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cb0c3f1b880b602639f44446ce2d504b6fbfcdc9 | abstract | 0 | 42 | Abstract | Let $E$ be a Moran set on $\mathbb{R}^1$ associated with a closed interval
$J$ and two sequences $(n_k)_{k=1}^\infty$ and
$(\mathcal{C}_k=(c_{k,j})_{j=1}^{n_k})_{k\geq1}$. Let $\mu$ be the infinite
product measure (Moran measure) on $E$ associated with a sequence
$(\mathcal{P}_k)_{k\geq1}$ of positive probability vecto... | {
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b203da03cad77dcbc9865e38b0f25352a648b7c0 | abstract | 1 | 42 | Abstract | For every $a\in\alpha_n$, we write
$I_a(\alpha,\mu):=\int_{P_a(\alpha_n)}d(x,\alpha_n)^rd\mu(x)$ and \[
\underline{J}(\alpha_n,\mu):=\min_{a\in\alpha_n}I_a(\alpha,\mu),\;
\overline{J}(\alpha_n,\mu):=\max_{a\in\alpha_n}I_a(\alpha,\mu). \] We show that
$\underline{J}(\alpha_n,\mu),\overline{J}(\alpha_n,\mu)$ and
$e^r_{n,... | {
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8b1a459be673f35be2ae6ae55b821be07c42e17a | subsection | 2 | 42 | Introduction | The quantization problem for probability measures has a deep background in information theory and engineering technology such as signal processing and data compression , , . Mathematically, this problem consists in the approximation of a given probability measure with discrete probability measures of finite support in ... | {
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4062f09e12ece510e5f173d7ac6e04d88a1778ed | subsection | 3 | 42 | Basic definitions and some known results | Let \nu be a Borel probability measure on \mathbb {R}^q. For x,y\in \mathbb {R}^q, we denote by d(x,y) the distance between x and y induced by a norm |\cdot | on \mathbb {R}^q, and for a subset \alpha of \mathbb {R}^q, let d(x,\alpha ):=\inf _{a\in \alpha }d(x,a). Set \mathcal {D}_{n}:=\lbrace \alpha \subset \mathbb {R... | {
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340715d672322c21f87589de781e2100b89d1757 | subsection | 4 | 42 | Basic definitions and some known results | Recall that the upper quantization dimension \overline{D}_{r}(\nu ) and the lower one \underline{D}_{r}(\nu ) for \nu of order r are defined by\overline{D}_{r}(\nu ):=\limsup _{n\rightarrow \infty }\frac{\log n}{-\log e_{n,r}(\nu )};\;\;\underline{D}_{r}(\nu ):=\liminf _{n\rightarrow \infty }\frac{\log n}{-\log e_{n,r}... | {
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09b2e595db82ed55279dce0369f893a94f44d273 | subsection | 5 | 42 | Basic definitions and some known results | Let t_i be the contraction ratio of f_i, 1\le i\le N, and s_r the unique solution of the equation\sum _{i=1}^N(p_is_i^r)^{\frac{s_r}{s_r+r}}=1.Assuming the OSC for (f_i)_{i=1}^N, Graf and Luschgy proved that\overline{D}_{r}(P)=\underline{D}_{r}(P)=s_r,\;0<\overline{Q}_r^{s}(P)\le \overline{Q}_r^{s}(P)<\infty .The above... | {
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d6dd2a507167acd01db03b8173729af1c3cdc616 | subsection | 6 | 42 | Asymptotic uniformity of the quantization error | A significant concern in quantization theory is, how much contribution each point of an n-optimal set make to the nth quantization error. This is closely connected with a famous conjecture of Gersho . In the study of this concern, Voronoi partitions play a crucial role. Recall that a Voronoi partition with respect to a... | {
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721daf5d3a0a96a5d26dc161a59b6a4cfc18650a | subsection | 7 | 42 | Asymptotic uniformity of the quantization error | Without the SSC, it turns out to be rather difficult to examine whether (REF ) holds or not. The main obstacle lies in the characterizations for Voronoi partitions with respect to n-optimal sets. Due to the lack of "gaps" among cylinder sets, the three-step procedure by means of partitioning, covering and packing, as d... | {
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27b042f5333a97421392b7e55739675e06e3b3eb | subsection | 8 | 42 | Statement of the main result | Let (n_k)_{k=1}^\infty be a sequence of integers with n_k\ge 2. For k\ge 1, let \mathcal {S}_k=(c_{k,j})_{j=1}^{n_k}, be a finite sequence of numbers such that\min _{1\le j\le n_k}c_{k,j}>0,\;c_{k,1}\cdots +c_{k,n_k}\le 1.We denote by \theta the empty word and set \Omega _0:=\lbrace \theta \rbrace . Write\Omega _k:=\lb... | {
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2db4ba6bd7562232a9ac8fa0fb8c45cee19f3690 | subsection | 9 | 42 | Statement of the main result | Then we have|J_\sigma |=c_\sigma :=c_{1,\sigma _1}\cdots c_{k,\sigma _k},\;{\rm for}\;\sigma =(\sigma _1,\ldots ,\sigma _k)\in \Omega _k,\;k\ge 1.Now let \Omega _k,k\ge 1, be endowed with discrete topology and \Omega _\infty be endowed with the corresponding product topology. For every k\ge 1, let (p_{k,j})_{j=1}^{n_k}... | {
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d89cb960b7a899497896052bbb6c4eb749629ce9 | subsection | 10 | 42 | Statement of the main result | Then we have\underline{J}(\alpha _n,\mu ),\;\overline{J}(\alpha _n,\mu ),\;e^r_{n,r}(\mu )-e^r_{n+1,r}(\mu )\asymp \frac{1}{n}e^r_{n,r}(\mu ).For the proof of Theorem REF , we will consider some auxiliary measures \nu _\sigma by pushing forward and pulling back the conditional measures of \mu on the cylinder sets J_\si... | {
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45266626edd04aa940b54ba51ec22e1a9e23511e | subsection | 11 | 42 | Preliminaries | For \sigma \in \Omega ^*, we write \sigma ^-:=\sigma |_{|\sigma |-1} if |\sigma |>1 and \sigma ^-=\theta if |\sigma |=1. We write \sigma \prec \omega if |\sigma |\le |\omega | and \sigma =\omega |_{|\sigma |}. Two words \sigma ,\omega \in \Omega ^* are called incomparable if neither \sigma \prec \omega nor \omega \prec... | {
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56dc551f28ddb39507c41499627e7ba841c6d002 | subsection | 12 | 42 | Preliminaries | Indeed, for every pair \sigma ,\tau \in \Lambda _{k,r}, we have\eta _r\mathcal {E}(\tau )\le \mathcal {E}(\sigma )\le \eta _r^{-1}\mathcal {E}(\tau ),\;\;{\rm implying}\;\;\mathcal {E}(\sigma )\asymp \mathcal {E}(\tau ).Using the assumption (REF ) and the arguments in the proof for , one can see that, there exists an i... | {
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c824dba1ed9543a93bf4b0e66c4d476f276b780a | subsection | 13 | 42 | Preliminaries | Then we haveK_\sigma \subset g_\sigma ^{-1}(J_\sigma )\;\;{\rm and}\;\;|K_\sigma |\le 1.Remark 2.2 One can see that \nu _\sigma is an amplification for \mu (\cdot |J_\sigma ). It will allow us to connect the integrals over J_\sigma with \mathcal {E}(\sigma ), while for suitably chosen k (cf. (REF )) and every \sigma \i... | {
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8bf4526a4c8e9af9170d9caf8f5f5881eff9ad59 | subsection | 14 | 42 | Preliminaries | Thene^r_{l,r}(\nu )-e^r_{l+1,r}(\nu )\le 3^r|K_\nu |^r(l+1)^{-1},\;l\ge 1.Let \beta _{l+1}\in C_{l+1,r}(\nu ) and let \lbrace P_b(\beta _{l+1})\rbrace _{b\in \beta _{l+1}} be a Voronoi partition with respect to \beta _{l+1}. There exists some b_0\in \beta _{l+1} with \nu (P_{b_0}(\beta _{l+1}))\le (l+1)^{-1}. We set \g... | {
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4d8690877aeadc5d844735c6a880454a0ba8be74 | subsection | 15 | 42 | Preliminaries | Let l\ge 1 and L\ge 1. We writeI_\rho (\beta ,\mu ):=\int _{J_\rho } d(x,\beta )^rd\mu (x),\;\rho \in \Omega ^*;\;\;\Psi _{l,L}:=\prod _{h=l+1}^{l+L}\Omega _h.Using Lemmas REF and REF , we are able to choose some constants which will be used in the characterization for the optimal sets. We haveLemma 2.6
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ec8f2db2da27fc4d0a417aa34d88da7f3c213a80 | subsection | 16 | 42 | Preliminaries | By the construction of E, there exists a \tau _0\in \Psi _{|\sigma |+5} such thatd(J_{\sigma \ast \omega \ast \tau _0},J_{\sigma \ast \omega }^c)\ge \underline{c}^5|J_\sigma |=\underline{c}^5c_\sigma ,\;
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0.0... | |
188aff8be4fb96e40e5883529679348832248c4e | subsection | 17 | 42 | Preliminaries | By Lemma REF , for M_1 as chosen in (1), there exists a number \zeta _{M_1,r}>0 depending on C and t, such thate_{M_1+1,r}^r(\nu _\tau )-e_{M_1+2,r}^r(\nu _\tau )>\zeta _{M_1,r}.Hence, by Lemma REF , for A:=3^r(M_1+6)([\zeta _{M_1,r}^{-1}\eta _r^{-1}]+1), and l\ge A,e^r_{l,r}(\nu _\sigma )-e^r_{l+M_1+6,r}(\nu _\sigma )... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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... | |
2b7ac84786b9b02b4735f4a2c1feec669beda0e2 | subsection | 18 | 42 | A characterization of the | Let M_i, i=1,2,3, be the integers as chosen in section 2. For every n\ge (M_2+2)\phi _{1,r}, there exists a unique integer k such that(M_2+2)\phi _{k,r}\le n<(M_2+2)\phi _{k+1,r}.In the remaining part of the paper, we always assume that n\ge (M_2+2)\phi _{1,r} and let k be the integer as chosen in (REF ). In this secti... | {
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{
"arxiv_id": "",
"doi": "10.1007/bfb0103945",
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"raw": "S. Graf and H. Luschgy, Foundations of quantization for probability dributions. Lecture Notes in Math. Vol. 1730, Springer-Verlag, 2000.",
"so... | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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... | |
ce1b646d8dcf06b9ac241f8e18a707f04adedffa | subsection | 19 | 42 | A characterization of the | If \widetilde{N}_\sigma \ge 1, then we have
I_\sigma (\beta ,\mu )\ge I_\sigma (\gamma ,\mu )=\mathcal {E}(\sigma )\int d(x,g_\sigma ^{-1}(\gamma ))^rd\nu _\sigma (x)\ge \mathcal {E}(\sigma )e^r_{\widetilde{N}_\sigma +2,r}(\nu _\sigma ).
If \widetilde{N}_\sigma =0, then we have \beta \subset J_\sigma ^c. By Lemma REF... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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0.03564012795686722,
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-0.060691967606544495,
... | |
0a8cba9fe50ed8dbc30940e1aa0e9fe071a7834e | subsection | 20 | 42 | A characterization of the | From the above analysis, we obtain{\rm card}\bigg (\alpha _n\setminus \bigcup _{\sigma \in \Lambda _{k,r}}J_\sigma \bigg )\le 2\phi _{k,r}.Using this and (REF ), we deduce{\rm card}\bigg (\alpha _n\cap \bigcup _{\sigma \in \Lambda _{k,r}}J_\sigma \bigg )\ge (M_2+2)\phi _{k,r}-2\phi _{k,r}=M_2\phi _{k,r}.Suppose L_\sigm... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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-0.... | |
24af7f065aa0284b1b67582c30425fecaee9e5da | subsection | 21 | 42 | A characterization of the | Then by Remark REF (i) and (iii), we haveI_\sigma (\alpha _n,\mu )\ge \mathcal {E}(\sigma )\max \lbrace e^r_{L_\sigma +2,r}(\nu _\sigma ),e^r_{M_1+1,r}(\nu _\sigma )\rbrace \ge \mathcal {E}(\sigma )e^r_{M_1+1,r}(\nu _\sigma ).This, together with the definition of \beta , yieldsI_\sigma (\alpha _n,\mu )-I_\sigma (\beta ... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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0.0... | |
c7f4cbbeaad8cc4c9ed9463a78aca858f410a6ac | subsection | 22 | 42 | A characterization of the | This will be used to give an upper estimate for \overline{J}(\alpha _n,\mu ).Lemma 3.3
For every \sigma \in \Lambda _{k,r} and \omega \in \Psi _{|\sigma |,3}, we have L_{\sigma \ast \omega }\ge 1.Suppose that L_{\sigma \ast \omega }=0 for some \sigma \in \Lambda _{k,r} and \omega \in \Psi _{|\sigma |,3}. We deduce a c... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.009866282343864441,
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-0... | |
c4b6c8b4f1fefbcc8fe317f5ce57c53f0e5a0532 | subsection | 23 | 42 | A characterization of the | Hence, by Remark REF (i), (ii),I_\sigma (\beta ,\mu )&\le &\int _{J_\sigma }d(x,\gamma _{L_\sigma -2}(\sigma ))^rd\mu (x)\\&=&p_\sigma \int _{J_\sigma } d(x,\gamma _{L_\sigma -2}(\sigma ))^rd\nu _\sigma \circ g_\sigma ^{-1}(x)\\&=&p_\sigma c_\sigma ^r\int d(x,g_\sigma ^{-1}(\gamma _{L_\sigma -2}(\sigma )))^rd\nu _\sigm... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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0.02... | |
d7ce9cf5493f41218659452afc7d61897a9ab8e9 | subsection | 24 | 42 | A characterization of the | It follows that\sum _{\rho \in \Lambda _{k,r},\rho \ne \sigma ,\tau }I_\tau (\beta ,\mu )\le \sum _{\rho \in \Lambda _{k,r},\rho \ne \sigma ,\tau }I_\tau (\alpha _n,\mu ).For x\in J_\sigma , we have d(x,\beta )\le d(x,\gamma _{L_\sigma -7}(\sigma )). By Remark REF (ii),(iii), we deduceI_\sigma (\beta ,\mu )-I_\sigma (\... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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0.02... | |
e894e44b24941439d52114cd38a292f1f40645ae | subsection | 25 | 42 | Proof of Theorem | Let n and k satisfy (REF ) and \alpha _n\in C_{n,r}(\mu ). By Lemma REF , for every \sigma \in \Lambda _{k,r} and \omega \in \Psi _{|\sigma |,3}, we have \alpha _n\cap J_{\sigma \ast \omega }\ne \emptyset . This implies that, for every a\in \alpha _n, we haveS_a:={\rm card}(\lbrace \sigma \in \Lambda _{k,r}:P_a(\alpha ... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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4cbcc7d6d0a52497b8af49518856abf67220cb88 | subsection | 26 | 42 | Proof of Theorem | By Lemmas REF and REF ,2M_1-2\le T_a:={\rm card}(\alpha _n\cap [\xi _2(\sigma ),\zeta _2(\tau )])={\rm card}(\alpha _n\cap G_a)\le 2M_3.Let g_\sigma be an arbitrary similitude of similarity ratio c_\sigma on \mathbb {R}^1. We define\lambda _a:=\mu (\cdot |G_a)\circ g_\sigma ,\; {\rm implying}\;\;\mu (\cdot |G_a)=\lambd... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0103945",
"end": 1326,
"openalex_id": "https://openalex.org/W1576475658",
"raw": "S. Graf and H. Luschgy, Foundations of quantization for probability dributions. Lecture Notes in Math. Vol. 1730, Springer-Verlag, 2000.",
"s... | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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78f92888d43e11a4875b374392632457e3f9474d | subsection | 27 | 42 | Proof of Theorem | Hence,\mu (G_a)=\mu (G_{a,\sigma })+\mu (G_{a,\tau }).Besides , the next lemma will be crucial for our lower estimate for \underline{J}(\alpha _n,\mu ). It is a variation of Proposition 12.12 in .Lemma 4.2 (see )
Let \nu be a Borel probability measure on \mathbb {R}^q with compact support K_\nu . Assume that there exis... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0103945",
"end": 152,
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"raw": "S. Graf and H. Luschgy, Foundations of quantization for probability dributions. Lecture Notes in Math. Vol. 1730, Springer-Verlag, 2000.",
"so... | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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d9ac3f48a31fbc48f7df560fa4550cc8b5bd821d | subsection | 28 | 42 | Proof of Theorem | Thus\mu (G_{a,\sigma })\ge (1-\overline{p}^3)p_\sigma ,\;\mu (G_{a,\tau })\ge (1-\overline{p}^3)p_\tau .Using (REF ), (REF ) and the definition of \lambda _{a}, we deduce\lambda _a(B(x,\epsilon ))&=&\mu (\cdot |G_a)\circ g_\sigma (B(x,\epsilon ))\\
&=&\frac{1}{\mu (G_a)}\mu (B(g_\sigma (x),c_\sigma \epsilon )\cap G_a)\... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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2e200ee8a41ca8b00db488d96da0d5a48b35f5e9 | subsection | 29 | 42 | Proof of Theorem | By (REF ), we have p_\sigma \ge p_\tau . So,\bigg (\frac{c_\tau }{c_\sigma }\bigg )^r\ge \eta _r\frac{p_\sigma }{p_\tau }\ge \eta _r.It follows that c_\tau ^{-1}c_\sigma \le \eta _r^{-1/r}. Using this and (REF ), we obtain\lambda _a(B(x,\epsilon ))\le C(1-\overline{p}^3)^{-1}(1+\eta _r^{-t/r})\epsilon ^t=:\chi _1\epsil... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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d2f193515a99daf9435cbebdd87341c04f1c54f7 | subsection | 30 | 42 | