{po l ynom ial of $k$})\\[8pt]
& \ qq u ad = \f rac{z ^{j+1}} { j+ 1 } \fr ac {1 }{k } + (\mbox {a polynom ial o f $k$}).
\ e nd {aligned}$ $]{ } Hence, ifweput [$$ \be g in{al ign ed}
& B ( k,z) : = (I)_{k} +( I I)_{k} + (III)_{ k } ... | +1} \binom{j}{r}\right\}
_ _ _ \frac{z^{j+1}}{k}
__ __ _ + (\mbox{polynomial_of $k$})\\[8pt]
_ & \qquad_=_\frac{z^{j+1}}{j+1}\frac{1}{k}
+ (\mbox{a polynomial of_$k$}).
_ \end{aligned}$$]{}_Hence,_if_we put [ $$\begin{aligned}
_ & B(k,z) :=_(I)_{k}+(II)_{k}+(III)_{k}+(IV)_{k}\\[4pt]
_ & \qquad - \fra... |
stick anymore to the large dijet relative rapidity region in the BFKL Pomeron manifestations hunting, since, from the one hand, we include the region of the moderate rapidity intervals into our consideration and, from the other hand, the resummation effects are quite pronounced at the moderate rapidity region.
We pre... | stick anymore to the large dijet relative rapidity area in the BFKL Pomeron materialization hunting, since, from the one hand, we admit the region of the moderate celerity intervals into our consideration and, from the early hand, the resummation effects are quite pronounced at the moderate celerity area.
We present... | stlck anymore to the large dijet relative rapidivy regikn in thd BFKL Pomeron manifestationd yuntibg, since, from the one fand, we ijclude tye rtgion of the modeczte raplbity jktervclw into our conxideration and, from the mtfex hand, the resummation effects are qtite prpnlunced at the iodegaee rziibity region.
We pre... | stick anymore to the large dijet relative in BFKL Pomeron hunting, since, from the of the moderate intervals into our and, from the other hand, the effects are quite pronounced at the moderate rapidity region. We present also in 2,3 estimations for NLO BFKL effects using the results of Ref. [@Cor95], where NLO to Lipat... | stick anymore to the large dijEt relative RapidIty RegIoN in tHe BFkL Pomeron manifEStatIons hunting, since, from thE one hAnD, We inCLuDe the Region oF ThE MOdeRaTe RapIdITy InterValS into ouR consideraTioN aNd, from the othER hAnd, the resuMmaTion effects aRe qUite prOnOunCEd at tHe mOderaTe rapiDIty regIon.
We pre... | stick anymore to the larg e dijet re lativ e r api di ty r egio n in the BFKLP omer on manifestations hunt ing,si n ce,f ro m the one ha n d, w e i nc lu deth e r egion of the mo derate rap idi ty intervals i n to our consi der ation and, f rom the o th erh and,the resu mmatio n effec ts are qu it e prono u ... | stick_anymore to_the large dijet relative_rapidity region_in_the BFKL_Pomeron_manifestations hunting, since,_from the one_hand, we include the_region of the_moderate_rapidity intervals into our consideration and, from the other hand, the resummation effects are_quite_pronounced at_the_moderate_rapidity region.
We pre... |
hat{\Psi}_{\ell}a(x_{i}).$
Chernozhukov, Newey, and Robins (2018) introduce machine learning methods for choosing the functions to include in the vector $A(x)$. This method can be combined with machine learning methods for estimating $E[q_{i}|x_{i}]$ to construct a double machine learning estimator of average surplus,... | hat{\Psi}_{\ell}a(x_{i}).$
Chernozhukov, Newey, and Robins (2018) introduce machine learning methods for choose the function to include in the vector $ A(x)$. This method can be combined with car learning methods for estimating $ E[q_{i}|x_{i}]$ to manufacture a double machine learn estimator of average excess, as s... | hat{\Osi}_{\ell}a(x_{i}).$
Chernozhukov, Nedey, and Robins (2018) introdnce macgine leafning methods for choosing tie fynctiins to include in the xector $A(x)$. This merhod xan be comujned wibk macglne lzacning methods fpr estimathng $E[q_{i}|x_{i}]$ to cmnrtxuct a double machine learning estimwtor of agerage surplus,... | hat{\Psi}_{\ell}a(x_{i}).$ Chernozhukov, Newey, and Robins (2018) introduce methods choosing the to include in can combined with machine methods for estimating to construct a double machine learning of average surplus, as shown in Chernozhukov, Hausman, and Newey (2018). In parametric moment functions like those in equ... | hat{\Psi}_{\ell}a(x_{i}).$
Chernozhukov, NEwey, and RobIns (2018) inTroDucE mAchiNe leArning methods fOR choOsing the functions to incLude iN tHE vecTOr $a(x)$. ThiS method CAn BE ComBiNeD wiTh MAcHine lEarNing metHods for estImaTiNg $E[q_{i}|x_{i}]$ to conSTrUct a double MacHine learning EstImator Of AveRAge suRplUs,... | hat{\Psi}_{\ell}a(x_{i}).$
Chernozh ukov, Ne wey ,andRobi ns (2018) intr o duce machine learning meth ods f or choo s in g the functi o ns t o i nc lu dein th e vec tor $A(x)$ . This met hod c an be combin e dwith machi nelearning met hod s fores tim a ting$E[ q_{i} |x_{i} ] $ to c onstructad oublem achinel e ar... | hat{\Psi}_{\ell}a(x_{i}).$
Chernozhukov, Newey,_and Robins_(2018) introduce machine learning_methods for_choosing_the functions_to_include in the_vector $A(x)$. This_method can be combined_with machine learning_methods_for estimating $E[q_{i}|x_{i}]$ to construct a double machine learning estimator of average surplus,... |
'}$
$D^{(\mathrm{e}) \pm}_{m m'} := 0, D^{(\mathrm{h}) \pm}_{m m'} := 0$ Input $| \Psi^N_{\mathrm{gs}} \rangle$ to $\mathcal{C}_{m m'}$ and measure the ancillae $| q_1^{\mathrm{A}} \rangle \otimes | q_0^{\mathrm{A}} \rangle :=$ observed ancillary state $E :=$ QPE$(| \widetilde{\Psi} \rangle, \mathcal{H})$ Find $E$ amo... | ' } $
$ D^{(\mathrm{e }) \pm}_{m m' }: = 0, D^{(\mathrm{h }) \pm}_{m m' }: = 0 $ Input $ | \Psi^N_{\mathrm{gs } } \rangle$ to $ \mathcal{C}_{m m'}$ and measure the ancillae $ | q_1^{\mathrm{A } } \rangle \otimes | q_0^{\mathrm{A } } \rangle: = $ observed ancillary state $ east: = $ QPE$(| \widetilde{\Psi } \rangle, ... | '}$
$D^{(\mahhrm{e}) \pm}_{m m'} := 0, D^{(\mathrm{h}) \po}_{m m'} := 0$ Input $| \Psi^N_{\mathcm{gs}} \rahgle$ to $\oathcal{C}_{m m'}$ and measure the enciolae $| q_1^{\mathrm{A}} \rangle \otimer | q_0^{\mathrl{A}} \ranglw :=$ ouserved ancillarb state $C :=$ QPS$(| \widztmlde{\Psi} \rangle, \kathcal{H})$ Fhnd $E$ amo... | '}$ $D^{(\mathrm{e}) \pm}_{m m'} := 0, D^{(\mathrm{h}) := Input $| \rangle$ to $\mathcal{C}_{m $| \rangle \otimes | \rangle :=$ observed state $E :=$ QPE$(| \widetilde{\Psi} \rangle, Find $E$ among $\{ E_\lambda^{N - 1} \}_\lambda$ ${ \mathcode`+=\numexpr\mathcode`+ + "1000\relax \mathcode`*=\numexpr\mathcode`* "1000\r... | '}$
$D^{(\mathrm{e}) \pm}_{m m'} := 0, D^{(\mathrm{h}) \pm}_{m m'} := 0$ INput $| \Psi^N_{\maThrm{gS}} \raNglE$ tO $\matHcal{c}_{m m'}$ and measure tHE ancIllae $| q_1^{\mathrm{A}} \rangle \otiMes | q_0^{\mAtHRm{A}} \rANgLe :=$ obsErved anCIlLARy sTaTe $e :=$ QPe$(| \wIDeTilde{\psi} \Rangle, \mAthcal{H})$ FinD $E$ aMo... | '}$
$D^{(\mathrm{e}) \pm} _{m m'} := 0, D ^{( \ma th rm{h }) \ pm}_{m m'} :=0 $ In put $| \Psi^N_{\mathrm {gs}} \ r angl e $to $\ mathcal { C} _ { m m '} $and m e as ure t heancilla e $| q_1^{ \ma th rm{A}} \rang l e\otimes |q_0 ^{\mathrm{A} } \ rangle : =$o bserv edancil lary s t ate $E :=$ QPE$ (| \widet i lde{... | '}$
$D^{(\mathrm{e}) \pm}_{m_m'} :=_0, D^{(\mathrm{h}) \pm}_{m m'}_:= 0$_Input_$| \Psi^N_{\mathrm{gs}}_\rangle$_to $\mathcal{C}_{m m'}$_and measure the_ancillae $| q_1^{\mathrm{A}} \rangle_\otimes | q_0^{\mathrm{A}}_\rangle_:=$ observed ancillary state $E :=$ QPE$(| \widetilde{\Psi} \rangle, \mathcal{H})$ Find $E$ amo... |
on adjacency matrices in the natural way: If $A$ is the adjacency matrix of a graph $G$, then $\pi(A)$ is the adjacency matrix of $\pi(G)$ and $\pi(A)$ is obtained by simultaneously permuting with $\pi$ both rows and columns of $A$.
\[def:weak\_iso\] Let $(G,{\kappa_1})$ and $(H,{\kappa_2})$ be $k$-color graph colori... | on adjacency matrices in the natural way: If $ A$ is the adjacency matrix of a graph $ G$, then $ \pi(A)$ is the adjacency matrix of $ \pi(G)$ and $ \pi(A)$ is obtained by simultaneously permute with $ \pi$ both row and columns of $ A$.
\[def: weak\_iso\ ] Let $ (G,{\kappa_1})$ and $ (H,{\kappa_2})$ be $ k$-color gr... | on adjacency matrices in tme natural way: Iy $A$ is vhe adjzcency mxtrix of a graph $G$, then $\pi(A)$ ms tye adhacency matrix of $\pi(G)$ xnd $\pi(A)$ id obtainwd bb simultaneously permutiky witg $\pi$ yovh rows and colomns of $A$.
\[def:feak\_iso\] Let $(G,{\kdpoa_1})$ and $(H,{\kappa_2})$ be $k$-color graph colori... | on adjacency matrices in the natural way: is adjacency matrix a graph $G$, matrix $\pi(G)$ and $\pi(A)$ obtained by simultaneously with $\pi$ both rows and columns $A$. \[def:weak\_iso\] Let $(G,{\kappa_1})$ and $(H,{\kappa_2})$ be $k$-color graph colorings with $G=([n],E_1)$ and We say that $(G,{\kappa_1})$ and $(H,{\... | on adjacency matrices in the nAtural way: IF $A$ is tHe aDjaCeNcy mAtriX of a graph $G$, then $\PI(A)$ is The adjacency matrix of $\pi(g)$ and $\pI(A)$ IS obtAInEd by sImultanEOuSLY peRmUtIng WiTH $\pI$ both RowS and colUmns of $A$.
\[def:WeaK\_iSo\] Let $(G,{\kappa_1})$ aND $(H,{\Kappa_2})$ be $k$-coLor Graph colori... | on adjacency matrices inthe natura l way : I f $ A$ istheadjacency matr i x of a graph $G$, then $\p i(A)$ i s the ad jacen cy matr i xo f $\ pi (G )$an d $ \pi(A )$is obta ined by si mul ta neously perm u ti ng with $\ pi$ both rows a ndcolumn sof$ A$.
\[d ef:we ak\_is o \] Let $(G,{\ka pp a _1})$a nd $(H, { ... | on_adjacency matrices_in the natural way:_If $A$_is_the adjacency_matrix_of a graph_$G$, then $\pi(A)$_is the adjacency matrix_of $\pi(G)$ and_$\pi(A)$_is obtained by simultaneously permuting with $\pi$ both rows and columns of $A$.
\[def:weak\_iso\] Let_$(G,{\kappa_1})$_and $(H,{\kappa_2})$_be_$k$-color_graph colori... |
x}}_k\|
&\leq& \frac{2}{\eta}\sum_{k=1}^K \varphi(f({{\bf x}}_k))-\varphi(f({{\bf x}}_{k+1})) \nonumber\\
&=& \frac{2}{\eta}(\varphi(f({{\bf x}}_1))-\varphi(f({{\bf x}}_{K+1}))) \nonumber\\
&\leq& \frac{2}{\eta}\varphi(f({{\bf x}}_1)).
\end{aligned}$$ So, we get $$\begin{aligned}
\|{{\bf x}}_{K... | x}}_k\|
& \leq & \frac{2}{\eta}\sum_{k=1}^K \varphi(f({{\bf x}}_k))-\varphi(f({{\bf x}}_{k+1 }) ) \nonumber\\
& = & \frac{2}{\eta}(\varphi(f({{\bf x}}_1))-\varphi(f({{\bf x}}_{K+1 }) )) \nonumber\\
& \leq & \frac{2}{\eta}\varphi(f({{\bf x}}_1) ).
\end{aligned}$$ So, we get $ $ \begin{aligned... | x}}_k\|
&\leq& \frac{2}{\eta}\sum_{k=1}^K \vxrphi(f({{\bf x}}_k))-\varpku(f({{\bf x}}_{n+1})) \nonujber\\
&=& \frac{2}{\eta}(\varphi(f({{\bf x}}_1))-\varphi(f({{\uf x}}_{J+1}))) \nonymber\\
&\leq& \frac{2}{\eta}\xarphi(f({{\bf x}}_1)).
