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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
64c3db49db426357cb6a943951a5a9beaa72c806 | subsection | 41 | 74 | A Class of Examples | In
particular, \Gamma _{{U}} is a principal groupoid with unit space
\Gamma _{{U}}^{(0)} = \coprod U_{i}, and is equivalent to the space X (see
).Let \Sigma _{c} be the groupoid extension equal as a topological
space to G\times \Gamma _{{U}} endowed with the operations\bigl (g,(i,x,j)\bigr )\bigl (h,(j,x,k)\bigr ) = \... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.049666453152894974,
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0.020986899733543396,
0.02564217709004879,
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-0.032815881073474884,
-0.006235017441213131,... | |
52e36cd9e74a2f837903b14e902b4f9765f8e562 | subsection | 42 | 74 | A Class of Examples | We equip \Sigma _{c}
with the Haar system \lambda =\lbrace \lambda ^{(i,x)}\rbrace given by\lambda ^{(i,x)}(f)=\sum _{j}\int _{G} f\bigl (g,(i,x,j)\bigr )\,d\mu (g).Then using Lemma REF ,f*f^{\prime }\bigl (g,&(i,x,j)\bigr )\\
&=\sum _{k}\int _{G} f\bigl (h,(i,x,k)\bigr )
f^{\prime }\bigl ( g-h-c_{iki}(x)+c_{kij}(x),(k... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0... | |
2fff3a549be4c793a269cf40b750c23ad523fa78 | subsection | 43 | 74 | A Class of Examples | Then \operatorname{Ind}((i,x),\tau ) is
equivalent to the representation L on \ell ^{2}(I(x)) where
L(f) is given by multiplication by the matrix A=(a_{jk}) witha_{jk}=\tau {(c_{ijk}(x))} \int _{G}f(g,(j,x,k))\tau (g)\,d\mu (g).We have (\Sigma _{c})_{(i,x)}=\lbrace \,(g,(j,x,i):j\in I(x)\,\rbrace . Then
\operatorname{I... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.0012620905181393027,... | |
a4e6a0d9c16eb4baa0f61d0a445027bb6d3efd2a | subsection | 44 | 74 | A Class of Examples | ButU(&f_{1}*f_{2})(j) \\
&=
\int _{G} f_{1}*f_{2}(g-c_{iji}(x), (j,x,i))\tau (g)\,d\mu (g) \\
&= \sum _{k} \int _{G}\int _{G}
f_{1}(h,(j,x,k)) f_{2}(-h+g-c_{iji}(x) -c_{jki}(x),(k,x,i))
\tau (g)\,d\mu (h)\,d\mu (g) \\
{which, since c_{iji}(x)+c_{jki}(x) =
c_{ijk}(x)+c_{iki}(x), is}
&=\sum _{k} \int _{G}\int _{G}
f_{1}(... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.0402727909386158,
0.010464823804795742,
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0.03832016885280609,
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-0.01410310436040163,
-0.009442748501896858,
... | |
6d7f458bae20ad2776f32fa33120f7db001841fe | subsection | 45 | 74 | A Class of Examples | Given
a Čech 2-cocycle c \in Z^2(U, {G}), where
{U}= \lbrace U_i\rbrace _{i \in I} is a locally finite open cover of X, the
cover \lbrace \widehat{G}\times U_i\rbrace _{i \in I} of \widehat{G}\times X is locally
finite and supports a normalized 2-cocycle \nu ^{c} such that\nu ^{c}_{ijk}(\tau ,x)=\overline{\tau \bigl (c... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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-0.005425859242677689,
0... | |
a0391f7158e1f97e676dbe4e746dea5f64b1917b | subsection | 46 | 74 | The Raeburn–Taylor Algebra | For our computation of the Dixmier-Douady class of C^{*}(\Sigma _{c}),
we want a slight modification of the Raeburn–Taylor Algebra
based on their original construction in and
reproduced in *Proposition 5.40. Specifically,
given a normalized 2-cocycle
\nu =\lbrace \nu _{ijk}\rbrace \in Z^{2}({U},{S}) defined on a locall... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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-0.013281479477882385,
-0.0103... | |
052e33dcba32d48a1ce79ebb975486e869e865e9 | subsection | 47 | 74 | The Raeburn–Taylor Algebra | We define a multiplication on A_{1}(\nu ) by twisting the
usual matrix multiplication with \nu :(f_{ij} )(g_{lk}) = (h_{ij})whereh_{ik}(x)=\sum _{j} \overline{\nu _{ijk}(x)}
f_{ij}(x)g_{jk}(x).To see that the sum in (REF ) is meaningful, observe that it is
always a finite sum:h_{ik}(x) ={\left\lbrace \begin{array}{ll}
... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.05010664463043213,
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0.024366719648241997,
-0.0226578451693058,
... | |
5c7260bdec0374f11590ec8348ddab8fd61f51c6 | subsection | 48 | 74 | The Raeburn–Taylor Algebra | To see
that it is anti-multiplicative, we require
Lemma REF (e):(f*g)^{*}_{ij}(x)
&= \nu _{iji}(x) \overline{(f*g)_{ji}(x)}\\
&= \nu _{iji}(x) \sum _{k}\nu _{jki}(x) \overline{f_{jk}(x)}
\overline{g_{ki}(x)} \\
{which, using Lemma~\ref {lem-norm-coc}(e), is}
&= \sum _{k}\overline{\nu _{ikj}(x)} \nu _{iki}(x) \overline{... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.05391572415828705,
0.023113276809453964,
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0.05971311405301094,
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0.004840057343244553,
-0.027415549382567406,
-0.... | |
677773b40463170016a6bc22077b3b1f51347a14 | subsection | 49 | 74 | The Raeburn–Taylor Algebra | The groupoid is the
blow-up \Gamma _{{U}} associated to the cover {U} of X corresponding to
\nu
and the cocycle \varphi _{\nu } in Z^{2}(\Gamma _{{U}},\mathbf {T}) is given by\varphi _{\nu }\bigl ((i,x,j),(j,x,k)\bigr )=\overline{\nu _{ijk}(x)}.(The complex conjugate in (REF ) is missing from the formula
in .) Note th... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.06672380864620209,
0.02582266367971897,
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0.013651520945131779,
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-0.017016708850860596,
-0.0029035236220806837,
... | |
85be91b85a184eaf1b65d5378a53bc3b1bb70fc7 | subsection | 50 | 74 | The Dixmier–Douady Class of | Given a locally compact space X, a locally compact abelian group G
and a 2-cocycle c \in H^2(X, {G}), let \nu ^{c} be as in
(REF ). We can form the associated Raeburn–Taylor twisted
groupoid (\Gamma _{,\varphi _{\nu ^{c}}). We want to verify that C^{*}(\Gamma _{,\varphi _{\nu ^{c}}) is a
continuous-trace C^{*}-algebra ... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.029535064473748207,
0.017513561993837357,
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0.011159555986523628,
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-0.002894771983847022,
0.0022940782364457846,
0... | |
60b844e6bd2d4048d2307fc162bc3bf4e2d7ef16 | subsection | 51 | 74 | The Dixmier–Douady Class of | \end{}Since each summand is in C_{0}(\widehat{G}\times X), it
will suffice to see that h_{ik} is continuous on \widehat{G}\times X. Suppose that (\tau _{n},x_{n})\rightarrow (\tau _{0},x_{0}). It is
enough to show that
h_{ik}(\tau _{n},x_{n})\rightarrow h_{ik}(\tau _{0},x_{0}). For this, it
suffices to consider each su... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04593610391020775,
0.01098040770739317,
-0.03366612643003464,
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-0.012338651344180107,
0.... | |
06ef29c12d62dc65c3b40f219707b9ca4d9bc106 | subsection | 52 | 74 | The Dixmier–Douady Class of | Just as in the proof of \cite {rw:morita}*{Proposition~5.40},
A(c) is complete with respect to the norm
\begin{align}\Vert f\Vert =\sup _{(\tau ,x)}\Vert f\Vert _{(\tau ,x)},
\end{align}
and has spectrum identified (topologically) with GX.
