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c2813e28b62e40b2a3b404f17bedb2a0eab75f4a | subsection | 170 | 216 | Supplementary Material for Section | REF .
[Figure: Effects of varying population sizes for Example : (a) Equilibrium route flows on r_1; (b) Equilibrium population costs.]Lemma 9.1
The route flows f^{ij, \dagger }\in \mathcal {F}^{ij, \dagger } induce a unique edge load w^{ij, \dagger }.Proof of Lemma REF
Following () and (REF ), any edge load w^{ij, \d... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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f3274be42b5d14e112afa368f93d75ab8762c010 | subsection | 171 | 216 | Supplementary Material for Section | Since \underline{\lambda }^i is attainable on the set \mathcal {F}^{ij, \dagger }, there exists \tilde{f}^{ij, \dagger }\in \mathcal {F}^{ij, \dagger } such that:\begin{split}
\underline{\lambda }^i= \frac{1}{\widehat{J}^{i}(\tilde{f}^{ij, \dagger }) \stackrel{\text{(\ref {widehatj})}}{=}\frac{1}{\left(\sum _{\in \math... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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2249b63a7ca2b5ea292be3b4690c812b574aa1dc | subsection | 172 | 216 | Supplementary Material for Section | Therefore, (\ref {linear_program_lambli}) is a linear programming. Analogously, the threshold \bar{\lambda }^i is the optimal value of the following linear program:
\begin{equation}
\begin{split}
\max \quad &y \\
s.t. \quad & -|\lambda ^{-ij}| \sum _{\in \mathcal {R}} f_{}(^j_, \widehat{}^{-j}) \ge y \cdot \quad \foral... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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7a52b6aa0d5806b884157cad830890a7d4425165 | subsection | 173 | 216 | Supplementary Material for Section | Rearranging, we obtain: \frac{1}{\widehat{J}^{j}(\tilde{f}^{ij, \dagger }) < 1-|\lambda ^{-ij}| -\lambda ^{i}=\lambda ^{j}, and so such \tilde{f}^{ij, \dagger } also satisfies (\ref {prime:popu_i}_{j}). Since \tilde{f}^{ij, \dagger } is an optimal solution of (\ref {drop_i_j}), which minimizes the same objective functi... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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bfbe911bf2b8769f6f8ad43613cf373be6668e43 | subsection | 174 | 216 | Supplementary Material for Section | Hence, (\ref {prime:popu_i}_{j}) can be dropped in (\ref {opt_l}) without changing the optimal solution set.
}\left[\text{Regime } \Lambda _3^{ij}\right]: Analogous to the proof given for regime \Lambda _1^{ij}, we can argue that constraint (\ref {prime:popu_i}_{j}) is tight in any equilibrium for any \lambda in regime... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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bec397376233d15816322c00f7f613a06a653090 | subsection | 175 | 216 | Supplementary Material for Section | Such \tilde{f}^{ij, \dagger } also satisfies constraint (\ref {prime:popu_i}_{j}). Therefore, \tilde{f}^{ij, \dagger } satisfies all the constraints in (\ref {eq:Lprime}), and minimizes \widehat{\Phi }(f). So \tilde{f}^{ij, \dagger } is an equilibrium route flow, which implies that \mathcal {F}^{*}(\lambda )\cap \mathc... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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61dd1aeb82f8d1d1869678d201f6244505712ece | subsection | 176 | 216 | Supplementary Material for Section | If \underline{\underline{\lambda }}^{i}< \bar{\bar{\lambda }}^{i}, for any \lambda ^{i}\in (\underline{\underline{\lambda }}^{i}, \bar{\bar{\lambda }}^{i}), we can check that any f^{ij, \dagger }\in \mathcal {F}^{ij, \dagger } satisfies the constraint (\ref {prime:popu_i}_i): \frac{1}{\widehat{J}^{i}(f^{ij, \dagger }) ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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3210f4106f8a10d23ce59802d4d47987d8aee353 | subsection | 177 | 216 | Supplementary Material for Section | Moreover, if there are two populations, then the equilibrium strategy profile is unique in regime \Lambda _1^{12} or \Lambda _3^{12}, and can be written as follows:
\begin{}
\begin{align}
\text{In regime $\Lambda _1^{ij}$: }\quad q^{1*}_(^1)&=f^{*}_{}(^1, \widehat{}^2)-\min _{\widehat{}^1\in \mathcal {T}^1} f^{*}_{}(\w... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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9ed8893e44e20ec61a48aae1838fd651fce5fb51 | subsection | 178 | 216 | Supplementary Material for Section | Therefore, from (\ref {sub:x_sum}) and (\ref {sub:x_bound}), we obtain:
\begin{align*}
\lambda ^1 {(\ref {sub:x_sum})}{=} \sum _{\in \mathcal {R}} \chi _^1 \stackrel{(\ref {sub:x_bound})}{\ge } \sum _{\in \mathcal {R}}\max _{^1\in \mathcal {T}^1} \left(f_r^{*}(\widehat{}^1, \widehat{}^2)-f_r^{*}(^1, \widehat{}^2)\right... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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eb1e91a969f267235e6a0596b7bb274d51280d28 | subsection | 179 | 216 | Supplementary Material for Section | We know from Theorem \ref {l_behavior} that constraint (\ref {prime:popu_i}_i) is tight in equilibrium, and thus f^{*}(\lambda ) and f^{*}(\lambda ^{^{\prime }}) satisfy: \frac{1}{\widehat{J}^{i}(f^{*}(\lambda )) = \lambda ^{i}< \lambda ^{i^{\prime }} =\frac{1}{\widehat{J}^{i}(f^{*}(\lambda ^{^{\prime }})). Consequentl... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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3b060b48ebac9e9d2aa684805fa742daf5bbe6dd | subsection | 180 | 216 | Supplementary Material for Section | Thus, \Psi (\lambda ) as well as w^{*}(\lambda ) remain fixed in regime \Lambda _2^{ij}.
}}\left[\text{Regime $\Lambda _3^{ij}$}\right]: Following similar argument in regime \Lambda _1^{ij}, we conclude that \Psi (\lambda ) monotonically increases in the direction z^{ij} in regime \Lambda _3^{ij}. As a result, w^{*}(\l... | {
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"arxiv_id": "",
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"raw": "Anthony V Fiacco and Jerzy Kyparisis. Convexity and concavity properties of the optimal value function in parametric nonlinear programming. Journal ... | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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842d54fca453b89e07d0c256a79d0b1cc59c2972 | subsection | 181 | 216 | Supplementary Material for Section | From Theorem , we know that the constraint (_j) must be tight in equilibrium when \lambda is in regime \Lambda _3^{ij}. However, since \widehat{J}^{j}(f^{*})=0 for any \lambda , the constraint (_j) is tight only when \lambda ^{j}=0, i.e. \lambda ^{i}=1-|\lambda ^{-ij}|. This implies that the regime \Lambda _3^{ij} is i... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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111e275b25ecd38bd45136771d88c895082e3268 | subsection | 182 | 216 | Supplementary Material for Section | REF .
[Figure: Effects of varying population sizes for Example : (a) Equilibrium route flows on r_1; (b) Equilibrium population costs.]Lemma 9.1
The route flows f^{ij, \dagger }\in \mathcal {F}^{ij, \dagger } induce a unique edge load w^{ij, \dagger }.Proof of Lemma REF
Following () and (REF ), any edge load w^{ij, \d... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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ae6183374489764a264a2597b88b20e21f9570cd | subsection | 183 | 216 | Supplementary Material for Section | Since \underline{\lambda }^i is attainable on the set \mathcal {F}^{ij, \dagger }, there exists \tilde{f}^{ij, \dagger }\in \mathcal {F}^{ij, \dagger } such that:\begin{split}
\underline{\lambda }^i= \frac{1}{\widehat{J}^{i}(\tilde{f}^{ij, \dagger }) \stackrel{\text{(\ref {widehatj})}}{=}\frac{1}{\left(\sum _{\in \math... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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affa04390468715c27f3ba0929ff653170f3500c | subsection | 184 | 216 | Supplementary Material for Section | Therefore, (\ref {linear_program_lambli}) is a linear programming. Analogously, the threshold \bar{\lambda }^i is the optimal value of the following linear program:
\begin{equation}
\begin{split}
\max \quad &y \\
s.t. \quad & -|\lambda ^{-ij}| \sum _{\in \mathcal {R}} f_{}(^j_, \widehat{}^{-j}) \ge y \cdot \quad \foral... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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1e808cc565edf92f0c168d3e1aecbbdf9a634c68 | subsection | 185 | 216 | Supplementary Material for Section | Rearranging, we obtain: \frac{1}{\widehat{J}^{j}(\tilde{f}^{ij, \dagger }) < 1-|\lambda ^{-ij}| -\lambda ^{i}=\lambda ^{j}, and so such \tilde{f}^{ij, \dagger } also satisfies (\ref {prime:popu_i}_{j}). Since \tilde{f}^{ij, \dagger } is an optimal solution of (\ref {drop_i_j}), which minimizes the same objective functi... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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29f6a18729aa6e8d8ecb79f3fd19f613d76e226b | subsection | 186 | 216 | Supplementary Material for Section | Hence, (\ref {prime:popu_i}_{j}) can be dropped in (\ref {opt_l}) without changing the optimal solution set.
}\left[\text{Regime } \Lambda _3^{ij}\right]: Analogous to the proof given for regime \Lambda _1^{ij}, we can argue that constraint (\ref {prime:popu_i}_{j}) is tight in any equilibrium for any \lambda in regime... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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947faeddaf614116647a1fdf1b083e783639af8b | subsection | 187 | 216 | Supplementary Material for Section | Such \tilde{f}^{ij, \dagger } also satisfies constraint (\ref {prime:popu_i}_{j}). Therefore, \tilde{f}^{ij, \dagger } satisfies all the constraints in (\ref {eq:Lprime}), and minimizes \widehat{\Phi }(f). So \tilde{f}^{ij, \dagger } is an equilibrium route flow, which implies that \mathcal {F}^{*}(\lambda )\cap \mathc... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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0d08eb61c41b41bb57f97dc5164a16ddc17d1231 | subsection | 188 | 216 | Supplementary Material for Section | If \underline{\underline{\lambda }}^{i}< \bar{\bar{\lambda }}^{i}, for any \lambda ^{i}\in (\underline{\underline{\lambda }}^{i}, \bar{\bar{\lambda }}^{i}), we can check that any f^{ij, \dagger }\in \mathcal {F}^{ij, \dagger } satisfies the constraint (\ref {prime:popu_i}_i): \frac{1}{\widehat{J}^{i}(f^{ij, \dagger }) ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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7768b422d0e2461bdd32dea59e50d88845cf556c | subsection | 189 | 216 | Supplementary Material for Section | Moreover, if there are two populations, then the equilibrium strategy profile is unique in regime \Lambda _1^{12} or \Lambda _3^{12}, and can be written as follows:
\begin{}
\begin{align}
\text{In regime $\Lambda _1^{ij}$: }\quad q^{1*}_(^1)&=f^{*}_{}(^1, \widehat{}^2)-\min _{\widehat{}^1\in \mathcal {T}^1} f^{*}_{}(\w... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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0.0006957370205782354,
... | |
8a64d13179c422f2336aee87aab4aafa784918aa | subsection | 190 | 216 | Supplementary Material for Section | Therefore, from (\ref {sub:x_sum}) and (\ref {sub:x_bound}), we obtain:
\begin{align*}
\lambda ^1 {(\ref {sub:x_sum})}{=} \sum _{\in \mathcal {R}} \chi _^1 \stackrel{(\ref {sub:x_bound})}{\ge } \sum _{\in \mathcal {R}}\max _{^1\in \mathcal {T}^1} \left(f_r^{*}(\widehat{}^1, \widehat{}^2)-f_r^{*}(^1, \widehat{}^2)\right... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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de05e49d53eb9a6e206fc83e5e50f9ffa8ab9d35 | subsection | 191 | 216 | Supplementary Material for Section | We know from Theorem \ref {l_behavior} that constraint (\ref {prime:popu_i}_i) is tight in equilibrium, and thus f^{*}(\lambda ) and f^{*}(\lambda ^{^{\prime }}) satisfy: \frac{1}{\widehat{J}^{i}(f^{*}(\lambda )) = \lambda ^{i}< \lambda ^{i^{\prime }} =\frac{1}{\widehat{J}^{i}(f^{*}(\lambda ^{^{\prime }})). Consequentl... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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19d9a2682fde7f7f47c22956e7c1edae7fe43369 | subsection | 192 | 216 | Supplementary Material for Section | Thus, \Psi (\lambda ) as well as w^{*}(\lambda ) remain fixed in regime \Lambda _2^{ij}.