\end{aoigntd}$$ So, we get $$\begii{zligned}
\|{{\bf w}}_{K... | x}}_k\| &\leq& \frac{2}{\eta}\sum_{k=1}^K \varphi(f({{\bf x}}_k))-\varphi(f({{\bf x}}_{k+1})) \nonumber\\ x}}_1))-\varphi(f({{\bf \nonumber\\ &\leq& x}}_1)). \end{aligned}$$ So, x}}_*\| \sum_{k=1}^K\|{{\bf x}}_{k+1}-{{\bf x}}_k\|+\|{{\bf x}}_*\| \\ &\leq& x}}_1))+\|{{\bf x}}_1-{{\bf x}}_*\| \\ & < \rho. \end{aligned}$$... | x}}_k\|
&\leq& \frac{2}{\eta}\sum_{k=1}^K \varphi(f({{\bF x}}_k))-\varphi(f({{\Bf x}}_{k+1})) \nOnuMbeR\\
&=& \fRac{2}{\eTa}(\vaRphi(f({{\bf x}}_1))-\varphi(F({{\Bf x}}_{K+1}))) \Nonumber\\
&\leq& \frac{2}{\eta}\varpHi(f({{\bf X}}_1)).
\eND{aliGNeD}$$ So, we Get $$\begiN{AlIGNed}
\|{{\Bf X}}_{K... | x}}_k\|
&\leq& \fra c{2}{\eta} \sum_ {k= 1}^ K\var phi( f({{\bf x}}_k) ) -\va rphi(f({{\bf x}}_{k+1} )) \n on u mber \ \ &=& \f r ac { 2 }{\ et a} (\v ar p hi (f({{ \bf x}}_1) )-\varphi( f({ {\ bf x}}_{K+1} ) )) \nonumber \\ &\leq& \f rac{2} {\ eta } \varp hi( f({{\ bf x}} _ 1)).
\end{ali gn e d}$$ ... | x}}_k\|
_ _ &\leq&_\frac{2}{\eta}\sum_{k=1}^K \varphi(f({{\bf_x}}_k))-\varphi(f({{\bf_x}}_{k+1})) \nonumber\\
__ _ &=& \frac{2}{\eta}(\varphi(f({{\bf_x}}_1))-\varphi(f({{\bf x}}_{K+1}))) \nonumber\\
_ __&\leq& \frac{2}{\eta}\varphi(f({{\bf x}}_1)).
\end{aligned}$$ So, we get $$\begin{aligned}
\|{{\bf x}}_{K... |
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YAML Metadata Warning:empty or missing yaml metadata in repo card
Check out the documentation for more information.
Dataset Inference V2: Detect Datasets, Not Strings
This repository contains data from 22 different domains of the PILE, divided into train and val sets. The data is in the form of a JSON file, with each entry containing the raw text, as well as various kinds of perturbations applied to it. The dataset is used to facilitate privacy research in language models, where the perturbed data can be used as reference detect the presence of a particular dataset in the training data of a language model.
Quick Links
- Website: The landing page for Dataset Inference V2
- arXiv Paper: Detailed information about the Dataset Inference V2 project, including the dataset, results, and additional resources.
- GitHub Repository: Access the source code, evaluation scripts, and additional resources for Dataset Inference.
- Dataset on Hugging Face: Direct link to download the various versons of the PILE dataset.
- Summary on Twitter: A concise summary and key takeaways from the project.
Applicability π
The dataset is in text format and can be loaded using the Hugging Face datasets library. It can be used to evaluate any causal or masked language model for the presence of specific datasets in its training pool. The dataset is not intended for direct use in training models, but rather for evaluating the privacy of language models. Please keep the validation sets, and the perturbed train sets private, and do not use them for training models.
Loading the Dataset
To load the dataset, use the following code:
from datasets import load_dataset
dataset = load_dataset("pratyushmaini/llm_dataset_inference", subset = "wikipedia", split = "train")
Available perturbations:
We use the NL-Augmenter library to apply the following perturbations to the data:
synonym_substitution: Synonym substitution of words in the sentence.butter_fingers: Randomly changing characters from the sentence.random_deletion: Randomly deleting words from the sentence.change_char_case: Randomly changing the case of characters in the sentence.whitespace_perturbation: Randomly adding or removing whitespace from the sentence.underscore_trick: Adding underscores to the sentence.
Codebase
The code for training the models and the availability of all fine-tuned models can be found at our GitHub repository.
Citing Our Work
If you find our codebase and dataset beneficial, please cite our work:
@misc{di2024,
title={Dataset Inference V2: Detect Datasets, Not Strings},
author={},
year={2024},
archivePrefix={arXiv},
primaryClass={cs.LG}
}
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