Moreover, we have the following analogue of the Raeburn--Taylor
result.
\begin{}... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04716998711228371,
0.03151784837245941,
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-0.013302790932357311,
0.... | |
e35004bd4ab76373aef344c01fb1ed7a6299a95c | subsection | 53 | 74 | The Dixmier–Douady Class of | Then as above, we get
p(n,i)\in A(\nu ^{c}) such that\pi _{(i,(\tau ,x))}(p(n,i))is a rank-one projection for all (\tau ,x)\in F_{n,i}. Similarly,
let \phi _{(n,i)(m,j)} \in C_{c}^{+}(W_{(n,i)(m,j)}) be identically
one on F_{(n,i)(m,j)}. | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.024093693122267723,
0.024139469489455223,
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... | |
f02055d7f36cf7b1a5275dd6ffd829df6b42d7a7 | subsection | 54 | 74 | The Dixmier–Douady Class of | Then we get v((n,i),(m,j))\in A(\nu ^{c})
withv((n,i),(m,j))_{rs}(\tau ,x)=
{\left\lbrace \begin{array}{ll}
\phi _{(n,i)(m,j)}(\tau ,x)&\text{if $r=i$ and $s=j$, and} \\ 0
&\text{otherwise.}
\end{array}\right.}Then check that\pi _{(i,(\tau ,x))} \bigl (v((n,i),(m,j))v((n,i),(m,j))^{*}\bigr ) =
\pi _{(i,(\tau ,x))} \big... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03600570186972618,
0.04238298535346985,
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0.... | |
5196a6cce2559b9646521a99cdb631d80acb2fb0 | subsection | 55 | 74 | The Dixmier–Douady Class of | With these modifications in place, we can prove
Theorem REF .
[Proof of Theorem REF ]
By Lemma , it suffices to produce an isomorphism
\Phi :C^{*}(\Sigma _{c})\rightarrow A(\nu ^{c}) that intertwines each
\operatorname{Ind}((i,x),\tau ) with \pi _{(i, (\tau ,x))}. We use the Fourier
Transform. If f\in C_{c}(\Sigma _{c}... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.018356362357735634,
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-0.030212467536330223,
0.016479527577757835,... | |
10255b6b8aeca5bba30ab96eaf4439e1aed61750 | subsection | 56 | 74 | The Dixmier–Douady Class of | Since
finite sums of such functions are dense in the inductive limit
topology, we deduce that \Phi (f)\in A(\nu ^{c}) for all f.Note\Phi (f^{*}(i,(\tau ,x),j))
&= \int _{G}\tau (g)
\overline{f(-g-c_{iji}(x),(j,x,i))}
\,d\mu (g) \\
&= \overline{\int _{G} \tau (g) f(g-c_{iji}(x),(j,x,i))\,d\mu (g)} \\
&= \overline{\tau (... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04313027113676071,
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0.014498853124678135,
0.015529034659266472,
0.016... | |
59c7a125b6de02412ec29ec939ecd73bc553d131 | subsection | 57 | 74 | The Dixmier–Douady Class of | It follows from Lemma REF , that
\operatorname{Ind}((i,x),\tau )(f) is equivalent to multiplication by the
matrix\bigl [ \tau (c_{ijk})(x)\Phi (f) \bigr ].Since \tau (c_{ijk}(x))=\overline{\nu ^{c}_{ijk}(\tau ,x)}, we see that\operatorname{Ind}((i,x),\tau )(f)=\pi _{(i,(\tau ,x))}(\Phi (f)).This shows immediately that ... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.017895327880978584,
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0.011243697255849838,
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-0.003487910609692335,
0... | |
c22509bbb8cfa7db8b1b008a900bc466dc74ed07 | subsection | 58 | 74 | Groupoids Associated to Local Homeomorphisms | In this section we extend our results from
Section to a more general setting. Let
\psi :Y\rightarrow X be a local homeomorphism and form the principal groupoidR(\psi )=\lbrace \,(x,y)\in Y\times Y:\psi (x)=\psi (y)\,\rbrace .Let G be a locally compact abelian group. Given a
unit-space-preserving groupoid extension\beg... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.030501890927553177,
0.02758750319480896,
-0.009124627336859703,
0.02758750319480896,
-0.019714077934622765,
0.... | |
481d3c9df6db51408c5df539f85393412c4dde4c | subsection | 59 | 74 | Groupoids Associated to Local Homeomorphisms | Equivalently, all locally trivial
principle G-bundles over the double-overlaps W_{ij} are trivial.
A special case where these assumptions automatically hold is when G
is a Lie group and X admits good covers in the sense that every open
over of X admits a refinement in which all nontrivial overlaps are
contractible. Thi... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.06970366835594177,
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0.... | |
8c78ddc1f73a6a1c4942934f5697db872c273a45 | subsection | 60 | 74 | Groupoids Associated to Local Homeomorphisms | 12 or
*Equation 1.3, the
C_{c}(\Sigma ({U}))-valued inner product on C_{c}(\Sigma _{U})
is given by\langle f_{1}\mathrel {,}f_{2}
\rangle _{{\hspace{-1.66656pt}}\copy
\scriptscriptstyle {\Sigma (U)}}(\gamma )=\int _{\Sigma }
\overline{f_{1}(\sigma ^{-1})} f_{2}(\sigma ^{-1}\gamma )
\,d\lambda ^{r(\gamma )}(\sigma ).Th... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.028973683714866638,
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0.0... | |
2f654ff0b354c45085817e1c86fa382f93236958 | subsection | 61 | 74 | Groupoids Associated to Local Homeomorphisms | Since the representations are irreducible, W must be a
unitary and the representations must be equivalent.We also will need to examine the case of blowing up the unit space
\Sigma ^{(0)} with respect to a locally finite cover {U}=\lbrace U_{i}\rbrace . This
gives us the equivalent groupoid\Sigma ^{\prime }=\lbrace \,(i... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03961551561951637,
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0... | |
9d592e9e3e27c73990ccc15401500171dd85b29f | subsection | 62 | 74 | Groupoids Associated to Local Homeomorphisms | This is verified in the next lemma which is analogous
to Lemma REF .Lemma 6.2 With {U} and \Sigma ^{\prime } as above, we have
\mathop {Z\mathord {\mathop {\text{--}}}}\!\operatorname{Ind}\nolimits (\operatorname{Ind}^{\Sigma ^{\prime }}((i,y),\tau ) equivalent to
\operatorname{Ind}^{\Sigma }(y,\tau ) for all y\in U_{i... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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... | |
03d78b314024790fba61cf48a3776982c063a260 | subsection | 63 | 74 | Groupoids Associated to Local Homeomorphisms | Suppose that \psi :Y\rightarrow X is a local homeomorphism,
and let \Sigma be a groupoid extension as in
(REF ). Then C^{*}(\Sigma ) has continuous trace with
spectrum identified with \widehat{G}\times X via
(\tau ,x)\mapsto \operatorname{Ind}((i,y),\tau ) for any y\in \Phi ^{-1}(x).
Furthermore, there is a locally fin... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.06445252150297165,
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0.01142109651118517,
0.0174... | |
8c287d7dbb1aaa67e0ca9fdc7edb07ae01be74f5 | subsection | 64 | 74 | Groupoids Associated to Local Homeomorphisms | If we letR^{\prime } = \lbrace \,(i,(x,y),j):\text{$\psi (x)=\psi (y)$, $x\in V_{i}$ and
$y\in V_{j}$}\,\rbrace ,then we obtain a generalised twist\begin{}[column sep=3cm]
G \times \coprod V_{j} [r,"\iota ^{\prime }", hook]
[dr,shift left, bend right = 15] [dr,shift right,
bend right = 15]&\Sigma ^{\prime } [r,"\pi ^{\... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.036063097417354584,
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-0.030739061534404755,
0... | |
b9eb61407446ca5dcb22f0eddc77f5f8d1cb044f | subsection | 65 | 74 | Groupoids Associated to Local Homeomorphisms | There is a
groupoid homomorphism \tau :R^{\prime }\rightarrow \Gamma _{W} such that\tau (i,(x,y),j)=(i,\psi (x),j)and\tau ^{-1}(i,w,j)=(i,(x,y),j)\quad \text{if $x\in V_{i}$ and $y\in V_{j}$ satisfy $\psi (x) = w = \psi (y)$.}So, defining
\tilde{\varphi } := \varphi \circ (\tau ^{-1} \times \tau ^{-1}) \in Z^{2}(\Gamma... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04272150248289108,
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-0.018370246514678,
0.00073713663732633,
-0.00532... | |
60b2371920a72ae976ba681d8b676171aaaf9413 | subsection | 66 | 74 | Extensions and Cocycles | Let G be a locally compact Hausdorff abelian group and let \Gamma
be a locally compact Hausdorff groupoid with Haar system
\lbrace \lambda ^{u}\rbrace _{u\in \Gamma ^{(0)}}. Following
, we
define an extension (or twist) by G over \Gamma to be a
central groupoid extension
\Gamma ^{(0)}\times G\xrightarrow{}\Sigma \xri... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.044590141624212265,
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-0.030413467437028885,
... | |
5ec763da3f0a273ffac4587f486f114717661f7f | subsection | 67 | 74 | Extensions and Cocycles | Hence there is a subsequence
\lbrace \sigma _{n_{k}}\rbrace _{k\ge 1} such that
\sigma _{k_{n}}\rightarrow \sigma \in K and
g_{k_{n}}\sigma _{k_{n}}\rightarrow \sigma ^{\prime }\in K. It follows that
\pi (\sigma )=\pi (\sigma ^{\prime }). Hence there exists g\in G such
that \sigma ^{\prime }=g\sigma . Therefore\iota (r... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.011818... | |
0e81446dbf6892668c44fd585b6d10d1a55a6123 | subsection | 68 | 74 | Extensions and Cocycles | The collection
T_{\Gamma }(G) of proper isomorphism classes of twists by G forms
an abelian group under * with neutral element [\Gamma \times G].Remark 7.2 Let \Sigma be an extension by G over a principal groupoid
\Gamma ; that is, the map \gamma \mapsto (r(\gamma ),s(\gamma )) is
injective; equivalently, I(\Gamma ) = ... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.0038663120940327644,
... | |
f258d7e37e0e43d02dfee676cbaa796fcc65b3aa | subsection | 69 | 74 | Extensions and Cocycles | So
\psi (\gamma _{0},\gamma _{1}) := \varphi (\gamma _{0}, \gamma _{1}) -
(d^{1}f)(\gamma _{0}, \gamma _{1}) defines a continuous normalized
2-cocycle cohomologous to \varphi .Notation 7.4
Recall from *Lemma I.1.14 that given a
normalized 2-cocycle \varphi on \Gamma , there is an extension
\Sigma ({\Gamma }, {\varphi ... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.0... | |
f1cd131bb7bcb525da4887950094779905e28352 | subsection | 70 | 74 | Extensions and Cocycles | By
replacing \tau with the map
\gamma \mapsto \tau (r(\gamma ))^{-1}\tau (\gamma ), we can assume
without loss of generality that \tau (u)=u for all u\in \Sigma ^{(0)}. Then
\tau (\gamma _{1})\tau (\gamma _{2})\tau (\gamma _{1}
\gamma _{2})^{-1}\in \pi ^{-1}(\Gamma ^{(0)})=\iota (\Gamma ^{(0)}\times G) for all (\gamma ... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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3b26754bce7906f4930d0b4be369cc23ad7a8042 | subsection | 71 | 74 | Extensions and Cocycles | The map \psi : \Sigma ({\Gamma }, {\varphi }) \rightarrow \Sigma
defined by
\psi (g,\gamma ) := g\cdot \tau (\gamma ) =
\iota (r(\gamma ),g)\tau (\gamma ) is a homeomorphism and a groupoid
morphism.bottu:diff82book
author=Bott, Raoul,
author=Tu, Loring W.,
title=Differential forms in algebraic topology,
series=Graduat... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.024094007909297943,
0.022384995594620705,
-0.03723203018307686,
-0.00988785084336996,
0.022064555436372757,
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-0... | |
cda0c43206c323e2d770a74be98b62a576e0e45c | subsection | 72 | 74 | Extensions and Cocycles | Soc.,
volume=137,
number=4,
pages=13231332,
review=MR2465655,kum:lnim85inproceedings
author=Kumjian, Alexander,
title=Diagonals in algebras of continuous trace,
date=1985,
booktitle=Operator algebras and their connections with topology and ergodic
theory,
series=Lecture Notes in Mathematics,
volume=1132,
publisher=Spri... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04367689788341522,
-0.007527473848313093,
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0.008156988769769669,
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0.035497017204761505,
... | |
8c5a8e34024e1c9d5150500ecf31ef0fc2d844ad | subsection | 73 | 74 | Extensions and Cocycles | (3),
pages=109134,pal:aom61article
author=Palais, Richard S.,
title=On the existence of slices for actions of non-compact Lie