}}\left[\text{Regime $\Lambda _3^{ij}$}\right]: Following similar argument in regime \Lambda _1^{ij}, we conclude that \Psi (\lambda ) monotonically increases in the direction z^{ij} in regime \Lambda _3^{ij}. As a result, w^{*}(\l... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf00938592",
"end": 641,
"openalex_id": "https://openalex.org/W2076370241",
"raw": "Anthony V Fiacco and Jerzy Kyparisis. Convexity and concavity properties of the optimal value function in parametric nonlinear programming. Journal ... | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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79b9dc9d90a50bc1451d18533968b4e2b1527ed8 | subsection | 193 | 216 | Supplementary Material for Section | From Theorem , we know that the constraint (_j) must be tight in equilibrium when \lambda is in regime \Lambda _3^{ij}. However, since \widehat{J}^{j}(f^{*})=0 for any \lambda , the constraint (_j) is tight only when \lambda ^{j}=0, i.e. \lambda ^{i}=1-|\lambda ^{-ij}|. This implies that the regime \Lambda _3^{ij} is i... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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2653c0207429acf88697f3b05f195ec96561bb34 | subsection | 194 | 216 | Supplementary material for Section | Proof of Proposition REF
Firstly, we prove that for any \lambda \in \Lambda ^{\dagger }, \mathcal {F}^{*}(\lambda ) \subseteq \mathcal {F}^{\dagger }. From the definition of \Lambda ^{\dagger } in (), we know that for any \lambda \in \Lambda ^{\dagger }, there exists at least one route flow f^{\dagger }\in \mathcal {F}... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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a5dff764c98ab655b4cb0d45e922a80746cc4b29 | subsection | 195 | 216 | Supplementary material for Section | Additionally, for any \lambda \in \operatornamewithlimits{arg\,min}_{\lambda } \Psi (\lambda ), we have \Psi (\lambda )=\min _{\lambda } \Psi (\lambda )=\widehat{\Phi }(f^{\dagger }). Since \mathcal {F}^{\dagger } includes all route flows that satisfy (REF )-() and attain the minimum value of \Psi (\lambda ), any equil... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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72d05cd797ee9f155cf7d4936df1f8b5ca893b04 | subsection | 196 | 216 | Supplementary material for Section | From the definition of \Lambda ^{\dagger } in (), we know that for any \lambda \in \Lambda ^{\dagger }, there exists at least one route flow f^{\dagger }\in \mathcal {F}^{\dagger } satisfying the constraints in (REF ), and hence such f^{\dagger } is a feasible solution of the optimization problem (REF ); thus \widehat{... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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832e140079d15acdb31cbef85190408ea439887e | subsection | 197 | 216 | Supplementary material for Section | Since \mathcal {F}^{\dagger } includes all route flows that satisfy (REF )-() and attain the minimum value of \Psi (\lambda ), any equilibrium route flow f^{*}\in \mathcal {F}^{*}(\lambda ) for \lambda \in \operatornamewithlimits{arg\,min}_{\lambda } \Psi (\lambda ) must be in \mathcal {F}^{\dagger }. Hence, such \lamb... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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b8453ab8e3afdb7aca393bf4ea253c4e1e3eca8c | subsection | 198 | 216 | Supplementary material for Section | Additionally, since f^{\dagger } is an optimal solution of (REF ), which has the same objective function as (REF ) but without the constraints (), we conclude that \widehat{\Phi }(f^{\dagger }) \le \Psi (\lambda ) for any feasible \lambda (including \lambda \in \Lambda ^{\dagger }). Thus, \Psi (\lambda )=\widehat{\Phi ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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e5d453f71232d53fa5d64b1c009c6858f290debf | subsection | 199 | 216 | Supplementary material for Section | \operatornamewithlimits{arg\,min}_{\lambda } \Psi (\lambda )\subseteq \Lambda ^{\dagger }. We can therefore conclude that \Lambda ^{\dagger }= \operatornamewithlimits{arg\,min}_{\lambda } \Psi (\lambda ).From Lemma REF , we know that the function \Psi (\lambda ) is convex in \lambda . Additionally, the set of feasible ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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... | |
a32d77ff608ec3a1e4648829082d537c3295bac6 | subsection | 200 | 216 | Supplementary material for Section | Since the set \mathcal {F}^{\dagger } in (REF ) contains all route flows such that the induced edge load is w^{\dagger }, we can conclude that the set of equilibrium route flow \mathcal {F}^{*}(\lambda ) \subseteq \mathcal {F}^{\dagger } for any \lambda \in \Lambda ^{\dagger }.Next, we prove that \Lambda ^{\dagger }= \... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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b95983547f30200a604a477b1e64c3f23d3f3dbf | subsection | 201 | 216 | Supplementary material for Section | Consequently, the set \Lambda ^{\dagger }= \operatornamewithlimits{arg\,min}_{\lambda } \Psi (\lambda ) is convex and non-empty.Finally, we show that w^{*}(\lambda )=w^{\dagger } if and only if \lambda \in \Lambda ^{\dagger }. From the first part of the proof, we know that \mathcal {F}^{*}(\lambda ) \subseteq \mathcal ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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4a26f510edc40e870cd4d20caebd735464a0971a | subsection | 202 | 216 | Extension to Networks with Multiple Origin-destination Pairs | In this section, we extend our model to networks with multiple origin-destination pairs, and show that all the results presented in the paper still hold. Consider a network with a set of origin-destination (o-d) pairs \mathcal {K}. Each o-d pair k \in \mathcal {K} is connected by the set of routes \mathcal {R}_k. The s... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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7ea76f5104098e3841ee2bd2ae75ebd4b07585b2 | subsection | 203 | 216 | Extension to Networks with Multiple Origin-destination Pairs | A strategy profile q is feasible if it satisfies:\sum _{r \in R_k} q_{r,k}^{i}(t^i)&=\lambda _k^i \cdot D_k, \quad \forall i \in I, \quad \forall t^i \in i, \quad \forall k \in \mathcal {K}, \\
q_{r,k}^{i}(t^i) & \ge 0, \quad \forall r \in \mathcal {R}_k, \quad \forall i \in I, \quad \forall t^i \in i, \quad \forall k ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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a2e88cc69fc3f9fba7b3990b86a2f3677235926a | subsection | 204 | 216 | Extension to Networks with Multiple Origin-destination Pairs | Firstly, we can check that the following function of q is a weighted potential function of the Bayesian congestion game with K o-d pairs:
\begin{align*}
\Phi (q)= \sum _{e \in \mathcal {E}}\sum _{s \in \mathcal {S}} \sum _{t \in \pi (s, t) \int _{0}^{\sum _{i \in I}\sum _{k \in \mathcal {K}}\sum _{r \in \left\lbrace \m... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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0.01153621170669794,... | |
f002116e319de31cd945a19611ce9eb46dd97e6a | subsection | 205 | 216 | Extension to Networks with Multiple Origin-destination Pairs | We can show that Theorem (\ref {l_behavior}) holds: three regimes (one or two may be empty) can be distinguished by whether or not the information impact all the travelers between o-d pair k who subscribe to TIS i (resp. j), i.e. whether or not (\ref {IIC}) is tight at the optimum of (\ref {opt_l}).
