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d9317a4399e4291901a76787a7549164f5eb3d00 | subsection | 42 | 45 | A relation between branching Markov processes and evolution equations | Popul. Biol.}, 52(3):179--197, 1997.
}\bibitem {Che2015}
X.~Chen.
A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk.
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J.~Coville and L.~Dupaigne.
Propagation speed of trave... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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57d1117942cf6789323e71b72f4b7e797f4a6b8c | subsection | 43 | 45 | A relation between branching Markov processes and evolution equations | Media,} 8, 275--279, 2013.
}\bibitem {INW1968i}
N.~Ikeda, M.~Nagasawa, S.~Watanabe.
\href {https://projecteuclid.org/euclid.kjm/1250524137}{Branching Markov processes I.}
{\em J. Math. Kyoto Univ.} 8 (2), 233--278, 1968.
}\bibitem {INW1968ii}
N.~Ikeda, M.~Nagasawa, S.~Watanabe.
\href { https://projecteuclid.org/euclid.... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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e1d97b83113360b278db0118a79872cd85342f11 | subsection | 44 | 45 | A relation between branching Markov processes and evolution equations | \end{align}\bibitem {Sev1951}
B.A.~Sevast^{\prime }yanov, The theory of branching random processes. (Russian)
{\em Uspehi Matem. Nauk (N.S.)} 6(46), 47--99, 1951.
}\bibitem {Sha1988}
M.~Sharpe.
\textit {General theory of Markov processes.}
{\em Academic press}, 133, 1988.
\end{}\bibitem {Shi2015}
Z.~Shi.
\textit {Branc... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
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] | 2,018 | en | Mathematics | [
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3dd9b66f54b36d6459a106a39ff89637533290ca | abstract | 0 | 17 | Abstract | We design a reversible version of truly concurrent process algebra CTC which
is called RCTC. It has good properties modulo several kinds of strongly
forward-reverse truly concurrent bisimulations and weakly forward-reverse truly
concurrent bisimulations. These properties include monoid laws, static laws,
new expansion ... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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a86171db4c656d15685cc873b0172281b919aba4 | subsection | 1 | 17 | Introduction | Process algebras are well-known formal theory based on the so-called interleaving bisimilarity, such as CCS and ACP . We did some works on truly concurrent process algebra, which is called CTC .Reversible computation is another interesting topic, there are researches on reversible computation by use of communication ke... | {
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} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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f9e64b6e4042e84906daecc4be06c0c52931688f | subsection | 2 | 17 | Backgrounds | In this subsection, we introduce the preliminaries on truly concurrent process algebra CTC , which is based on the truly concurrent bisimulation semantics. | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
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] | 2,018 | en | Computer Science | [
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8e7c8c0b85f75e9eb234d88275935799ad643c8f | subsection | 3 | 17 | CTC | CTC is a calculus of truly concurrent systems. It includes syntax and semantics:Its syntax includes actions, process constant, and operators acting between actions, like Prefix, Summation, Composition, Restriction, Relabelling.
Its semantics is based on labeled transition systems, Prefix, Summation, Composition, Restr... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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7c449ff7f6f7369bb5ab4802f92ba8c4e85b0897 | subsection | 4 | 17 | Operational Semantics | The semantics of CTC is based on truly concurrent bisimulation/rooted branching truly concurrent bisimulation equivalences, for the conveniences, we introduce some concepts and conclusions on them.Definition 2.1 (Prime event structure with silent event)
Let \Lambda be a fixed set of labels, ranged over a,b,c,\cdots an... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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1458f61de3135205d17009bebe1316027ddd52f2 | subsection | 5 | 17 | Operational Semantics | We let \hat{C}=C\backslash \lbrace \tau \rbrace .A consistent subset of X\subseteq \mathbb {E} of events can be seen as a pomset. Given X, Y\subseteq \mathbb {E}, \hat{X}\sim \hat{Y} if \hat{X} and \hat{Y} are isomorphic as pomsets. In the following of the paper, we say C_1\sim C_2, we mean \hat{C_1}\sim \hat{C_2}.Defi... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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a2223e7acfa9fedb8a2643d6742e2ff1c32dc59d | subsection | 6 | 17 | Operational Semantics | A pomset bisimulation is a relation R\subseteq \mathcal {C}(\mathcal {E}_1)\times \mathcal {C}(\mathcal {E}_2), such that if (C_1,C_2)\in R, and C_1\xrightarrow{}C_1^{\prime } then C_2\xrightarrow{}C_2^{\prime }, with X_1\subseteq \mathbb {E}_1, X_2\subseteq \mathbb {E}_2, X_1\sim X_2 and (C_1^{\prime },C_2^{\prime })\... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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e029d30db1e1f87975dbe6701565d4fa68861e45 | subsection | 7 | 17 | Operational Semantics | When PESs \mathcal {E}_1 and \mathcal {E}_2 are weak step bisimilar, we write \mathcal {E}_1\approx _s\mathcal {E}_2.Definition 2.7 (Posetal product)
Given two PESs \mathcal {E}_1, \mathcal {E}_2, the posetal product of their configurations, denoted \mathcal {C}(\mathcal {E}_1)\overline{\times }\mathcal {C}(\mathcal {... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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53ddfe01c399ef5862128ead6a44cc68e22323a8 | subsection | 8 | 17 | Operational Semantics | We say that R is downward closed when for any (C_1,f,C_2),(C_1^{\prime },f,C_2^{\prime })\in \mathcal {C}(\mathcal {E}_1)\overline{\times }\mathcal {C}(\mathcal {E}_2), if (C_1,f,C_2)\subseteq (C_1^{\prime },f^{\prime },C_2^{\prime }) pointwise and (C_1^{\prime },f^{\prime },C_2^{\prime })\in R, then (C_1,f,C_2)\in R.F... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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c3c9e59c0f6565d9eb672e9e8c35b4f398887243 | subsection | 9 | 17 | Operational Semantics | \mathcal {E}_1,\mathcal {E}_2 are hereditary history-preserving (hhp-)bisimilar and are written \mathcal {E}_1\sim _{hhp}\mathcal {E}_2.Definition 2.10 (Weak (hereditary) history-preserving bisimulation)
A weak history-preserving (hp-) bisimulation is a weakly posetal relation R\subseteq \mathcal {C}(\mathcal {E}_1)\o... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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4bbddb0b52a6e3da77caa159814245674371644c | subsection | 10 | 17 | Forward-reverse Truly Concurrent Bisimulations | Definition 3.1 (Forward-reverse (FR) pomset transitions and forward-reverse (FR) step)
Let \mathcal {E} be a PES and let C\in \mathcal {C}(\mathcal {E}), \emptyset \ne X\subseteq \mathbb {E}, \mathcal {K}\subseteq \mathbb {N}, and X[\mathcal {K}] denotes that for each e\in X, there is e[m]\in X[\mathcal {K}] where (m\... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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5c88b6d2d61c04f8c78fa8adcc36681a2a472bbc | subsection | 11 | 17 | Forward-reverse Truly Concurrent Bisimulations | When the events in X are pairwise concurrent, we say that C{X}C^{\prime } is a weak forward step and C^{\prime }0055{{}{X[\mathcal {K}]} C is a weak reverse step.
}
}We will also suppose that all the PESs in this paper are image finite, that is, for any PES \mathcal {E} and C\in \mathcal {C}(\mathcal {E}), and a\in \La... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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7ffe5c1a1c33859fb5425e959022f7bf085eb9c5 | subsection | 12 | 17 | Forward-reverse Truly Concurrent Bisimulations | When PESs \mathcal {E}_1 and \mathcal {E}_2 are FR step bisimilar, we write \mathcal {E}_1\sim _s^{fr}\mathcal {E}_2.
}
}\begin{}[Weak forward-reverse (FR) pomset, step bisimulation]
Let \mathcal {E}_1, \mathcal {E}_2 be PESs. A weak FR pomset bisimulation is a relation R\subseteq \mathcal {C}(\mathcal {E}_1)\times \ma... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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acf05e31a71c1fbd09e49354777b0cf7406259d1 | subsection | 13 | 17 | Forward-reverse Truly Concurrent Bisimulations | When PESs \mathcal {E}_1 and \mathcal {E}_2 are weak FR step bisimilar, we write \mathcal {E}_1\approx _s^{fr}\mathcal {E}_2.
}
}\begin{}[Forward-reverse (FR) (hereditary) history-preserving bisimulation]
An FR history-preserving (hp-) bisimulation is a posetal relation R\subseteq \mathcal {C}(\mathcal {E}_1)\overline{... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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da32819b26db13c24de7d1a06261e23cf06d7eb4 | subsection | 14 | 17 | Forward-reverse Truly Concurrent Bisimulations | \mathcal {E}_1,\mathcal {E}_2 are weak FR history-preserving (hp-) bisimilar and are written \mathcal {E}_1\approx _{hp}^{fr}\mathcal {E}_2 if there exists a weak FR hp-bisimulation R such that (\emptyset ,\emptyset ,\emptyset )\in R.
}A weak FR hereditary history-preserving (hhp-) bisimulation is a downward closed wea... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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0eeeba1d010c13d8d02a679e804c58b063f302ab | subsection | 15 | 17 | Forward-reverse Truly Concurrent Bisimulations | And we follow the conventions of process algebra.
}\begin{}[Syntax]
Reversible truly concurrent processes RCTC are defined inductively by the following formation rules:
\end{}\begin{}
\item A\in \mathcal {P};
\item \textbf {nil}\in \mathcal {P};
\item if P\in \mathcal {P}, then the Prefix \alpha .P\in \mathcal {P} and ... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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cff98c1796f76109b2ba4b506f84aac4aa230e29 | subsection | 16 | 17 | Forward-reverse Truly Concurrent Bisimulations | And the predicate \xrightarrow{}\alpha [m] represents successful forward termination after execution of the action \alpha , the predicate 0055{{}{\alpha [m]}\alpha represents successful reverse termination after execution of the event \alpha [m], the the predicate \textrm {Std(P)} represents that p is a standard proces... | {
"cite_spans": []
} | 1805.03575 | Reversible Truly Concurrent Process Algebra | [
"Yong Wang"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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296e2af187e7a0edf8507263e8ea4f532f6a571a | abstract | 0 | 63 | Abstract | The problem of computing a linear combination of sources over a multiple
access channel is studied. Inner and outer bounds on the optimal tradeoff
between the communication rates are established when encoding is restricted to
random ensembles of homologous codes, namely, structured nested coset codes
from the same gene... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
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28426cc340b58b1dae62eb57843319126562530e | subsection | 1 | 63 | Introduction | Consider a multiple access channel (MAC) with two senders and one receiver, in which the receiver wishes to
reliably estimate a linear function of the transmitted sources
from the senders (see Figure REF ). One trivial approach to this computation problem involves two steps: first recover the individual sources and the... | {
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"raw": "R. Ahlswede, “Multiway communication channels,” in Proc. 2nd Int. Symp. Inf. Theory, Tsahkadsor, Armenian SSR, 1971, pp. 23–52.",
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... | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
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99fc008cca692dc3e65d5ad1d0a71bfe70ebd5eb | subsection | 2 | 63 | Introduction | With mathematical rate expressions in single-letter mutual information terms and with physical rate performances
better than those of lattice codes,
homologous codes have a potential to bringing a deeper understanding of the fundamental limits of the computation problem.Several open questions remain, however. What is t... | {
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Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
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91f8067029897dd8a75fba690ac1f5fe6055487d | subsection | 3 | 63 | Introduction | For a length-n sequence (row vector) x^n=(x_1,x_2,\ldots ,x_n) \in \mathcal {X}^n, we define its type
as \pi (x \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } x^n) = {\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }\lbrace i \colon x_i = x \rbrace \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }}/{n} f... | {
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} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
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"Young-Han Kim"
] | [
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885a72ece048755a26a68558a38644a2e9326a65 | subsection | 4 | 63 | Introduction | We use \epsilon _n \ge 0 to denote a generic sequence of n that tends to zero as n \rightarrow \infty , and use \delta _i(\epsilon ) \ge 0, i \in \mathbb {Z}^+, to denote a continuous function of \epsilon that tends to zero as \epsilon \rightarrow 0. Throughout the paper, information measures are in logarithm base q. | {
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} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
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cb392c905697a5724ca2654d84ef06adca7db1c5 | subsection | 5 | 63 | Formal Statement of the Problem | Consider the two-sender finite-field input memoryless multiple access channel (MAC)(\mathcal {X}_1\times \mathcal {X}_2, p(y\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }x_1, x_2), \mathcal {Y})in Figure REF , which consists of two sender alphabets \mathcal {X}_1 = \mathcal {X}_2 = \mathbb {F}_q, a receiver alph... | {
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Homologous Codes | [
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5948dcdcf60d8d702100f2a1a1a8ac60070906cb | subsection | 6 | 63 | Formal Statement of the Problem | More specifically, given a pair of symbol-by-symbol mappings \varphi _j \mathbb {F}_q\rightarrow \mathcal {X}_j, j=1,2, consider the virtual channel with finite field inputs, p(y \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } v_1,v_2) = p_{Y \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } X_1,X_2}(y \mathc... | {
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} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
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"Young-Han Kim"
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fb6316e6b78dbbe9408185ab921f808290e131b9 | subsection | 7 | 63 | Formal Statement of the Problem | Suppose that the codewords x_1^n(m_1), m_1 \in \mathbb {F}_q^{nR_1}, and x_2^n(m_2), m_2 \in \mathbb {F}_q^{nR_2} that constitute the codebook are generated according to the following steps:Let {\hat{R}}_j = D(p_{X_j} \Vert \mathrm {Unif}(\mathbb {F}_q))+\epsilon , j=1,2, where D(\cdot \Vert \cdot ) is the Kullback–Lei... | {
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} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
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36632a52f5d35f5eb6391faf49501dcf502fa267 | subsection | 8 | 63 | Formal Statement of the Problem | The random code ensemble generated in this manner is referred to as an (n, nR_1,nR_2; p, \epsilon ) random homologous code ensemble, where p is the given input pmf and \epsilon >0 is the parameter used in steps 1 and 3 in codebook generation. A rate pair (R_1,R_2) is said to be achievable by the (p,\epsilon )-distribut... | {
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} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
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ae8f17bc79fd6e65031c5dbb8ac509fa2fa29575 | subsection | 9 | 63 | Main Result | In this section, we present a single-letter characterization of the optimal rate region when the target linear combination is in the following class.Definition 1
A linear combination W_{{\bf a}} = a_1 X_1 \oplus a_2 X_2 for some {\bf a}= [a_1 \; a_2] \in \mathbb {F}_q^2 \setminus { \lbrace \mathbf {0}\rbrace } is said... | {
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} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
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e976ea0666c1064e3b40865e4fa54fe03d49ce36 | subsection | 10 | 63 | Main Result | Let {R}_\mathrm {CF}(p) be the set of rate pairs (R_1,R_2) such thatR_j \le H(X_j) - H(W_{{\bf a}} \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } Y), \quad \forall j \in \lbrace 1,2\rbrace \text{ with } a_j \ne 0.Let {R}_\mathrm {MAC}(p) be the set of rate pairs (R_1,R_2) such thatR_1 &\le I(X_1;Y\mathchoice{{1m... | {
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Homologous Codes | [
"Pinar Sen",
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"Young-Han Kim"
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918b8769d0ccf1548c6d58ee8f337bdce3303645 | subsection | 11 | 63 | An Inner Bound | The computation performance of random homologous code ensembles was studied using a suboptimal joint typicality decoder in , . For completeness, we first describe the joint typicality decoding rule and then characterize the rate region achievable by the (p,\epsilon )-distributed random homologous code ensemble under th... | {
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Homologous Codes | [
"Pinar Sen",
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"Young-Han Kim"
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6e82ace1071a54b567a98e30520e4fb2d1729198 | subsection | 12 | 63 | An Inner Bound | We will omit the steps that were already established in , and instead provide detailed references.Upon receiving y^n, the \epsilon ^{\prime }-joint typicality decoder, \epsilon ^{\prime }>0, looks for a unique vector s \in \mathbb {F}_q^{\kappa } such thats = a_1[m_1 \; l_1\; \mathbf {0}] \oplus a_2 [m_2 \; l_2\; \math... | {
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Homologous Codes | [
"Pinar Sen",
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"Young-Han Kim"
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4e438a32b002a1786d13ab77a83ddd80a3922ad9 | subsection | 13 | 63 | An Inner Bound | Note that the region {R}_\mathrm {CF}(p) = {R}_\mathrm {CF}(p,\delta =0), as defined in (REF ) in Section . Similarly, let {R}_{j}(p) denote the region {R}_{j}(p,\delta =0) for j=1,2 in (REF ) and (REF ).We are now ready to state the rate region achievable by the random homologous code ensembles that combines the inner... | {
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Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
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636c28cfa57b3f8340c7e18fd1d64383e1cf61d2 | subsection | 14 | 63 | An Inner Bound | Then,H(M_j\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }X_j^n(M_j),\mathcal {C}_n) &= H(M_j \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } X_j^n(M_j),
G, D_1^n, D_2^n, (L_1(m_1) m_1 \in \mathbb {F}_q^{nR_1}),(L_2(m_2) m_2 \in \mathbb {F}_q^{nR_2})) \\
& \le H(M_j \mathchoice{{1mu}\vert {1mu}}{\vert }{\ver... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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54d5e9d07573209b9a66b6e7858e4ae484a5f0a7 | subsection | 15 | 63 | An Inner Bound | Since this condition is satisfied if (REF ) holds, the proof of (REF ) follows.The proof of (REF ) follows by taking the closure of the union of (REF ) over all \delta >0, which completes the proof of Theorem REF .The inner bound (REF ) in Theorem REF is valid for computing an arbitrary linear combination, which may no... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
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a9b6a39a16f2674a0519ebd72ec4e801c47a3fd8 | subsection | 16 | 63 | An Outer Bound | We first present an outer bound on the rate region {R}^*(p,\epsilon ) for a fixed input pmf p and \epsilon >0. We then discuss the limit of this outer bound as \epsilon \rightarrow 0 to establish an outer bound on the rate region {R}^*(p). Given an input pmf p and \delta >0, we define the rate region {R}^{**}(p,\delta ... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
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] | [
"cs.IT",
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2a91b1aeefb88fa106af07d8292bde780b428f45 | subsection | 17 | 63 | An Outer Bound | If a rate pair (R_1, R_2) is achievable by the (p,\epsilon )-distributed random homologous code ensemble for computing an arbitrary linear combination W_{{\bf a}}, then there exists a continuous \delta ^{\prime }(\epsilon ) that tends to zero monotonically as \epsilon \rightarrow 0 such that(R_1,R_2) \in {R}^{**}(p,\de... | {
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Homologous Codes | [
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5fda140758a65e173126bb79f746ede66877d6e3 | subsection | 18 | 63 | An Outer Bound | Then, for n sufficiently large,nR_j &= H(M_j\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }M_{j^c}, \mathcal {C}_n)
\\ & \overset{(a)}{\le } I(M_j;Y^n\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } M_{j^c}, \mathcal {C}_n) + n \epsilon _n
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Homologous Codes | [
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0.0017243946203961968,
-0.010819432325661182,
-0.00022484848159365356,
-0.03262615576386452,
0.0012503769248723984,
-0.0247824508696794... | |
090f068b01b06ce668a6b7bb17a7a9e439f6c3be | subsection | 19 | 63 | An Outer Bound | To further upper bound (REF ), we make a connection between the distribution of the random homologous codebook and the input pmf p as follows.Lemma 3
Let (X,Y) \sim p_{X,Y}(x,y) on \mathbb {F}_q\times \mathcal {Y} and \epsilon > 0. Let X^n(m) be the random codeword assigned to message m \in \mathbb {F}_q^{nR} by an (n... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05857287347316742,
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0... | |
5436e7d0f280378c9a4f7fdeebac5368d591766c | subsection | 20 | 63 | An Outer Bound | Combining (REF ) with Lemma REF (with p(x) \leftarrow p(x_1)p(x_2)), we havenR_j &\le 1 + nR_j \operatorname{\textsf {P}}(E_n = 0) + n ( I( X_j ;Y \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } X_{j^c}) + \delta _2(\epsilon )) + n \epsilon _n
\\
& \overset{(d)}{\le } n ( I( X_j ;Y \mathchoice{{1mu}\vert {1mu}}{... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.00... | |
fcae8ba6868682e78b48b462ee041bb998e8ff47 | subsection | 21 | 63 | An Outer Bound | Following arguments similar to (REF ), the first term in (REF ) can be bounded asI(M_1, M_2; Y^n\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }\mathcal {C}_n) &\le 1 + n(R_1+R_2) \operatorname{\textsf {P}}(E_n = 0) + \sum _{i=1}^n I(M_1, M_2; Y_i \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {C}_... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.0092... | |
c8a22261a8b6b001e89642b7e6c836e6a38832e0 | subsection | 22 | 63 | An Outer Bound | Letting n \rightarrow \infty in (REF ) and (REF ) establishesR_j & \le I(X_j; Y \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } X_{j^c}) + \delta _2(\epsilon ),
\\
R_j & \le I(X_1,X_2;Y) - \min \lbrace R_{j^c}, I(X_{j^c};W_{{\bf a}},Y) \rbrace + \delta _6(\epsilon ).The proof of (REF ) follows by taking a continu... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
c21642ce94c1b6482b272381d596ce87a883d40f | subsection | 23 | 63 | Optimal Achievable Rates for Broadcast Channels with Marton Coding | In this section, we apply the techniques developed in the previous sections to establish the optimal rate region for broadcast channels by Marton coding. Consider the two-receiver discrete memoryless broadcast channel (DM-BC) (\mathcal {X}, p(y_1,y_2\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }x), \mathcal {Y}_... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/tit.1978.1055892",
"end": 539,
"openalex_id": "https://openalex.org/W2144007657",
"raw": "T. M. Cover, “Broadcast channels,” IEEE Trans. Inf. Theory, vol. 18, no. 1, pp. 2–14, Jan. 1972.",
"source_ref_id": "0c2964958a9fb5c905b... | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.029075097292661667,
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0.02080281265079975,
-0.035836391150951385,
-0... | |
2402be49ad956697752e766f6cf70b93d66797cb | subsection | 24 | 63 | Optimal Achievable Rates for Broadcast Channels with Marton Coding | If there are more than one such pair of (l_1,l_2), choose one of them uniformly at random; otherwise, choose one uniformly at random from [2^{n{\hat{R}}_1}] \times [2^{n{\hat{R}}_2}].We refer to the random tuple \mathcal {C}_n = ((U_1^n(m_1,l_1) m_1 \in [2^{nR_1}], l_1 \in [2^{n{\hat{R}}_1}]), (U_2^n(m_2,l_2) m_2 \in [... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.016249656677246094,
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0.023726025596261024,
-0.021193215623497963,
-0.0... | |
b37fb207309321004b9d83a77c8fceb730a7e90a | subsection | 25 | 63 | Optimal Achievable Rates for Broadcast Channels with Marton Coding | Given pmf p=p(u_1,u_2) and function x(u_1,u_2), the optimal rate region {R}_\mathrm {BC}^*(p), when it exists, is defined as{R}_\mathrm {BC}^*(p) = \mathrm {cl}\left[ \bigcup _{\alpha \in [0 \; 1]} \lim _{\epsilon \rightarrow 0} {R}_\mathrm {BC}^*(p, \alpha , \epsilon ) \right].We are now ready to state main result of ... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.018405167385935783,
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0.0290728397667408,
-0.01417014840990305,
-0... | |
20934a1a0808bd97957d29a68d1e3e3f35d5de41 | subsection | 26 | 63 | Optimal Achievable Rates for Broadcast Channels with Marton Coding | For the converse, given a fixed pmf p=p(u_1,u_2), \alpha \in [0 \; 1], and \epsilon >0, we define the rate region {R}_\mathrm {BC}^{**}(p,\alpha ,\delta ) as the set of rate pairs (R_1,R_2) such thatR_1 &\le I(U_1; Y_1,U_2) - \alpha I(U_1;U_2) + \delta ,\\
R_1 &\le I(U_1,U_2;Y_1) - \min \lbrace R_2;I(U_2;Y_1,U_1)-\over... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04588473588228226,
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... | |
472d278620df5d6aa1c85a421afba7ebaecea893 | subsection | 27 | 63 | Optimal Achievable Rates for Broadcast Channels with Marton Coding | By Fano's inequality,H(M_j \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } Y_j^n, \mathcal {C}_n= \text{\footnotesize $\mathcal {C}$} _n) \le 1 + nR_j P_e^{(n)}(\text{\footnotesize $\mathcal {C}$} _n)\quad j=1,2.Taking the expectation over Marton random codebook \mathcal {C}_n, it follows thatH(M_j \mathchoice{{1... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1017/cbo9781139030687",
"end": 1005,
"openalex_id": "https://openalex.org/W4300840296",
"raw": "A. El Gamal and Y.-H. Kim, Network Information Theory. Cambridge: Cambridge University Press, 2011.",
"source_ref_id": "47d8c071334445c... | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.016619950532913208,
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-0.018588704988360405,
-0.01764248125255108... | |
a6a462aac3e978b0125c2ddf4c56540914299ff5 | subsection | 28 | 63 | Optimal Achievable Rates for Broadcast Channels with Marton Coding | For n sufficiently large, we havenR_1 &= H(M_1\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }M_2, \mathcal {C}_n)
\\ & \overset{(a)}{\le } I(M_1;Y_1^n\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } M_2, \mathcal {C}_n) + n \epsilon _n
\\ & \le I(M_1, {\tilde{E}}_n ;Y_1^n\mathchoice{{1mu}\vert {1mu}}{\vert }... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0.031128985807299614,... | |
f046181e65b6025ad0f7611fe49a5e94f498104c | subsection | 29 | 63 | Optimal Achievable Rates for Broadcast Channels with Marton Coding | Following arguments similar to (REF ), the first term in (REF ) can be bounded asI(M_1, M_2; Y_1^n\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }\mathcal {C}_n) &\le 1 + n(R_1+R_2) \operatorname{\textsf {P}}({\tilde{E}}_n = 0) + \sum _{i=1}^n I(M_1, M_2; Y_{1i} \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert ... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0... | |
95e6a6be50c18791fbb972e5a95664e40e91b773 | subsection | 30 | 63 | Optimal Achievable Rates for Broadcast Channels with Marton Coding | This lemma is a version of Lemma REF for Marton random code ensembles.Lemma 5
For every \epsilon ^{\prime } > \epsilon and for n sufficiently large,I(M_2; Y_1^n \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }\mathcal {C}_n) \ge n [ \min \lbrace R_2, I(U_2;Y_1,U_1) - \overline{\alpha }I(U_1;U_2), I(U_1,U_2;Y_1) \... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/tit.1979.1056046",
"end": 1657,
"openalex_id": "https://openalex.org/W1978188352",
"raw": "K. Marton, “A coding theorem for the discrete memoryless broadcast channel,” IEEE Trans. Inf. Theory, vol. 25, no. 3, pp. 306–311, 1979.",
... | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.0007098347996361554,
-0.025170769542455673,
... | |
fda5e78ef6acad445037b3e3b489ff7f0b09b871 | subsection | 31 | 63 | Discussion | For the linear computation problem, the outer bound on the optimal rate region presented in Section is valid for any computation, not only for natural computation. The inner bound presented in Theorem REF , however, matches with this outer bound only for natural computation. It is an interesting but difficult problem ... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.020777398720383644,
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0.006407127249985933,
-0.024133512750267982,... | |
a1b9814b2eee89dae58c8e37c5e839fe2fcdf25a | subsection | 32 | 63 | Discussion | If T = (X_1,X_2), (REF ) reduces to the rate region {R}_\mathrm {MAC}(p) in Section . Thus, we can conclude that this general outer bound recovers as extreme special cases the components of the outer bound in Theorem REF that was established for a random ensemble of homologous codes. Whether and when both outer bounds ... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
dd8ad743fb08a4800c3fe2f37b4a0abdf96e94f3 | subsection | 33 | 63 | Proof of Proposition | Fix pmf p = p(x_1)p(x_2). We first show that [{R}_{CF}(p) \cup {R}_\mathrm {MAC}(p) ] \subseteq {R}^*(p).
Suppose that the rate pair (R_1, R_2) \in {R}_{CF}(p). Then, for every j \in \lbrace 1,2\rbrace such that a_j \ne 0, the rate pair (R_1,R_2) satisfiesR_j &\le H(X_j) - H(W_{{\bf a}} \mathchoice{{1mu}\vert {1mu}}{\v... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0... | |
ba41c44049e528d5c8b7d743cc988db8c4d4c594 | subsection | 34 | 63 | Proof of Proposition | Then, (R_1,R_2) satisfiesR_j &\le I(X_1,X_2;Y) - I(X_{j^c};W_{{\bf a}},Y) \\
&= H(X_j) - H(W_{{\bf a}}\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }Y),for each j \in \lbrace 1,2\rbrace with a_j \ne 0. Then, (R_1,R_2) \in {R}_{CF}(p). It is easy to see that the rate pair (R_1,R_2) \in {R}^*(p) that satisfies R_{j... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.009405205957591534,
0... | |
67a98a1bb6e2cd1a2d1473db60ca393a93dfe249 | subsection | 35 | 63 | Body | Lemma 6
Let G be an nR \times n random matrix over \mathbb {F}_q with R < 1 where each element is drawn i.i.d. \mathrm {Unif}(\mathbb {F}_q). Then,\lim _{n \rightarrow \infty } n \operatorname{\textsf {P}}(G \textrm { is not full rank}) = 0.Probability of choosing nR linearly independent rows can be written as\operato... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0... | |
86374ca5bfcea4cb7ea0cc9edffb3ecdbebf38e5 | subsection | 36 | 63 | Proof of Lemma | Fix pmf p = p(x_1)p(x_2). We will show that if the condition in (REF ) holds, then {R}_{CF}(p) \cup {R}_1(p) \cup {R}_2(p) = {R}_{CF}(p) \cup {R}_\mathrm {MAC}(p). By definition of the rate regions {R}_1(p), {R}_2(p) and {R}_\mathrm {MAC}(p), it is easy to see that {R}_{CF}(p) \cup {R}_1(p) \cup {R}_2(p) \subseteq {R}_... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.004208182450383902,
-0.01392553560435772,
-0.04614884406328201,
-0.007420593872666359,
-0.... | |
1e532e4f564c9b6ef0bdd0de60a5ef64cfd38dee | subsection | 37 | 63 | Proof of Lemma | By condition (REF ), we haveI(X_{j^c};W_{{\bf a}},Y) &= I(X_1,X_2;Y) - H(X_j) + H(W_{{\bf a}}\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }Y) \\
&= I(X_1,X_2;Y) - H(X_j) + \min _{{\bf b}\ne 0} H(W_{{\bf b}}\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }Y) \\
&\le I(X_1,X_2;Y) - H(X_j) + \min _{{\bf b}\in \... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.0... | |
c27b876f64341a1e10c5d57e45a22ea6371fbebb | subsection | 38 | 63 | Proof of Lemma | Then,H(M_j\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }Y^n,M_{j^c},\mathcal {C}_n) = I(M_j; W^n_{{\bf a}} \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } Y^n, M_{j^c}, \mathcal {C}_n) + H(M_j\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }W^n_{{\bf a}},Y^n,M_{j^c},\mathcal {C}_n).To bound the first t... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
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d68b3bdd5e82a829bc5dab1a27b7d67bf1c52b71 | subsection | 39 | 63 | Proof of Lemma | Then,\operatorname{\textsf {P}}(X_i = x, Y_i = y \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } X^n \in {\mathcal {T}_{\epsilon }^{(n)}}(X)) &= \operatorname{\textsf {P}}(X_i = x\mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } X^n \in {\mathcal {T}_{\epsilon }^{(n)}}(X)) \operatorname{\textsf {P}}(Y_i = y \... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
94f9b495f581fe3bfab49aea3f31e0ecadb5a5aa | subsection | 40 | 63 | Proof of Lemma | Then, we have\operatorname{\textsf {P}}(U^n(L)=u^n ) &= \sum _{l} \sum _{\mathsf {G}} \operatorname{\textsf {P}}( L=l, G = \mathsf {G}, D^n = u^n \ominus l \mathsf {G})
\\
&\overset{(a)}{=} \sum _{l} \sum _{\mathsf {G}} \operatorname{\textsf {P}}( L=l, G = \sigma (\mathsf {G}), D^n = v^n \ominus l \sigma (\mathsf {G}))... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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ef5c7befa1772de945d644110cab06862484e7b0 | subsection | 41 | 63 | Proof of Lemma | Then, for every type \Theta within the set {\mathcal {T}_{\epsilon }^{(n)}}(X), we have\operatorname{\textsf {P}}(X_i = x \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } X^n \in {\mathcal {T}_{\epsilon }^{(n)}}(X,\Theta )) &= \sum _{x^n \in {\mathcal {T}_{\epsilon }^{(n)}}(X,\Theta ) \atop \textrm {s.t. } x_i = x... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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3389922b0432722bfc510e826e834192dfc3e665 | subsection | 42 | 63 | Proof of Lemma | Combining this observation with the fact that \Theta is the type of a typical sequence, we get(1-\epsilon ) p(x) \le \operatorname{\textsf {P}}(X_i = x \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } X^n \in {\mathcal {T}_{\epsilon }^{(n)}}(X,\Theta )) \le (1+\epsilon ) p(x), \quad \forall x \in \mathcal {X}.Sinc... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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960deedfa45998915177841165eca75fea3c21b7 | subsection | 43 | 63 | Proof of Lemma | First, by Lemma REF , we haveI(M_{j^c}; Y^n \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }\mathcal {C}_n) \ge I(M_{j^c}; W_{{\bf a}}^n, Y^n \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }\mathcal {C}_n) - n \epsilon _n.Therefore, it suffices to prove that for n sufficiently large,I(M_{j^c}; W_{{\bf a}}^n, ... | {
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Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
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82058f8cd0755a42f9d747f7ebf9c903acbee7ff | subsection | 44 | 63 | Proof of Lemma | By the symmetry of the codebook generation, for each m \in \mathbb {F}_q^{nR_{j^c}}, m \ne M_{j^c}, we have\operatorname{\textsf {P}}( m \in \mathcal {L},&\, E_n = 1)
\\
&= \operatorname{\textsf {P}}( m \in \mathcal {L}, E_n = 1 \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {M}_1, \mathcal {M}_2)
\\
&=... | {
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{
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"raw": "S. H. Lim, C. Feng, A. Pastore, B. Nazer, and M. Gastpar, “A joint typicality approach to algebraic network information theory,” IEEE Trans.... | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
5becd69fa5df2fdba6abd234e16a5b69705ccfd4 | subsection | 45 | 63 | Proof of Lemma | Since \operatorname{\textsf {P}}(E_n = 1) tends to one as n \rightarrow \infty , for n sufficiently large we have \operatorname{\textsf {P}}(E_n=1) \ge q^{-\epsilon }. Therefore, for n sufficiently large, the conditional probability is bounded as follows\operatorname{\textsf {P}}( m \in \mathcal {L}\mathchoice{{1mu}\ve... | {
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"source_ref_id": "47d8c071334445c... | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
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4af62175bb977478f8e54cfa6e4296c9a04048fe | subsection | 46 | 63 | Proof of Lemma | Then, for n sufficiently large, we haveH(M_{j^c} \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {C}_n, & W_{{\bf a}}^n, Y^n)
\\&= H(M_{j^c} \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {C}_n, W_{{\bf a}}^n, Y^n, E_n, F_n) + I ( M_{j^c}; E_n, F_n \mathchoice{{1mu}\vert {1mu}}{\vert }{\v... | {
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"source_ref_id": "309e3b9027fbe16e2c53... | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
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e540106f59cadea34a3924904472f076e340fe04 | subsection | 47 | 63 | Proof of Lemma | Substituting back givesI(M_{j^c}; W_{{\bf a}}^n, Y^n \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {C}_n) &= H(M_{j^c} \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {C}_n) - H(M_{j^c} \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {C}_n, W_{{\bf a}}^n, Y^n)
\\
&= nR_{j^... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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0.0124... | |
74efceb7155f4c5e0c66b426f9dd744476f8a96c | subsection | 48 | 63 | Proof of Lemma | First, by (the averaged version of) Fano's lemma in (REF ), we haveI(M_2; Y_1^n \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }\mathcal {C}_n) \ge I(M_2; M_1, Y_1^n \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {C}_n) - n \epsilon _n.Therefore, it suffices to prove that for n sufficiently large,I... | {
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Homologous Codes | [
"Pinar Sen",
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"Young-Han Kim"
] | [
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805e0c18ec9deb87a09bbe30f4924125763cc81e | subsection | 49 | 63 | Proof of Lemma | By the symmetry of the codebook generation, for each m_2 \ne M_2 \in [2^{nR_2}] we start with& \operatorname{\textsf {P}}( m_2 \in \mathcal {L}, {\tilde{E}}_n = 1) \\ &= \operatorname{\textsf {P}}( m_2 \in \mathcal {L}, {\tilde{E}}_n=1 \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {M}_1,\mathcal {M}_2)... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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a21dba77f6217d604779dfb36b22808d27e3cca7 | subsection | 50 | 63 | Proof of Lemma | Two summation terms on the right hand side of (REF ) can be bounded using techniques similar to those in the achievability proof (see Appendix ) for Theorem REF to get\operatorname{\textsf {P}}( m_2 \in \mathcal {L}, {\tilde{E}}_n = 1) \le 2^{-n(I(U_2;Y_1,U_1)-\overline{\alpha }I(U_1;U_2) - 4\delta (\epsilon ^{\prime }... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
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7139380fa253a7cce617a057eaddfd024fcc9e9f | subsection | 51 | 63 | Proof of Lemma | Since \epsilon ^{\prime } > \epsilon and \operatorname{\textsf {P}}({\tilde{E}}_n=1) tends to one as n \rightarrow \infty , by the conditional typicality lemma in , \operatorname{\textsf {P}}({\tilde{F}}_n=1) tends to one as n \rightarrow \infty . | {
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Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
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64bab40a65cc3f6edc222588a13f0d2748a27ae1 | subsection | 52 | 63 | Proof of Lemma | Then, for n sufficiently large, we haveH(M_2 \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {C}_n, M_1, Y_1^n)
&= H(M_2 \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {C}_n, M_1, Y_1^n, {\tilde{E}}_n,{\tilde{F}}_n) + I ( M_2; {\tilde{E}}_n,{\tilde{F}}_n \mathchoice{{1mu}\vert {1mu}}{\ver... | {
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Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
"cs.IT",
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e2ffe5d68cd5427ce61dbc1b6b150d7de26fbbf2 | subsection | 53 | 63 | Proof of Lemma | Substituting back givesI(M_2; M_1, Y_1^n \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {C}_n)
&= H(M_2 \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {C}_n) - H(M_2 \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } \mathcal {C}_n, M_1, Y_1^n)
\\
&= nR_2 - H(M_2 \mathchoice{{1mu}\ver... | {
"cite_spans": []
} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
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12b2947db02b50c4d50e95fda1b0f7b559c43d46 | subsection | 54 | 63 | Proof of Achievability for Theorem | Let \alpha \in [0 \; 1] and \epsilon >0. Consider an (n,nR_1,nR_2;p,\alpha ,\epsilon ) Marton random code ensemble. We use the nonunique simultaneous joint typicality decoding rule in to establish the achievability. Let \epsilon ^{\prime } > \epsilon . Upon receiving y_j^n at receiver j=1,2, the \epsilon ^{\prime }-jo... | {
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Homologous Codes | [
"Pinar Sen",
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"Young-Han Kim"
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1babe75b8af584b4d168cd2cde6d67256edbe635 | subsection | 55 | 63 | Proof of Achievability for Theorem | It suffices to consider decoder 1, which declares an error if one or more of the following events occur\mathcal {E}_0 &= \lbrace (U_1^n(M_1,l_1),U_2^n(M_2,l_2)) \notin {\mathcal {T}_{\epsilon }^{(n)}}(U_1,U_2) \textrm { for every } (l_1,l_2) \in [2^{n{\hat{R}}_1}] \times [2^{n{\hat{R}}_2}]\rbrace , \\
\mathcal {E}_1 &=... | {
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Homologous Codes | [
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a70107a0e3ce4095c49f2810e7076e7074ffa79b | subsection | 56 | 63 | Proof of Achievability for Theorem | First, by the symmetric codebook generation,\operatorname{\textsf {P}}(\mathcal {E}_{2} \cap \mathcal {E}_0^c) &\le \operatorname{\textsf {P}}(\mathcal {E}_2) \\
&= \operatorname{\textsf {P}}(\mathcal {E}_2 \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert } M_1=M_2=1) \\
&\le \operatorname{\textsf {P}}( (U_1^n(m_1,l... | {
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Homologous Codes | [
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eab9ae1aaafaf5e3b1e52e4643bf9d1170e9d7d9 | subsection | 57 | 63 | Proof of Achievability for Theorem | Letting {\hat{R}}_1 = \alpha (I(U_1;U_2) + 10\epsilon H(U_1,U_2)), we haveR_1 \le \max \lbrace 0, I(U_1;Y_1) - \alpha I(U_1;U_2) - 2\delta (\epsilon ^{\prime })\rbrace .Secondly, we can decompose the event \mathcal {E}_2 = \mathcal {E}_{21} \cup \mathcal {E}_{22} such that\mathcal {E}_{21} &= \lbrace (U_1^n(m_1,l_1),U_... | {
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} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
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46defe743e30cc668eccd7bbe18b20fa8bd33b2d | subsection | 58 | 63 | Proof of Achievability for Theorem | Substituting {\hat{R}}_1 + {\hat{R}}_2 = I(U_1;U_2) + 10 \epsilon H(U_1,U_2), it follows that \operatorname{\textsf {P}}(\mathcal {E}_{22}) tends to zero as n \rightarrow \infty if R_1 + R_2 \le I(U_1,U_2;Y_1)- 3\delta (\epsilon ^{\prime }).We next bound the probability \operatorname{\textsf {P}}(\mathcal {E}_{21} \cap... | {
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} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
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7cbc33e34e1ac41edfe35582d078bea3608ce5d7 | subsection | 59 | 63 | Proof of Achievability for Theorem | By the symmetric codebook generation,\operatorname{\textsf {P}}(\mathcal {E}_{21} \cap \mathcal {E}_0^c) = \operatorname{\textsf {P}}(\mathcal {E}_{21} \cap \mathcal {E}_0^c \mathchoice{{1mu}\vert {1mu}}{\vert }{\vert }{\vert }\mathcal {M}_1,\mathcal {M}_2 ),which can be bounded as&\operatorname{\textsf {P}}(\mathcal {... | {
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Homologous Codes | [
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94f6cf89a3898931a7515b6c0377b96edde927bc | subsection | 60 | 63 | Proof of Achievability for Theorem | Letting {\hat{R}}_1 = \alpha (I(U_1;U_2) + 10\epsilon H(U_1,U_2)) and {\hat{R}}_2 = \overline{\alpha }(I(U_1;U_2) + 10\epsilon H(U_1,U_2)) results in R_1 \le I(U_1;Y_1,U_2) - \alpha I(U_1;U_2) - 4\delta (\epsilon ^{\prime }) and R_1 \le I(U_1,U_2;Y_1)-4\delta (\epsilon ^{\prime }).Combining with (REF ), the probability... | {
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} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
"Sung Hoon Lim",
"Young-Han Kim"
] | [
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a35e6195095b6f34784290ed6509e7c0b9cc0135 | subsection | 61 | 63 | Proof of Achievability for Theorem | Taking \epsilon \rightarrow 0 and then taking the closure implies{R}_{\mathrm {BC},1}(p,\alpha ) \cap {R}_{\mathrm {BC},2}(p,\alpha ) \; \subseteq \; {R}_\mathrm {BC}^*(p,\alpha ).The achievability proof follows from the next lemma that provides an equivalent characterization for the rate region in Theorem REF .Lemma 1... | {
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} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
"Pinar Sen",
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] | [
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7435d9eddbe7fa6bf97d5a9e65da092e85f3acfa | subsection | 62 | 63 | Proof of Achievability for Theorem | If instead R_2 \ge \min \lbrace I(U_2;Y_1,U_1)-\overline{\alpha }I(U_1;U_2), I(U_1,U_2;Y_1)\rbrace , thenR_1 &\le I(U_1;Y_1,U_2) - \alpha I(U_1;U_2),\\
R_1 &\le I(U_1,U_2;Y_1) - \min \lbrace I(U_2;Y_1,U_1)-\overline{\alpha }I(U_1;U_2), I(U_1,U_2;Y_1)\rbrace = \max \lbrace 0, I(U_1;Y_1) - \alpha I(U_1;U_2) \rbrace .Ther... | {
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} | 1805.03338 | On the Optimal Achievable Rates for Linear Computation With Random
Homologous Codes | [
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f68c719d8cc76cb58a723b3b79513cf19d96ec31 | abstract | 0 | 39 | Abstract | Marginal structural models (MSMs) allow for causal analysis of longitudinal
data. The MSMs were originally developed as discrete time models. Recently,
continuous-time MSMs were presented as a conceptually appealing alternative for
survival analysis. In applied analyses, it is often assumed that the
theoretical treatme... | {
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} | 1802.01946 | The additive hazard estimator is consistent for continuous-time marginal
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8e2bcd87fdadadb001b53a1fa75b7b21f300aae5 | subsection | 1 | 39 | Outline | MSMs can be used to obtain causal effect estimates in the presence of confounders, which e.g. may be time-dependent . The procedure is particularly appealing because it allows for a sharp distinction between confounder adjustment and model selection : first, we adjust for observed confounders by weighing the observed d... | {
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"raw": "James M Robins, Miguel Angel Hernan, and Babette Brumback. Marginal structural models and causal inference in epidemiology, 2000.",
... | 1802.01946 | The additive hazard estimator is consistent for continuous-time marginal
structural models | [
"Pål Christie Ryalen",
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a68c50497c0fa67bdb316934dfb357ef05823c31 | subsection | 2 | 39 | Motivation | We will present a strategy for dealing with confounding and dependent censoring in continuous time. Confounding, which may be time-varying, will often be a problem when analysing observational data, e.g. coming from health registries. The underlying goal is to assess the effect a treatment strategy has on an outcome.We... | {
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2a878521c035fd85656512bb470e6b4a62d74c26 | subsection | 3 | 39 | Hypothetical scenarios and
likelihood ratios | We consider observational event-history data where n i.i.d. subjects are followed over the study period [0,T]. Let N^{i,A} \text{ and } N^{i,D} respectively be counting processes that jump when treatment A and outcome D of interest occur for subject i. Furthermore, let Y^{i,A},Y^{i,D} be the at-risk processes for A and... | {
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structural models | [
"Pål Christie Ryalen",
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b2cc4f67597eb40b54026450de1d195efb71f4f3 | subsection | 4 | 39 | Hypothetical scenarios and
likelihood ratios | We stress that we consider the intensity process marginalized over \mathcal {L}, and thereby it is defined with respect to \mathcal {F}_t ^{i, \mathcal {V}_0}, and not \mathcal {F}_t ^{i, \mathcal {V}_0 \cup \mathcal {L}}. In other words, we assume that the hazard for event D with respect to the filtration \mathcal {F}... | {
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"Pål Christie Ryalen",
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96c73ad67252c045707122de23c4befd7952c0d5 | subsection | 5 | 39 | Re-weighted additive hazard regression | Our main goal is to estimate the cumulative coefficient function in (REF ), i.e.B_t := \int _0^t b_s dsfrom the observational data distributed according to P = P^1 \otimes \dots \otimes P^n. If we had known all the true likelihood ratios, we could try to estimate (REF ) by re-weighting each individual in Aalen's additi... | {
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20f1571f5c06d0c830cdcf232b19f5eee31d67d7 | subsection | 6 | 39 | Parameters that are transformations of cumulative hazards | It has recently been emphasised that the common interpretation of hazards in survival analysis as the causal risk of death during (t, t + \Delta ] for an individual that is alive at t, is often not appropriate; see e.g. . An example in shows that this can also be a problem in RCTs; if N is a counting process that jumps... | {
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a2f0c10bfb3e8e94ac897864bc4f101bf47e7653 | subsection | 7 | 39 | Parameters that are transformations of cumulative hazards | Nevertheless, some commonly studied parameters cannot be written on the form (REF ), such as the median survival, and the hazard ratio.In we showed that \eta ^{(n)} provides a consistent estimator of \eta if\lim _{ n \rightarrow \infty } P ( \sup _{t\le T} | B_t^{(n)} - B_t | \ge \epsilon ) = 0 for every \epsilon >0, i... | {
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2ffb1558369453bf962b4f12381fa094fb0b2f76 | subsection | 8 | 39 | Consistency and P-UT property | The consistency and P-UT property of B^{(n)} introduced in Section REF is stated as a Theorem below. A proof can be found in the Appendix.Theorem 1 (Consistency of weighted additive hazard regression)
Suppose thatThe conditional density of R^{(i,n)}_t
given { \mathcal {F}_t^{i, \mathcal {V}_0 \cup \mathcal {L} } }
doe... | {
"cite_spans": []
} | 1802.01946 | The additive hazard estimator is consistent for continuous-time marginal
structural models | [
"Pål Christie Ryalen",
"Mats Julius Stensrud",
"Kjetil Røysland"
] | [
"stat.ME"
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a6d1149cffefedc76c0190ae64ee8653b09dad93 | subsection | 9 | 39 | Consistency and P-UT property | Condition \ref {eq:consistW}) states that the weight estimator converges to the theoretical weights R_t^i, in a not very strong sense. The uniform boundedness of \lbrace R^i \rbrace _i is a positivity condition similar to the positivity condition required for standard inverse probability weighting. | {
"cite_spans": []
} | 1802.01946 | The additive hazard estimator is consistent for continuous-time marginal
structural models | [
"Pål Christie Ryalen",
"Mats Julius Stensrud",
"Kjetil Røysland"
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"stat.ME"
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82f0e7522f447bf5f111ff2f3c2b67880eb78020 | subsection | 10 | 39 | Causal validity and a consistent estimator the individual likelihood ratios | We can model the individual likelihood ratio in many settings where the underlying model is causal. To do this, we assume that each subject is represented by the outcomes of r baseline variables Q_1, \dots Q_r, and d counting processes N^1, \dots , N^d. Moreover, we let \mathcal {F}_t denote the filtration that is gene... | {
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{
"arxiv_id": "",
"doi": "10.1007/440.1432-2064",
"end": 820,
"openalex_id": "https://openalex.org/W4236747242",
"raw": "J. Jacod. Multivariate point processes: Predictable projection, radon-nikodym derivatives, representation of martingales. Probability Theory and ... | 1802.01946 | The additive hazard estimator is consistent for continuous-time marginal
structural models | [
"Pål Christie Ryalen",
"Mats Julius Stensrud",
"Kjetil Røysland"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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4eac59fdc0b32b88671d9616a451e6ad36f8ac89 | subsection | 11 | 39 | Causal validity and a consistent estimator the individual likelihood ratios | If the intervention is aimed at N^j, changing the intensity from \lambda ^j to \tilde{\lambda }^j, then the likelihood ratio takes the formR_t = \big ( \prod _{s \le t} \theta _s^{\Delta N_s^j}\big ) \exp \big ( \int _0^t \lambda _s^j -
\tilde{\lambda }_s^j ds \big ),where \theta _t := \frac{\tilde{\lambda }_t^j }{\lam... | {
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{
"arxiv_id": "",
"doi": "10.3150/10-bej303",
"end": 342,
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"raw": "K. R\\text{ø}ysland. A martingale approach to continuous-time marginal structural models. Bernoulli, 2011.",
"source_ref_id": "393a1ebd7fdbc8a5... | 1802.01946 | The additive hazard estimator is consistent for continuous-time marginal
structural models | [
"Pål Christie Ryalen",
"Mats Julius Stensrud",
"Kjetil Røysland"
] | [
"stat.ME"
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24f3ba38272d9ee0252b371af3da22e1d0edf5a8 | subsection | 12 | 39 | Estimation of weights using additive hazard regression | Suppose we have a causal model as described in the beginning of Section , allowing us to obtain a known form of the likelihood ratio R^i. To model the hypothetical scenario, we need to rely on estimates of the likelihood ratio. In the following, we will only focus on a causal model where we replace the intensity of tre... | {
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structural models | [
"Pål Christie Ryalen",
"Mats Julius Stensrud",
"Kjetil Røysland"
] | [
"stat.ME"
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2458cd469d3718fef93b725dbf3000ea64faeb2e | subsection | 13 | 39 | Estimation of weights using additive hazard regression | To proceed, we assume that \lambda ^{i,A} and \tilde{\lambda }^{i,A} satisfy the additive hazard model, i.e. that
there are vector valued functions h_t and \tilde{h}_t, and covariate processes Z_t and \tilde{Z}_t that are adapted to \mathcal {F}_t^{i,\mathcal {V}_0 \cup \mathcal {L}} and \mathcal {F}_t^{i,\mathcal {V}_... | {
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structural models | [
"Pål Christie Ryalen",
"Mats Julius Stensrud",
"Kjetil Røysland"
] | [
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92a9b311696e40146595dfaf08a3e42769e0118f | subsection | 14 | 39 | Estimation of weights using additive hazard regression | Our candidate for \theta ^{(i,n)}_t, when t > 0, depends on the choice of an increasing sequence \lbrace \kappa _n\rbrace _n with \lim \limits _{n\longrightarrow \infty } \kappa _n = \infty such that \sup _{n} \frac{\kappa _n }{\sqrt{n} } < \infty . This estimator takes the form\theta ^{(i,n)}_t &= {\left\lbrace \begin... | {
"cite_spans": []
} | 1802.01946 | The additive hazard estimator is consistent for continuous-time marginal
structural models | [
"Pål Christie Ryalen",
"Mats Julius Stensrud",
"Kjetil Røysland"
] | [
"stat.ME"
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69ba7d2d1a6b3358bd7ed718397d8276806ed42f | subsection | 15 | 39 | Estimation of weights using additive hazard regression | For each i,
\lim _{ \delta \rightarrow 0} P \big ( \inf \limits _{t \le T} |
\tilde{Z}_t^{i\intercal } \tilde{h}_t | \le \delta \big ) =
0,
E\big [\sup \limits _{s \le T} |Z_{s}^i|^3_3 \big ] < \infty and
E\big [\sup \limits _{s \le T} |\tilde{Z}_{s}^i|^3_3 \big ] < \infty for every i
\lim _{a \rightarrow \infty... | {
"cite_spans": []
} | 1802.01946 | The additive hazard estimator is consistent for continuous-time marginal
structural models | [
"Pål Christie Ryalen",
"Mats Julius Stensrud",
"Kjetil Røysland"
] | [
"stat.ME"
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2fb1d8e6e07e04063603ebc44432dedca9633909 | subsection | 16 | 39 | Software | We have developed R software for estimation of continuous-time MSMs that solve ordinary differential equations, in which additive hazard models are used to model both the time to treatment and the time to the outcome of interest. Our procedure involves two steps: first, we estimate continuous-time weights using fitted ... | {
"cite_spans": []
} | 1802.01946 | The additive hazard estimator is consistent for continuous-time marginal
structural models | [
"Pål Christie Ryalen",
"Mats Julius Stensrud",
"Kjetil Røysland"
] | [
"stat.ME"
] | 2,018 | en | Statistics | [
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