groups,
date=1961,
journal=Ann. of Math.,
volume=73,
pages=295323,ren:groupoidbook
author=Renault, Jean,
title=A groupoid approach to C^{*}-algebras,
series=Lecture Notes in Mathematics,
publish... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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-... | |
3fd04cfd0fc73048dd6267b513fe9acee04172b9 | abstract | 0 | 33 | Abstract | We derive an algorithm to compute satisfiability bounds for arbitrary
{\omega}-regular properties in an Interval-valued Markov Chain (IMC)
interpreted in the adversarial sense. IMCs generalize regular Markov Chains by
assigning a range of possible values to the transition probabilities between
states. In particular, we... | {
"cite_spans": []
} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
] | [
"cs.SY"
] | 2,018 | en | Computer Science | [
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0.... | |
a47ea5d4ad40e9761284145bafbc31bdd298100f | subsection | 1 | 33 | INTRODUCTION | Markov Chains have been extensively used as an intuitive yet powerful mathematical tool for modeling systems evolving through time in a stochastic fashion. They allow us to answer critical questions about the behavior of the underlying systems, often specified in terms of symbolic temporal logics, and derive appropriat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/tac.2014.2298143",
"end": 343,
"openalex_id": "https://openalex.org/W2059470663",
"raw": "X. Ding, S. L. Smith, C. Belta, and D. Rus, “Optimal control of Markov decision processes with linear temporal logic constraints,” IEEE Transa... | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
] | [
"cs.SY"
] | 2,018 | en | Computer Science | [
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4cb037c3ff621008eb91ee2f9027d4757bd9b28a | subsection | 2 | 33 | INTRODUCTION | Constructing the Cartesian product of a Markov Chain with a DRA enables to compute the probability that the stochastic evolution of the Markov Chain's state fulfills the property encoded in the DRA. In particular, it was shown that this probability is equal to that of reaching special sets of states called accepting Bo... | {
"cite_spans": []
} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
] | [
"cs.SY"
] | 2,018 | en | Computer Science | [
0.010886525735259056,
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0.... | |
75428939e91f7dde354409cb65137cff7a765d30 | subsection | 3 | 33 | PRELIMINARIES | An Interval-Valued Markov Chain (IMC) is a 5-tuple \mathcal {I} = (Q, {, \widehat{T}, \Pi , L) where:
\begin{}
\item Q is a finite set of states,
\item {: Q \times Q \rightarrow [0, 1] maps pairs of states to a lower transition bound so that {_{Q_{j} \rightarrow Q_{\ell }} := {(Q_{j}, Q_{\ell }) denotes the lower boun... | {
"cite_spans": []
} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
] | [
"cs.SY"
] | 2,018 | en | Computer Science | [
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0.0033593857660889626,
-0.014742380939424038,
... | |
5d16cfe234b118f35041cb84589e507dbf9ebb93 | subsection | 4 | 33 | PRELIMINARIES | \\
}A Markov Chain \mathcal {M} is said to be \textit {induced} by IMC \mathcal {I} if for all Q_j,Q_\ell \in Q,
\begin{align}
{(Q_j,Q_\ell ) \le T(Q_j, Q_\ell ) \le \widehat{T}(Q_j,Q_\ell ) \; \;.
}
\end{align}An IMC \mathcal {I}_{2} with transition functions {_{2} and \widehat{T}_{2} is said to be \textit {induced} b... | {
"cite_spans": []
} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
] | [
"cs.SY"
] | 2,018 | en | Computer Science | [
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a1c5f85ac5dedb01084ef2ae229a2cf1c07ab878 | subsection | 5 | 33 | PRELIMINARIES | An element (E_{i}, F_{i}) \in Acc, with E_{i}, F_{i} \subset S, is called a \textit {Rabin Pair}.
\end{}
}The probability of satisfying \omega -regular property \phi starting from initial state Q_i in IMC \mathcal {I} under adversary \mathcal {\nu } is denoted by \mathcal {P}_{\mathcal {I}[\mathcal {\nu }]}(Q_i \models... | {
"cite_spans": []
} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
] | [
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4526109e16d7316f93a973b4bfd9c063978e1384 | subsection | 6 | 33 | PRELIMINARIES | Then, we construct the product \mathcal {I} \otimes \mathcal {A}, which is itself an IMC.\\
}\begin{}
Let \mathcal {I} = (Q, {, \widehat{T}, \Pi , L) be an Interval-valued Markov Chain and \mathcal {A} = (S, 2^{\Pi }, \delta , s_0, Acc) be a Deterministic Rabin Automaton. | {
"cite_spans": []
} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
] | [
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60a2c45cefee8a51341ea29d06c9588237713b2c | subsection | 7 | 33 | PRELIMINARIES | The \textit {product} \mathcal {I} \otimes \mathcal {A} = (Q \times S, {, \widehat{T^{\prime }}, Acc^{\prime }, L^{\prime }) is an Interval-valued Markov Chain where:
\begin{}
\item Q \times S is a set of states,\\
\item {_{ \left<Q_{j},s\right> \rightarrow \left<Q_{\ell },s^{\prime }\right>} =
{\left\lbrace \begin{arr... | {
"cite_spans": []
} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
] | [
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9390a1867f62d4794f5336e96231fa40ab1ecb1b | subsection | 8 | 33 | PRELIMINARIES | \; F_{i} \in L^{\prime }(\left<Q_{j},s_{\ell } \right>) \; \Bigg ) \\
& \wedge \Bigg ( \; \forall \left<Q_{j},s_{\ell } \right> \in B \; . \; E_{i} \notin L^{\prime }(\left<Q_{j},s_{\ell } \right>) \; \Bigg ).
\end{align}
\end{}\mbox{}\\
}In words, every state in a BSCC B is reachable from any state in B, and every sta... | {
"cite_spans": []
} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
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740fe8ccceaf86b24b0d9c2c3b8f0328d5fa9b9c | subsection | 9 | 33 | PRELIMINARIES | Indeed, for any two states Q_j and Q_{\ell } in \mathcal {I} and some states s, s^{\prime }, s^{\prime \prime } and s^{\prime \prime \prime } in \mathcal {A}, we allow T_{ \left<Q_{j},s\right> \rightarrow \left<Q_{\ell },s^{\prime }\right>} and T_{ \left<Q_{j},s^{\prime \prime }\right> \rightarrow \left<Q_{\ell },s^{\p... | {
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Markov Chains | [
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8063a486984f783f469ab13a5dd294ec1ab7d235 | subsection | 10 | 33 | PRELIMINARIES | However, the set of accepting and non-accepting states may not be fixed in product IMCs and varies as a function of the assumed values for each transition. Specifically, U^A and U^N are determined by transitions that can be turned ``on" or ``off", i.e. those whose lower bound is zero and upper bound non-zero, as seen i... | {
"cite_spans": []
} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
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b27178b01bbcc9dad8c1bdd627d0035e5237248a | subsection | 11 | 33 | BOUNDING THE SATISFIABILITY OF | In , the authors discussed an algorithm for computing the probability bounds of reaching any fixed set of states in an IMC. We remarked in the previous section that, in general, the set of accepting and non-accepting states in a product IMC may depend on the assumed transition values. This is however not always the cas... | {
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Markov Chains | [
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f9812ff6e08ee34d866432ba228a26d2ee2e4295 | subsection | 12 | 33 | BOUNDING THE SATISFIABILITY OF | Specifically, we prove that all product IMCs induce a worst-case NASIMCs containing the largest set of non-accepting states and in which the probability of reaching an accepting BSCC is minimized from any initial state. Then, we show that the converse best-case ASIMCs are always induced by product IMCs with one Rabin p... | {
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} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
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d2630e1b159e741e271d947a16cb5175a1be3cbd | subsection | 13 | 33 | Lower Bound Computation | A key observation is that any infinite sequence of states in a Markov Chain eventually reaches a BSCC.Lemma 1 For any infinite sequence of states \pi = q_{0}q_{1}q_{2}\ldots in a Markov Chain, there exists an index i \ge 0 such that q_i belongs to a BSCC.The following corollary relies on the fact that a BSCC is either ... | {
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bb70b6767563824e9cd94f3ead95528cb0bdf238 | subsection | 14 | 33 | Lower Bound Computation | Let {, \widehat{T}, {_{1}, \widehat{T}_{1}, {_{2}, \widehat{T}_{2} and {_{3}, \widehat{T}_{3} be the transition bounds functions in the product IMCs \mathcal {I} \otimes \mathcal {A}, (\mathcal {I} \otimes \mathcal {A})_{1}, (\mathcal {I} \otimes \mathcal {A})_{2} and (\mathcal {I} \otimes \mathcal {A})_{3} respectivel... | {
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} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
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765885c4f09e938d3190e6fc23ffafb4ba1ee6e1 | subsection | 15 | 33 | Lower Bound Computation | Set {_{3}(Q_{i}, Q_{j}) = {_{2}(Q_{i}, Q_{j}) and \widehat{T}_{3}(Q_{i}, Q_{j}) = \widehat{T}_{2}(Q_{i}, Q_{j}) for all Q_{i} \in U^N_2 \setminus D and for all Q_{j} \in Q \times S. Set {_{3}(Q_{i}, Q_{j}) > 0 for all Q_{i}, Q_{j} \in D such that {_{1}(Q_{i}, Q_{j}) > 0 and {_{2}(Q_{i}, Q_{j}) > 0. Set \widehat{T}_{3}(... | {
"cite_spans": []
} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
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e22f68890a3e14675629f7fd9a28dfef0334c318 | subsection | 16 | 33 | Lower Bound Computation | There exists a non-empty set of NASIMCs [\mathcal {I} \otimes \mathcal {A}]^{N}_{\ell } \subseteq [\mathcal {I} \otimes \mathcal {A}]^{N} such that, for all (\mathcal {I} \otimes \mathcal {A})_{\ell } \in [\mathcal {I} \otimes \mathcal {A}]^{N}_{\ell } with transition functions {_{\ell } and \widehat{T}_{\ell }, {_{\el... | {
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Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
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83f9ed9359b515a1ba3086b6bd0d8af4c988a43a | subsection | 17 | 33 | Lower Bound Computation | For any NASIMC (\mathcal {I} \otimes \mathcal {A})_2 with non-accepting states U^N_2 induced by \mathcal {I} \otimes \mathcal {A}, there exists a NASIMC (\mathcal {I} \otimes \mathcal {A})_1 with non-accepting states U^N_1 induced by \mathcal {I} \otimes \mathcal {A} such that, for any initial state \left<Q_i, s_0\righ... | {
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} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
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9f1821cc684e8dc03673e3950f4d9c0bcc5b1f5b | subsection | 18 | 33 | Lower Bound Computation | Since U^N_2 \subseteq U^N_1, if \left<Q_i,s_0\right> \notin U^N_1, it must be true that \mathcal {\widehat{P}}_{(\mathcal {I} \otimes \mathcal {A})_1}(\left<Q_i,s_0\right> \models \Diamond U^N_1) \ge \mathcal {\widehat{P}}_{(\mathcal {I} \otimes \mathcal {A})_2}(\left<Q_i,s_0\right> \models \Diamond U^N_2)
\end{}
}\mbo... | {
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} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
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46ca0b9fe2bbd37d2b200d135503b38bb83e9dd7 | subsection | 19 | 33 | Lower Bound Computation | If \left<Q_i, s_0\right> \notin U^{N}_{\ell }, the inequality follows from the fact that, by the definition of [\mathcal {I} \otimes \mathcal {A}]^{N}_{l}, for any Markov Chain induced by (\mathcal {I} \otimes \mathcal {A})^{^{\prime }}, there exists a Markov Chain induced by (\mathcal {I} \otimes \mathcal {A})_{\ell }... | {
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Markov Chains | [
"Maxence Dutreix",
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c50dec18fe0eec399762db50e78f607e4ffffd2a | subsection | 20 | 33 | Lower Bound Computation | By Lemma 3 and Lemma 4, the maximum value for \mathcal {P}_{(\mathcal {M} \otimes \mathcal {A})_{\nu }}(\left<Q_i, s_0\right> \models \Diamond U^N) is reached for some (\mathcal {M} \otimes \mathcal {A})_{\nu } induced by the NASIMC (\mathcal {I} \otimes \mathcal {A})_{\ell }. | {
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} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
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844a4901b88711aa198e65462560e65f93195d75 | subsection | 21 | 33 | Upper Bound Computation | One could think of a similar approach here and compute a satisfiability upper bound by maximizing the probability of transition to an accepting BSCC. However, due to the acceptance condition of Rabin Automata, the analogous version of Lemma 2 for accepting states does not always hold true, as shown in Fig. 2. We conseq... | {
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} | 1809.06352 | Satisfiability Bounds for {\omega}-regular Properties in Interval-valued
Markov Chains | [
"Maxence Dutreix",
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5ac6be09c250686db8b05b40ae10c6822bedbf4c | subsection | 22 | 33 | Product IMC with one Rabin pair | The theorem and lemma in this section are similar to the ones in section IV A, and are provided without proof due to space constraints.We denote by U^A_{u} the largest set of accepting states induced by \mathcal {I} \otimes \mathcal {A}. We define the set of best case ASIMCs [\mathcal {I} \otimes \mathcal {A}]^{A}_{u} ... | {
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13c0ee49c7520374574e29f1ac247f0df51efcdd | subsection | 23 | 33 | Product IMC with more than one Rabin pairs | We previously observed that product IMCs with more than one Rabin pair don't necessarily induce a unique largest set of accepting states. Instead, we exploit the fact that \omega -regular expressions, and consequently DRAs, are closed under complementation . The following theorem states that any \omega -regular propert... | {
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Markov Chains | [
"Maxence Dutreix",
"Samuel Coogan"
] | [
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da8d4f3687615e31fb9c6c24d7a10c4be96a0696 | subsection | 24 | 33 | Product IMC with more than one Rabin pairs | By theorem 1, we have that\mathcal {P}_{\mathcal {I}[\mathcal {\nu }_{\ell }]}(Q_i \models \lnot \phi ) = 1 - \mathcal {\widehat{P}}_{(\mathcal {I} \otimes \mathcal {\overline{A}})_{\ell }}(\left<Q_i, s_0\right> \models \Diamond U^N_{\ell }) \;\; .Therefore, the first inequality reduces to\mathcal {\widehat{P}}_{\mathc... | {
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Markov Chains | [
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36fe26f519288b8862bea62f5a21ec5a410876df | subsection | 25 | 33 | Search Algorithm | [t!]InputInput
OutputOutput
Lower and upper bound probabilities of satisfying \phi in \mathcal {I}, {_{\mathcal {I}}(Q_{i} \models \phi ) and \widehat{\mathcal {P}}_{\mathcal {I}}(Q_{i} \models \phi ), for all initial states Q_{i}.}
\mbox{}\\Construct a DRA A corresponding to ; \mbox{}\\ Generate the product I A; \mbo... | {
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71ff533fccb0a62e39c4cd4b08f51f29f2962d20 | subsection | 26 | 33 | Search Algorithm | The resulting product IMC induces a directed graph with a vertex for each state and an edge for all non-zero transitions. In this graph, all SCCs are enumerated. Then, for each SCC and if necessary, we remove the states that prevent it from being a BSCC and accepting (or non-accepting) if allowed by \mathcal {I} \otime... | {
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c4f3e6254485dff4dbdb9afa4f6551e81bc84b55 | subsection | 27 | 33 | Search Algorithm | \item \underline{Search for U^{N}_{l}}: For all such F_{i}^{\prime }s, check whether some state in C^{j} maps to the corresponding non-accepting set E_{i}. If this is the case for all such F_{i}^{\prime }s, C^{j} is a non-accepting BSCC. Otherwise, the unmatched F_{i} states cannot belong to a non-accepting BSCC. Treat... | {
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069d32b40d9de292f2f4a6918b1bd21852565e55 | subsection | 28 | 33 | Search Algorithm | Matrices \widehat{T} and { respectively contain the upper and lower probabilities of transition from state to state:
}\begin{}
\begin{}{c || c c c c c c}
{ & q_0 & q_1 & q_2 & q_3 & q_4 & q_5\\
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6b2f14f7ba4ee820efaa5ec20b127da294054a63 | subsection | 29 | 33 | Search Algorithm | We demonstrated its application through a case study. In future works, we will seek to exploit the mechanisms unveiled in this paper and apply them to Bounded-parameter Markov Decision Processes, the controllable counterparts of IMCs, e.g. to minimize or maximize the probability of occurrence of some behavior in a syst... | {
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517bd1dfbb301b56f0e9839b4b2247df1adcb3a7 | subsection | 30 | 33 | CASE STUDY | We now apply the concepts developed in the previous sections to a case study. Our system of interest is an agent moving stochastically on a two-dimensional grid shown in Fig. 3. The grid is divided into 6 locations, representing 6 different states the agent can visit. We assume the system to be evolving in a discrete-t... | {
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8827ff260d1cd11d445c74b8f3022453b407bb64 | subsection | 31 | 33 | CASE STUDY | We aim to bound the probability of satisfying \omega -regular properties \phi _{1} and \phi _{2}, represented by automata \mathcal {A}_{1} and \mathcal {A}_{2} in Fig. 4, from every initial state q. In natural language, these properties respectively translate to1) “The agent visits a green state infinitely many times w... | {
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64af6aaeedb0c841f97ac210b6b207baf544dfe9 | subsection | 32 | 33 | CONCLUSIONS | We derived an efficient automaton-based technique for bounding the probability of satisfying any \omega -regular property in an IMC interpreted as an IMDP. We demonstrated its application through a case study. In future works, we will seek to exploit the mechanisms unveiled in this paper and apply them to Bounded-param... | {
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7729bb486ed0f3e43385ecb5178683779b079e0a | abstract | 0 | 52 | Abstract | GANs excel at learning high dimensional distributions, but they can update
generator parameters in directions that do not correspond to the steepest
descent direction of the objective. Prominent examples of problematic update
directions include those used in both Goodfellow's original GAN and the
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ced99e7c8cdc7b1fc6970a4b5e00a889bf752535 | subsection | 1 | 52 | Introduction | 5mm
Generative adversarial networks (GANs) excel at learning generative models of complex
distributions, such as images , ,
textures , , , and
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042a665736d9a0e6ca8531e285856251442ff2ce | subsection | 2 | 52 | Introduction | As we see later, popular methods such as WGAN-GP are affected by this issue.Therefore we set out to answer a simple but fundamental question:
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4dd05efd5a4bbd5ad5f47e7d1dcaeca37b58f062 | subsection | 3 | 52 | Introduction | This divergence enforces not just correct values for the critic, but also
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6ce4f3c9042ab244dec645771275b24c592631b0 | subsection | 4 | 52 | Notation, Definitions and Assumptions | In an adversarial divergence is defined:Definition 1 (Adversarial Divergence)
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8c7b5090a5893eeaa7124238ae808e4d72113a22 | subsection | 5 | 52 | Notation, Definitions and Assumptions | Then define\tau :\mathcal {P}(X)\times \mathcal {P}(X)\times \mathcal {F}&\rightarrow \mathbb {R}\cup \lbrace +\infty \rbrace \\
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9ac3e2ec8eab227d593730d1b362afe03407b556 | subsection | 6 | 52 | Notation, Definitions and Assumptions | \tau is called a strict adversarial divergence if for any \mathbb {P},\mathbb {P}^*\in \mathcal {P}(X),\tau (\mathbb {P}^*\Vert \mathbb {P})=\inf _{\mathbb {P}^{\prime }\in \mathcal {P}(X)}\tau (\mathbb {P}^*\Vert \mathbb {P}^{\prime })\Rightarrow \mathbb {P}^*=\mathbb {P}In order to analyze GANs that minimize a critic... | {
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b31ed25df7e7c577c5168f0128b6ba811a4dbcf0 | subsection | 8 | 52 | Requirements Derived From Related Work | With the concept of an Adversarial Divergence now formally defined, we can investigate
existing GAN methods from an Adversarial Divergence minimization standpoint.
During the last few years, weaknesses in existing GAN frameworks have been highlighted and new frameworks have been
proposed to mitigate or eliminate these ... | {
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610f592ebc99c0f7575676b65bb3deec12852818 | subsection | 9 | 52 | Requirements Derived From Related Work | The Wasserstein distance \tau _W is the weakest divergence in the class of strict adversarial divergences ,
leading to the following requirement:Requirement 1 (Equivalence to \tau _W)
An adversarial divergence \tau is said to fulfill Requirement REF if \tau is a strict adversarial divergence
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c1371f26a481f0d9c8f2535ec4ebf168fd8f3821 | subsection | 10 | 52 | Requirements Derived From Related Work | Although this method has impressive experimental results, it is not yet ideal.
showed that an optimal critic for \tau _I has undefined gradients
on the support of the generated distribution \mathbb {Q}_\theta . Thus,
the update direction \nabla _\theta \mathbb {E}_{\mathbb {Q}_\theta }[f^*] is undefined;
even if a dir... | {
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8fd35c074049fe4c18bb28a7b75eddd15ef7bf4a | subsection | 11 | 52 | Correct Update Rule Requirement | In the previous section, we stated a bare minimum requirement for an update rule (namely that it is well defined). In this section, we'll go further
and explore criteria for a “good” update rule. For example in Lemma REF in Section of Appendix,
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a4b89a9f9dc0197d2371210757a270eb228dc968 | subsection | 12 | 52 | Correct Update Rule Requirement | REF
together with Theorem 1 from is that for every f^*\in \operatorname{OC}_{\tau }(\mathbb {P},\mathbb {Q}_{\theta _0})\nabla _\theta \tau (\mathbb {P}\Vert \mathbb {Q}_\theta )|_{\theta _0}
&=\nabla _\theta \tau (\mathbb {P}\Vert \mathbb {Q};f^*)\\
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b4efb823a6c4e6cc69164329b740bd774b71cd0e | subsection | 13 | 52 | Correct Update Rule Requirement | GAN learning happens in mini-batches, therefore \nabla _\theta \mathbb {E}_{\mathbb {P}\otimes \mathbb {Q}_\theta }[r_{f^*}] isn't calculated directly, but estimated
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1d95167142813a7cdaae05954ae7f6cea4027e59 | subsection | 14 | 52 | Correct Update Rule Requirement | In the same way,
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6a27b6c4f2b418e371b6f7733c1b3e4d7a9b1dca | subsection | 15 | 52 | Correct Update Rule Requirement | REF we see that\nabla _\theta \tau (\mathbb {P}\Vert \mathbb {Q}_\theta )|_{\theta _0}&=-\nabla _\theta (\mathbb {E}_{\mathbb {Q}_\theta }[m_2(f^*)]+\mathbb {E}_{\mathbb {P}\otimes \mathbb {Q}_\theta }[r_{f^*}])|_{\theta _0}\\
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80070a5bc16b851e1e4d3f22d7a138f084089181 | subsection | 16 | 52 | Penalized Wasserstein Divergence | We now attempt to find an adversarial
divergence that fulfills all four requirements. We start by formulating an adversarial divergence \tau _P and a corresponding update rule
than can be shown to comply with Requirements REF and REF .
Subsequently in Section , \tau _P will be refined to make its update rule
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d1e997d7348c5d087faccfbe2abe3c2e07834c6d | subsection | 17 | 52 | Penalized Wasserstein Divergence | Set\tau _P(\mathbb {P}\Vert \mathbb {Q};f):=\,&\mathbb {E}_{x\sim \mathbb {P}}[f(x)]-\mathbb {E}_{x^{\prime }\sim \mathbb {Q}}[f(x^{\prime })] \\
-\lambda \,&\mathbb {E}_{x\sim \mathbb {P},x^{\prime }\sim \mathbb {Q}}\left[\frac{(f(x)-f(x^{\prime }))^2}{\Vert x-x^{\prime }\Vert }\right].Define the penalized Wasserstein... | {
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d491d12b802de6095752c5fdf5cf3d02ea35c7ea | subsection | 18 | 52 | Penalized Wasserstein Divergence | See Figure REF for a simple example.
[Figure: Comparison of \tau _P update rule given different optimal critics.Consider the simple example of divergence \tau _P from Definition between Dirac measureswith update rule \frac{1}{2} \frac{d}{d\theta }\mathbb {E}_{\delta _\theta }[f](the update rule is from Lemma in Appendi... | {
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468ef670f84edc2812dd5133950929d42253832f | subsection | 19 | 52 | Penalized Wasserstein Divergence | In response, we propose the First Order Penalized Wasserstein Divergence. | {
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912b01085ec23d1e4b6b11b4abb7ef4fdca1aabf | subsection | 20 | 52 | First Order Penalized Wasserstein Divergence | As was seen in the last section, since \tau _P only constrains the value of optimal critics on the supports of \mathbb {P} and \mathbb {Q}_\theta ,
the gradient \nabla _\theta \mathbb {E}_{\mathbb {Q}_\theta }[f^*] is not well defined. A natural method to refine \tau _P to achieve
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ed0bef52b8bb341733e6cac5430926443887f303 | subsection | 21 | 52 | First Order Penalized Wasserstein Divergence | Set \mathcal {F}=C^1(X), \lambda ,\mu >0 and&\tau _F(\mathbb {P}\Vert \mathbb {Q};f):=\mathbb {E}_{x\sim \mathbb {P}}[f(x)]-\mathbb {E}_{x^{\prime }\sim \mathbb {Q}}[f(x^{\prime })]\\
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ea770dab99f41c23b4fd982c4fc25a4f301c0854 | subsection | 22 | 52 | First Order Penalized Wasserstein Divergence | Therefore updates to \theta that reduce \tau _F(\mathbb {P}\Vert \mathbb {Q}^{\prime }_\theta ) also reduce
\tau _F(\mathbb {P}\Vert \mathbb {Q}_\theta ).Conveniently, as is shown in Lemma REF in Appendix, Section ,
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45c761b6cf54e81b37eb39d9cef0b9d757ae5775 | subsection | 23 | 52 | Image Generation | We begin by testing the FOGAN on the CelebA image generation task ,
training a generative model with the DCGAN architecture and obtaining Fréchet Inception Distance (FID) scores
competitive with state of the art methods without doing a tuning parameter search.
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392325d328a10352a1aa56f4f2defa678d492405 | subsection | 24 | 52 | One Billion Word | Finally, we use the First Order Penalized Wasserstein Divergence to train a character level generative language model on the
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vector into a 32\times C matrix, where C is the number of possible characters.
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... | |
4f52057d0395ad80da6b39d14412d81cc4c2fbf8 | subsection | 25 | 52 | Proof of Things | [Proof of Theorem REF ]
The proof of this theorem is split into smaller lemmas that are proven individually.That \tau _P is a strict adversarial divergence which is equivalent to \tau _W is proven in Lemma REF ,
thus showing that \tau _P fulfills Requirement REF .
\tau _P fulfills Requirement REF by design.
The exist... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.026657938957214355,
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0.009552300907671452,
-0.024536900222301483,
0... | |
9acc61076b5ea63d8c85453d162db1ead0f1400f | subsection | 26 | 52 | Proof of Things | Now consider \gamma \in (0,1), then&\mathbb {E}_{x\sim \mathbb {P},x^{\prime }\sim \mathbb {Q}}\left[\frac{(\gamma (f(x)-f(x^{\prime })) + (1-\gamma )(\hat{f}(x)-\hat{f}(x^{\prime })))^2}{\Vert x-x^{\prime }\Vert }\right] \\
\le &\mathbb {E}_{x\sim \mathbb {P},x^{\prime }\sim \mathbb {Q}}\left[\frac{\gamma (f(x)-f(x^{\... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.01820835843682289,
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0.03269262611865997,
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0.010370980016887188,
-0.02060459554195404,
-... | |
a67c612963f7f88a232c036a731f4b538fe1454c | subsection | 27 | 52 | Proof of Things | REF and REF , then f\in \operatorname{OC}_{\tau _P}(\mathbb {P},\mathbb {Q})Since in Lemma REF it was shown that the the mapping f\mapsto \tau _P(\mathbb {P}\Vert \mathbb {Q},f) is concave,
f\in \operatorname{OC}_\tau (\mathbb {P},\mathbb {Q}) if and only if f\in C^1(X) and f is a local maximum of \tau _P(\mathbb {P}\V... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.030316919088363647,
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0.018965238705277443,
-0.009177466854453087,
0.01627989485859871,
0.006053466349840164,
... | |
64073234b49e03c31192008795b579aa29604892 | subsection | 28 | 52 | Proof of Things | Then there exists a f\in \mathcal {F}=C^1(X) such that\forall x^{\prime }\in \Omega :\quad \mathbb {E}_{x\sim \mathbb {P}}\left[\frac{f(x)-f(x^{\prime })}{\Vert x-x^{\prime }\Vert }\right]=\frac{1}{2\lambda }and\forall x\in \operatorname{supp}(\mathbb {P}):\quad \mathbb {E}_{x^{\prime }\sim \mathbb {Q}}\left[\frac{f(x)... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.03955157473683357,
0.02070659212768078,
-0.036865975707769394,
-0.03985675796866417,
-0.008888423442840576,
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-0.0028400991577655077,
0.0021896802354604006,
-0.023621080443263054,
-0.013916295021772385,
-0.... | |
62fcfc0cf91c7422db767d389c0d312ca05f437f | subsection | 29 | 52 | Proof of Things | REF\forall x\in \operatorname{supp}(\mathbb {P}):\quad f(x)=\frac{\mathbb {E}_{x^{\prime }\sim \mathbb {Q}}[\frac{f(x^{\prime })}{\Vert x-x^{\prime }\Vert }]+\frac{1}{2\lambda }}{\mathbb {E}_{x^{\prime }\sim \mathbb {Q}}[\frac{1}{\Vert x-x^{\prime }\Vert }]}.Now it's clear that if the mapping T:\mathcal {F}\rightarrow ... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.029346106573939323,
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-0.020662833005189896,
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0.04593436419963837,
0.0071915509179234505,
0.006020301021635532,
0.0032619500998407602,
0.005016917362809181,... | |
a343c4ef5d50f2d163df5f31906b3d5f38fa3331 | subsection | 30 | 52 | Proof of Things | REF and REF and \tau _P(\mathbb {P}\Vert \mathbb {Q};f^*)=\tau _P(\mathbb {P}\Vert \mathbb {Q}).Define the mapping S:\mathcal {F}\rightarrow \mathcal {F} byS(f)(x)=\frac{f(x)}{2\lambda \mathbb {E}_{\tilde{x}\sim \mathbb {P},x^{\prime }\sim \mathbb {Q}}\left[\frac{f(\tilde{x})-f(x^{\prime })}{\Vert \tilde{x}-x^{\prime }... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.01878903992474079,
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0.0474381297826767,
0.010462946258485317,
0.013225591741502285,
0.008120040409266949,
0.0024... | |
350d57ee74c98418c0b07d8d1a0ad1ad80a717b5 | subsection | 31 | 52 | Proof of Things | REF\mathbb {E}_{\tilde{x}\sim \mathbb {P},x^{\prime }\sim \mathbb {Q}}\left[\frac{T(f)(\tilde{x})-T(f)(x^{\prime })}{\Vert \tilde{x}-x^{\prime }\Vert }\right]
&=\mathbb {E}_{\tilde{x}\sim \mathbb {P}}\left[\mathbb {E}_{x^{\prime }\sim \mathbb {Q}}\left[\frac{T(f)(\tilde{x})}{\Vert \tilde{x}-x^{\prime }\Vert }\right]\ri... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.027339264750480652,
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-0.02732400968670845,
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0.02215212769806385,
0.02273186668753624,
-0.01476045697927475,
0.01215163... | |
e329d83d9b64333ec4f63f5e18bde0030e04e035 | subsection | 32 | 52 | Proof of Things | REF we get\mathbb {E}_{x\sim \mathbb {P},x^{\prime }\sim \mathbb {Q}}\left[\frac{f_1(x^{\prime })-f_2(x^{\prime })}{\Vert x-x^{\prime }\Vert }\right]
=\mathbb {E}_{x\sim \mathbb {P},x^{\prime }\sim \mathbb {Q}}\left[\frac{f_1(x)-f_2(x)}{\Vert x-x^{\prime }\Vert }\right].Now since for every f\in \mathcal {F} it holds by... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.03911891207098961,
0.00954324472695589,
-0.010504435747861862,
-0.046015072613954544,
-0.006491847801953554,
0.006480405107140541,
0.05458949878811836,
0.0389358289539814,
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0.02277868054807186,
-0.02012396603822708,
0.0006732145557180047,
-0.012014877051115036,
0.00... | |
61adb1b2d99eb82466cab616095d4fbd13012a69 | subsection | 33 | 52 | Proof of Things | If f_1,f_2\in S(\mathcal {F}) then\sup _{x\in \operatorname{supp}(\mathbb {P})} |T(f_1)(x)-T(f_2)(x)|
&=\sup _{x\in \operatorname{supp}(\mathbb {P})}\left|\frac{
\mathbb {E}_{x^{\prime }\sim \mathbb {Q}}\left[\frac{f_1(x^{\prime })-f_2(x^{\prime })}{\Vert x-x^{\prime }\Vert }\right]
}{
\mathbb {E}_{x^{\prime }\sim \mat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.5555/1756006.1859901",
"end": 1494,
"openalex_id": "https://openalex.org/W2124331852",
"raw": "Sriperumbudur, B. K., Gretton, A., Fukumizu, K., Schölkopf, B., and Lanckriet, G. R. Hilbert space embeddings and metrics on probability measu... | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.05186380818486214,
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-0.01723197102546692,
... |
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