\end{align*}Fourthl... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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d2c78a080c95df681fd2e79d815b246913628817 | subsection | 206 | 216 | Extension to Networks with Multiple Origin-destination Pairs | We denote the strategy profile as q=(q_{r,k}^{i}(t^i))_{r \in \mathcal {R}_k, i \in I, t^i \in {i}, k \in \mathcal {K}}, where q_{r,k}^{i}(t^i) is the amount of travelers in population i who take route r between o-d pair k when the signal is t^i. A strategy profile q is feasible if it satisfies:\sum _{r \in R_k} q_{r,k... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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88237dad9d5a56384cead82014ec33eb5e856428 | subsection | 207 | 216 | Extension to Networks with Multiple Origin-destination Pairs | A feasible strategy profile q^{*} is a BWE if for any k \in \mathcal {K}, any i \in I, and any t^i\in {i}:
\begin{align*}
\forall \in \mathcal {R}_k, \quad q_{r,k}^{i*}(t^i)>0 \quad \Rightarrow \quad \mathbb {E}[c_{}({q^{*}})|t^i]\le \mathbb {E}[c_{^{\prime }}(q^{*})|t^i], \quad \forall ^{\prime } \in \mathcal {R}_k.
\... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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... | |
1f340d05902f41cfab02f6105b07a1cb10651d10 | subsection | 208 | 216 | Extension to Networks with Multiple Origin-destination Pairs | Particularly, the information impact constraint (\ref {sub:popu_i}) for o-d pair $k \in \mathcal {K}$ and population $i\in I$ now becomes:
\begin{align}
D_k - \sum _{r \in \mathcal {R}_k} \min _{t^i \in i} f_{r,k}(t^i, t^{-i}) \le \lambda _k^i k, \quad \forall t^{-i} \in {-i}, \quad \forall i \in I, \quad \forall k \in... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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... | |
23351095e011ff6dd57337e4957e0a7232818b70 | subsection | 209 | 216 | Extension to Networks with Multiple Origin-destination Pairs | Consider a network with a set of origin-destination (o-d) pairs \mathcal {K}. Each o-d pair k \in \mathcal {K} is connected by the set of routes \mathcal {R}_k. The set of all routes is \mathcal {R}= \cup _{k \in \mathcal {K}} \mathcal {R}_k. The demand of travelers between o-d pair k \in \mathcal {K} is D_k > 0. The i... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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9ee575b387b9d056612eb15a49e12ab83ac8079b | subsection | 210 | 216 | Extension to Networks with Multiple Origin-destination Pairs | A strategy profile q is feasible if it satisfies:\sum _{r \in R_k} q_{r,k}^{i}(t^i)&=\lambda _k^i \cdot D_k, \quad \forall i \in I, \quad \forall t^i \in i, \quad \forall k \in \mathcal {K}, \\
q_{r,k}^{i}(t^i) & \ge 0, \quad \forall r \in \mathcal {R}_k, \quad \forall i \in I, \quad \forall t^i \in i, \quad \forall k ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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26ede4f79996f160e49dbacd778ba3d8d19f1da5 | subsection | 211 | 216 | Extension to Networks with Multiple Origin-destination Pairs | Firstly, we can check that the following function of q is a weighted potential function of the Bayesian congestion game with K o-d pairs:
\begin{align*}
\Phi (q)= \sum _{e \in \mathcal {E}}\sum _{s \in \mathcal {S}} \sum _{t \in \pi (s, t) \int _{0}^{\sum _{i \in I}\sum _{k \in \mathcal {K}}\sum _{r \in \left\lbrace \m... | {
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4edbb9e533687745ca5a4569ab1ac84e090790a9 | subsection | 212 | 216 | Extension to Networks with Multiple Origin-destination Pairs | We can show that Theorem (\ref {l_behavior}) holds: three regimes (one or two may be empty) can be distinguished by whether or not the information impact all the travelers between o-d pair k who subscribe to TIS i (resp. j), i.e. whether or not (\ref {IIC}) is tight at the optimum of (\ref {opt_l}).
\end{align*}Fourthl... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
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"Asuman E. Ozdaglar"
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ee14fae82313fef01b53158db2d7a6f3dae77c11 | subsection | 213 | 216 | Extension to Networks with Multiple Origin-destination Pairs | We denote the strategy profile as q=(q_{r,k}^{i}(t^i))_{r \in \mathcal {R}_k, i \in I, t^i \in {i}, k \in \mathcal {K}}, where q_{r,k}^{i}(t^i) is the amount of travelers in population i who take route r between o-d pair k when the signal is t^i. A strategy profile q is feasible if it satisfies:\sum _{r \in R_k} q_{r,k... | {
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"Manxi Wu",
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5615c7a12d62ab0ef33d80e3bd79f0b0be449a8c | subsection | 214 | 216 | Extension to Networks with Multiple Origin-destination Pairs | A feasible strategy profile q^{*} is a BWE if for any k \in \mathcal {K}, any i \in I, and any t^i\in {i}:
\begin{align*}
\forall \in \mathcal {R}_k, \quad q_{r,k}^{i*}(t^i)>0 \quad \Rightarrow \quad \mathbb {E}[c_{}({q^{*}})|t^i]\le \mathbb {E}[c_{^{\prime }}(q^{*})|t^i], \quad \forall ^{\prime } \in \mathcal {R}_k.
\... | {
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"Manxi Wu",
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3e93bd4cbc68e11949a1dd831e68ad8c71fb6a89 | subsection | 215 | 216 | Extension to Networks with Multiple Origin-destination Pairs | Particularly, the information impact constraint (\ref {sub:popu_i}) for o-d pair $k \in \mathcal {K}$ and population $i\in I$ now becomes:
\begin{align}
D_k - \sum _{r \in \mathcal {R}_k} \min _{t^i \in i} f_{r,k}(t^i, t^{-i}) \le \lambda _k^i k, \quad \forall t^{-i} \in {-i}, \quad \forall i \in I, \quad \forall k \in... | {
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56e4486b95a09b94e3ef1f4e1f007098e5af0591 | abstract | 0 | 25 | Abstract | We present a full formalization in Martin-L\"of's Constructive Type Theory of
the Standardization Theorem for the Lambda Calculus using first-order syntax
with one sort of names for both free and bound variables and Stoughton's
multiple substitution. Our formalization is based on a proof by Ryo Kashima, in
which a noti... | {
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} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
for the Lambda Calculus using Multiple Substitution | [
"Martín Copes",
"Nora Szasz",
"Álvaro Tasistro"
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84b3d9b43c433633bda9ea1cc3049675249b2fe0 | subsection | 1 | 25 | Introduction | In a formalization of the Lambda Calculus in Martin-Löf's Constructive Type Theory is presented, which uses first-order syntax with one sort of names for both free and bound variables that does not identify \alpha -convertible terms, and a multiple substitution operation introduced by Stoughton in . The approach enabl... | {
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98bbe2dee84d4fa725f8817eb1eb0bad885e3756 | subsection | 2 | 25 | Preliminaries | In what follows we will introduce the main definitions and results in , that are previous to this work and are used in our formalization.
We present the definitions directly using Agda code along with informal explanations, while the proofs are written in English to ease their reading.
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4c786fdff14319f9e7efdb680724611560fe7f32 | subsection | 3 | 25 | Preliminaries | The fact that structural recursion is sufficient for stating this very concrete definition is a (very welcome) non-trivial consequence of the employment of multiple substitutions._∙_ : Λ → Σ → Λ(v x) ∙ σ = σ x(M · N) ∙ σ = (M ∙ σ) · (N ∙ σ)(ƛ x M) ∙ σ = ƛ y (M ∙ (σ ≺+ (x , v y)))where y = χ (σ , ƛ x M)Notice the last l... | {
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e95a74af38f05dc12f3b61348b5eb5f65c95437b | subsection | 4 | 25 | Preliminaries | This is proven in , and we just mention the corresponding lemma here:lemmaM∼M'→Mσ≡M'σ : {M M' : Λ}{σ : Σ} → M ∼α M' → M ∙ σ ≡ M' ∙ σFrom now on we present definitions and results not included in the library .Firstly, we have proven that this definition of alpha equivalence is decidable:_∼α?_ : ∀ A B -> Dec (A ∼α B)Give... | {
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f745480549d87ccd671999b64c6096c32ca2e2d1 | subsection | 5 | 25 | Preliminaries | We shall start by defining the contraction of the n-th redex as a relation between terms depending on the natural number n.data _β_@_ : Λ -> Λ -> ℕ -> Set whereouter-redex : ∀ {x A B} -> ((ƛ x A) · B) β (A [ x := B ]) @ 0appNoAbsL : ∀ {n A B C} -> A β B @ n -> ¬ isAbs A -> (A · C) β (B · C) @ nappAbsL : ∀ {n A B C} -> ... | {
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53d6550a95c21dac4c3eb2609d7c1d9329f21fdf | subsection | 6 | 25 | The Standardization Theorem | In the present section we show the formalization of the Standardization Theorem in Constructive Type Theory that follows the proof given by Kashima in .
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03dc4cc7dd6fc59b2121ca3b0fd14009cd41dc61 | subsection | 7 | 25 | Standard Reduction Sequences | A reduction sequence is a sequence of terms M_0, M_1,..., M_n such that M_{i+1} is obtained from M_i by the contraction of some redex, i.e., (\forall \ i \in {0...n{-}1}) \ M_{i}\ \longrightarrow _{\beta }M_{i+1}.
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for the Lambda Calculus using Multiple Substitution | [
"Martín Copes",
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842995db9b9774d47fb9035345c4e91ac4bdedc4 | subsection | 8 | 25 | Two Useful Reduction Relations | The next step is to capture the existence of a standard sequence as an inductively defined reduction relation between terms. To this end, Kashima introduces two auxiliary one-step reduction relations:\longrightarrow _{l} stands for leftmost reduction and corresponds to the contraction of the leftmost redex, i.e. the on... | {
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"Martín Copes",
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11f630a5039a7961be4bb22ffed7ec96ee936b38 | subsection | 9 | 25 | Two Useful Reduction Relations | The second one is a form of the substitution composition lemma:corollary1SubstLemma : {x y : V} {σ : Σ}{M N : Λ} → y #⇂ (σ , ƛ x M)→ ((M ∙ (σ ≺+ (x , v y))) [y := N]) ∼α (M ∙ (σ ≺+ (x , N)))corollary1Prop7 : {M N : Λ}{σ : Σ}{x : V}→ M ∙ (σ ≺+ (x , N ∙ σ)) ≡ (M [x := N]) ∙ σNow we prove that substitution preserves \lon... | {
"cite_spans": []
} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
for the Lambda Calculus using Multiple Substitution | [
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e1916397e94959e092dd9aba22e7291541df4535 | subsection | 10 | 25 | Two Useful Reduction Relations | We can easily extend the previous result to \twoheadrightarrow _{hap}:hap-subst : ∀{M N σ} -> M →→hap N -> (M ∙ σ) →→hap (N ∙ σ)By induction on M\twoheadrightarrow _{hap}N:Case refl: Direct using refl.
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for the Lambda Calculus using Multiple Substitution | [
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8e8f03f62ce9c45c67f276c955febeca7e681d18 | subsection | 11 | 25 | Standard Reduction | Using \twoheadrightarrow _{hap}, Kashima characterizes the existence of a standard sequence as a further reduction relation \twoheadrightarrow _{st}, which stands for standard reduction, as follows:data _→→st_ (L : Λ) : Λ -> Set wherest-var : ∀{x} -> L →→hap (v x) -> L →→st (v x)st-app : ∀{A B C D} -> L →→hap (A · B) -... | {
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} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
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a3afeb6221293f22490c56ffcb818e425da1a26a | subsection | 12 | 25 | Standard Reduction | From L \twoheadrightarrow _{hap}M and M \twoheadrightarrow _{hap}A\ B we conclude that L \twoheadrightarrow _{hap}A\ B by transitivity of \twoheadrightarrow _{hap}. Finally, from this plus A \twoheadrightarrow _{st}C and B \twoheadrightarrow _{st}D, we conclude that L \twoheadrightarrow _{st}C\ D using st-app.
For the... | {
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842b5daa4af51873f1f935d076ca3eccd916c01a | subsection | 13 | 25 | Standard Reduction | Therefore, from M\ \bullet \ \sigma \twoheadrightarrow _{hap}x\ \bullet \ \sigma \twoheadrightarrow _{st}x\ \bullet \ \sigma ^{\prime } we conclude that M\ \bullet \ \sigma \twoheadrightarrow _{st}x\ \bullet \ \sigma ^{\prime } using hap-st→st.
Case st-app: Assume M \twoheadrightarrow _{hap}A\ B,
\sigma \rightarrow _{... | {
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} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
for the Lambda Calculus using Multiple Substitution | [
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370b9c069847748e8a640b54f8e948e24247ec66 | subsection | 14 | 25 | Standard Reduction | Let z = \chi (\iota \downharpoonright \ ((ƛ x\ A) \bullet \ \sigma )\ ((ƛ x\ B)\ \bullet \ \ \sigma ^{\prime })). Due to the definition of the choice function \chi , z is fresh in every term and substitution involved. We can now prove that:
ƛ y_A\ (A\ \bullet \ \ \sigma \prec \hspace{-3.99994pt}+(x,\ y_A))\ \sim _{\alp... | {
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} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
for the Lambda Calculus using Multiple Substitution | [
"Martín Copes",
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f9a6292a3cc24617d2b30ac3903aabfe4f1b9bf1 | subsection | 15 | 25 | Standard Reduction | In addition, we know that N^{\prime }\ \bullet \ \ \sigma ^{\prime } \sim _{\alpha }N\ \bullet \ \ \sigma ^{\prime }, since they are equal (lemmaM∼M'→Mσ≡M'σ) and \sim _{\alpha } is reflexive. From these we obtain our goal using the st-alpha rule.The following lemma states that if there is a standard reduction to a term... | {
"cite_spans": []
} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
for the Lambda Calculus using Multiple Substitution | [
"Martín Copes",
"Nora Szasz",
"Álvaro Tasistro"
] | [
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e3f460700e23d5685c1864b40c9a6b7cabc49dff | subsection | 16 | 25 | Standard Reduction | For example, if M \longrightarrow _{\beta }N was constructed using the rule appAbsL then we know that M=A \ C, N=B \ C and (A\ C)\, \beta \, (B\ C)\, @\, (suc\ n), with A\, \beta \, B\, @\, n for some n. Since M is an application, L \twoheadrightarrow _{st}M must have been constructed using either the st-app constructo... | {
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} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
for the Lambda Calculus using Multiple Substitution | [
"Martín Copes",
"Nora Szasz",
"Álvaro Tasistro"
] | [
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520087b31bfecfdf690e422f77fa9aa6832966d7 | subsection | 17 | 25 | Standard Reduction | Since we know that M \longrightarrow _{\beta }N and M^{\prime } \sim _{\alpha }M we can use the \alpha -\beta diamond property of Section (lem-βα), to obtain a term K such that M^{\prime } \longrightarrow _{\beta }K and K \sim _{\alpha }N, so we prove our goal using the st-alpha rule.Finally, using this last result we... | {
"cite_spans": []
} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
for the Lambda Calculus using Multiple Substitution | [
"Martín Copes",
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f9d5fd6736b7154699459f86eeb377e0e92019da | subsection | 18 | 25 | Standard Sequences | The next results show the relation between the reduction relations \twoheadrightarrow _{l}, \twoheadrightarrow _{hap} and \twoheadrightarrow _{st} with the existence of a standard reduction sequence.
Firstly notice that, since leftmost reductions always involve the reduction of redexes at position 0, then any sequence ... | {
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} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
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"Martín Copes",
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48126f9c022bf69b0a84a200fab806339e6f4be5 | subsection | 19 | 25 | Standard Sequences | Similarly, the case st-abs also relies in this lemma, and the induction hypothesis: we know from hap→seqβst that there is a standard reduction sequence with lower bound 0 from M to \lambda x. A; the induction hypothesis tells us that there exists a natural number n such that there is a reduction sequence from A to B wi... | {
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} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
for the Lambda Calculus using Multiple Substitution | [
"Martín Copes",
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af3789808b4c600c06511d512129fa862d06d673 | subsection | 20 | 25 | Standard Sequences | The induction hypothesis gives us a standard reduction sequence from M to N^{\prime } and we can directly perform an alpha step to N by using the \alpha -step constructor from seq\beta -st.The Standardization Theorem finally follows from this last result and lemma β→st that states that M \twoheadrightarrow _{\beta }N i... | {
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} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
for the Lambda Calculus using Multiple Substitution | [
"Martín Copes",
"Nora Szasz",
"Álvaro Tasistro"
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1e79972e0afea082661322138d3acbd8664d328d | subsection | 21 | 25 | The Leftmost Reduction Theorem | A quite relevant corollary of the Standardization Theorem is the Leftmost Reduction Theorem, which states that if a term M has a normal form, then the leftmost-outermost reduction strategy will find it.
In the present section we show how this property can be derived from standardization. It is worth noticing that this ... | {
"cite_spans": []
} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
for the Lambda Calculus using Multiple Substitution | [
"Martín Copes",
"Nora Szasz",
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df9733e8bf03c9f9b519a013ee61928930d97506 | subsection | 22 | 25 | The Leftmost Reduction Theorem | From the induction hypothesis we have that n \equiv 0, and since {\tt countRedexes}\ C \equiv 0, n+ {\tt countRedexes}\ C \equiv 0.
Case appAbsR: we have that (C\, A)\, \beta \, (C\, B)\, @\, suc\,({\tt countRedexes}\, C + n) where A\, \beta \, B\, @\, n, C is an abstraction and (C\ B) is in normal form. However, this... | {
"cite_spans": []
} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
for the Lambda Calculus using Multiple Substitution | [
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"Álvaro Tasistro"
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84c08ce00dab7eef7890782bdb6aec2a881f0855 | subsection | 23 | 25 | The Leftmost Reduction Theorem | Finally, from A \twoheadrightarrow _{l}B^{\prime } and B^{\prime } \longrightarrow _{l}B we conclude that A \twoheadrightarrow _{l}B using rule append.Finally, if we have that M \twoheadrightarrow _{\beta }N, the Standardization Theorem lets us conclude that there exists a standard reduction sequence from M to N. There... | {
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} | 10.4204/EPTCS.274.3 | 1807.01871 | Formalization in Constructive Type Theory of the Standardization Theorem
for the Lambda Calculus using Multiple Substitution | [
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8e8a9fb71463c2ad6cb1e93c4951b0017df4a0c6 | subsection | 24 | 25 | Conclusions | In this work we have extended some metatheoretical results from by formalizing a proof of the Standardization Theorem in Lambda Calculus using Constructive Type Theory. We use a concrete approach to \lambda -terms and the notion of multiple substitution. The latter enables us to proceed by structural induction only, p... | {
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241a668dc1473e21bd6195346893f018851e81ba | abstract | 0 | 10 | Abstract | The small butterfly shaped structure of spinal cord (SC) gray matter (GM) is
challenging to image and to delinate from its surrounding white matter (WM).
Segmenting GM is up to a point a trade-off between accuracy and precision. We
propose a new pipeline for GM-WM magnetic resonance (MR) image acquisition and
segmentat... | {
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} | 1808.02408 | Spinal Cord Gray Matter-White Matter Segmentation on Magnetic Resonance
AMIRA Images with MD-GRU | [
"Antal Horvath",
"Charidimos Tsagkas",
"Simon Andermatt",
"Simon Pezold",
"Katrin Parmar",
"Philippe Cattin"
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94ca87ef4314ea1bfa47657db41e9f491dd4e7fc | subsection | 1 | 10 | Introduction | Cervical spinal cord (SC) segmentation in magnetic resonance (MR) images is a viable means for quantitatively assessing the neurodegenerative effects of diseases in the central nervous system.
While conventional MR sequences only allowed differentiation of the boundary between SC and cerebrospinal fluid (CSF), more rec... | {
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} | 1808.02408 | Spinal Cord Gray Matter-White Matter Segmentation on Magnetic Resonance
AMIRA Images with MD-GRU | [
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"Simon Andermatt",
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20eea32aa1932be0c5535a3ef505f4d2fd5d1273 | subsection | 2 | 10 | Method | The Multi-Dimensional Gated Recurrent Unit (MD-GRU) is a generalization of a bi-directional recurrent neural network (RNN), which is able to process images.
It achieves this task by treating each direction along each of the spatial dimensions independently as a temporal direction.
The MD-GRU processes the image using t... | {
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} | 1808.02408 | Spinal Cord Gray Matter-White Matter Segmentation on Magnetic Resonance
AMIRA Images with MD-GRU | [
"Antal Horvath",
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"Simon Andermatt",
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e0964fcd2b8e4ada7080cb027cf09b16d3719a42 | subsection | 3 | 10 | Dice Loss | A straightforward approximation of a DL for a multi-labelling problem isL_\text{D}= - \, \frac{1}{\sum _{l\in \mathcal {L}}\omega _l}\,\sum \limits _{l\in \mathcal {L}} \omega _l\, \frac{2\,\sum _{x\in X} p_{lx}\,r_{lx}}{\sum _{x\in X}p_{lx}+r_{lx}},with the image domain X, labels \mathcal {L}, predictions p, raters r,... | {
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} | 1808.02408 | Spinal Cord Gray Matter-White Matter Segmentation on Magnetic Resonance
AMIRA Images with MD-GRU | [
"Antal Horvath",
"Charidimos Tsagkas",
"Simon Andermatt",
"Simon Pezold",
"Katrin Parmar",
"Philippe Cattin"
] | [
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105f7500daa4265114c89f0d6d93731118667118 | subsection | 4 | 10 | Data | In the following subsections, we describe the images used for the experiments: healthy subjects scan-rescan AMIRA dataset (own), which we call the AMIRA dataset,
and the SCGM challenge datasethttp://cmictig.cs.ucl.ac.uk/niftyweb/program.php?p=CHALLENGE last accessed: 2024/12/20 13:07:22 , which we refer to as SCGM data... | {
"cite_spans": []
} | 1808.02408 | Spinal Cord Gray Matter-White Matter Segmentation on Magnetic Resonance
AMIRA Images with MD-GRU | [
"Antal Horvath",
"Charidimos Tsagkas",
"Simon Andermatt",
"Simon Pezold",
"Katrin Parmar",
"Philippe Cattin"
] | [
"cs.CV"
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f096b84ee4f9416fa9aee4bef3a88f802cc9841e | subsection | 5 | 10 | AMIRA Dataset | The first dataset used in this paper consists of 24 healthy subjects (14 female, 10 male, age 40\pm 11 years).
Each subject was scanned 3 times, remaining in the scanner between the first and second scan, and leaving the scanner and being repositioned between the second and third scan.
Each scan contains 12 axial cross... | {
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} | 1808.02408 | Spinal Cord Gray Matter-White Matter Segmentation on Magnetic Resonance
AMIRA Images with MD-GRU | [
"Antal Horvath",
"Charidimos Tsagkas",
"Simon Andermatt",
"Simon Pezold",
"Katrin Parmar",
"Philippe Cattin"
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cd7c98f772a1b1659915841b59f68ee2c23b9be0 | subsection | 6 | 10 | SCGM Dataset | The SCGM segmentation challenge data consists of 40 training datasets and 40 test datasets acquired at 4 different sites.
Both training and test datasets each have 10 samples of each site.
The 4 sites have different imaging protocols with different field of view, size and resolution.
Each dataset was manually segmented... | {
"cite_spans": []
} | 1808.02408 | Spinal Cord Gray Matter-White Matter Segmentation on Magnetic Resonance
AMIRA Images with MD-GRU | [
"Antal Horvath",
"Charidimos Tsagkas",
"Simon Andermatt",
"Simon Pezold",
"Katrin Parmar",
"Philippe Cattin"
] | [
"cs.CV"
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2925e4d003ddfed1dea20fd3f2ca35ad369af376 | subsection | 7 | 10 | Experiments and Results | In the following subsections, we describe our experiments, the chosen MD-GRU options, and show their results. | {
"cite_spans": []
} | 1808.02408 | Spinal Cord Gray Matter-White Matter Segmentation on Magnetic Resonance
AMIRA Images with MD-GRU | [
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36e23e8b50c558dfe57edc47dbff239ffb431817 | subsection | 8 | 10 | AMIRA segmentation model | We split the 24 subjects into 3 groups of 8 subjects each for 3 cross-validations: training on two groups and testing on a third group.
To handle over-fitting, of each training set we excluded one subject and used it for validation.We used the standard MD-GRUhttps://github.com/zubata88/mdgru last accessed: 2024/12/20 1... | {
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} | 1808.02408 | Spinal Cord Gray Matter-White Matter Segmentation on Magnetic Resonance
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50a29bcb37cc8adda443a020a80a1d2723e75d74 | subsection | 9 | 10 | AMIRA segmentation model | In our case, the evaluation scores did not show big differences for \lambda in a range from 0.25 to 0.75, when using the class weights \omega _l according to (REF ) for both DL and GDL.
MD-GRU with the trivial linear combinations \lambda =0 (only CEL) and \lambda =1 (only GDL) did not perform as good as true combinatio... | {
"cite_spans": []
} | 1808.02408 | Spinal Cord Gray Matter-White Matter Segmentation on Magnetic Resonance
AMIRA Images with MD-GRU | [
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b26f22f14e7c1882d09acb7a71f5f39db7aec5de | abstract | 0 | 41 | Abstract | In Computer Vision,object tracking is a very old and complex problem.Though
there are several existing algorithms for object tracking, still there are
several challenges remain to be solved. For instance, variation of illumination
of light, noise, occlusion, sudden start and stop of moving object, shading
etc,make the ... | {
"cite_spans": []
} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
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cec2d236f05a1173b6497be721668004aa98891f | subsection | 1 | 41 | Introduction | Object Tracking employs the idea of following an object as long as its movement can be captured by a camera in various environments under Variable Background and Static Background. Moving object detection and tracking pose a challenge in real world scenarios like automatic surveillance system, traffic monitoring, vehic... | {
"cite_spans": []
} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
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ba2bf58ab831d2d61186ecf152f3e880048a2bad | subsection | 2 | 41 | Introduction | Later using that sparse matrix they compress foreground and background targets, and perform tracking by using naive-Bayes classifier. In Moudgil et.al provide a benchmark dataset for long duration video sequence which they name as 'Track Long and Prosper(TLP)'. This dataset is important because most tracking algorithms... | {
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9fdd4113b877863de90954be9ec8a44406d94cf8 | subsection | 3 | 41 | Introduction | Hierarchical Annealed Particle Swarm Optimization for Articulated Object Tracking Xuan et.al show articulate object tracking by decomposing the search space into subspaces and then using particle swarms to optimize over these subspaces hierarchically. Monocular Video Human Motion Tracking based on Hybrid PSO Ben shows ... | {
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179acc1a77fc2beb34956656f438f56ee64aaa29 | subsection | 4 | 41 | Introduction | In case of unknown object , dual tracking approach simply needs to recalculate the dominant points on the contour of the unknown target object(objects) and no need to spend huge time to learn/train the unknown environment with unknown object form the beginning of tracking of the target object as we have seen in case of... | {
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ad6b88054d77e758366a249f512c9f4aee9942aa | subsection | 5 | 41 | Introduction | In sec 2.3.3 we experimentally demonstrate that the set of dominant points on the contour of the target object is basically a subset of interest points.Further note that the use of dominant points as good features for object tracking is an important and unique concept which is not used by classical KLT algorithm for ob... | {
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529b447b01788dc01f17a0716514ccfea2224317 | subsection | 6 | 41 | Basic concepts | In this paper we propose a dual approach for object tracking based on optical flow and swarm Intelligence. The optical flow based tracker i.e. KLT, tracks the dominant points of the target object from frame 1 to last frame; whereas swarm Intelligence based PSO (Particle Swarm Optimization) tracker simultaneously tracks... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
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fc95f7e06815b574aa32c8136e1b449ddc2df491 | subsection | 7 | 41 | Basic concepts | As the polygonally approximated target object is embedded and tightly captured within the frame of multiswarms ring(strips) so under any kind of environmental disturbances as stated earlier the tracking of the target object is not lost in the midway of any video sequence of tracking. Another specialty and uniqueness of... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
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06e7b9298932bb733a2820ceb4f92fb017706707 | subsection | 8 | 41 | Dominant Point Detection | For the detection of the dominant point on the contour of the target object we use the methods , and ,. We first perform contour tracking of the target object to find the Chain Code based on Freeman's Chain Code. Freeman Chain code gives us list of pixels around object body. Among those pixels we eliminate linear point... | {
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09aa74cc8dd1bbe21cd50ca21f47f9b4ffa207b8 | subsection | 9 | 41 | Feature selection | Before any tracking of moving object the most fundamental step is the selection of “trackable" features. First we have to determine the parameters to find out good features. According to Tomasi and Kanade 'a single pixel cannot be tracked until it has s a very distinctive brightness with respect to all of its neighbors... | {
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6a50577345568046e5fa6b901e71633d2d928777 | subsection | 10 | 41 | Selecting Dominant point(points) as good feature | The main reason for choosing dominant point as a trackable feature is that by definition dominant point itself holds maximum curvature information on the contour of a target object. So quite obviously a window centered at dominant point should always give us enough texture for tracking from one frame to another. The ar... | {
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a3158ba8119bbf4d899ffb1ff04d2dd3d883561e | subsection | 11 | 41 | Dominant points as subset of interest points | In section 2.2 we state that dominant point holds maximum curvature information on the contour of a target object and provides enough texture for tracking. In this subsection we further clarify this concept through a simple experiment as example that dominant points are the subset of interest points which are the key e... | {
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a19aebcd0869aad22a487e59bbdd7e97f6d56fcc | subsection | 12 | 41 | Concepts of tracking dominant points by KLT | The basic notion of tracking by KLT can be explained by looking at two images in an image sequence. Let us assume that the first image is captured at time t and the second image is captured at time t + . It is important to keep in mind that the incremental time depends on the frame rate of the video camera and should b... | {
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f74470706561a702eb360036cfbd353c4e1e7957 | subsection | 13 | 41 | Calculation Feature displacement | Now we have basic information to solve the displacement d mentioned above. The solution is explained in. According to, we can calculate displacement d from from image frame I to image frame J.Thus we obtain-\epsilon = \iint _W\Big [
J(x + \frac{d}{2}) - I(x - \frac{d}{2})
\Big ]^2 W(x) dxwhere x =[x \quad y]^T , the di... | {
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90e509a300f2b2ad7263ec5da984e921ec1c0887 | subsection | 14 | 41 | KLT algorithm | We summaries the KLT algorithm as follows -Step 1: Find the dominant points which satisfy min(\lambda _1,\lambda _2) > \lambda (see equation -(4).Step 2: For each dominant point compute displacement to next frame using the Lucas-Kanade method (see equation -(12)).Step 3: Store displacement of each dominant point, updat... | {
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cea7ac0088e8d273d34a6d22548b21b110ae974b | subsection | 15 | 41 | Particles Swarm Optimization (PSO) method for tracking | In 1995 James Kennedy and Russell Eberhart proposed an evolutionary algorithm that creates a ripple among Bio-inspired algorithms. This particular algorithm is called Particle Swarm Optimization (PSO). In a simple term it is a method of optimization for continuous non-linear function. This method is influenced by swarm... | {
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2f01830b349c9eb9f2f2e714fe60eee9f3907511 | subsection | 16 | 41 | Particles Swarm Optimization (PSO) method for tracking | Pseudo code of the basic PSO algorithm is given in appendix .In this paper the PSO based tracker tracks the dynamically approximated polygon of the target object and continuously supplements the tracking of the dominant points of the target object by KLT. | {
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3c38a03b36fd47d31c69da178114411b559f9c9b | subsection | 17 | 41 | Setting PSO parameters and Initialization | Because of dynamic nature, setting PSO parameters to right value is a crucial task. Below we discuss some of the major parameters.Multiswarms - In the proposed dual tracking algorithm one tracker is PSO based approach. In the basic concept of section 2.1 we have clearly explain a key feature of dual tracking algorithm ... | {
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0.015839185565710068,
0.0... | |
50ad9967b138bc1dcc8a37b2b5b795317754227d | subsection | 18 | 41 | Setting PSO parameters and Initialization | P_{gbest_{i}} = min
{\left\lbrace \begin{array}{ll}
\sqrt{(X_{D1} - X_i)^2 + (Y_{D1} - Y_i)^2} \\
\sqrt{(X_{D2} - X_i)^2 + (Y_{D2} - Y_i)^2}
\end{array}\right.}
where (X_{D1},Y_{D1}) is the position of the 1st dominant point and (X_{D2},Y_{D2}) position of the 2nd dominant point and X_{i},Y_{i} is the coordinate of th... | {
"cite_spans": []
} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
"Rajesh Misra",
"Kumar S. Ray"
] | [
"cs.CV",
"cs.NE"
] | 2,018 | en | Computer Science | [
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-0.0008037316729314625,
-0.04591335356235504,
-0.00706329895183444,
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0.0032454542815685272,
0.035534005612134933,
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0.036419302225112915,
0.025444667786359787,
-0.06380246579647064,
-0.011371491476893425,
0.0061322106048464775... |
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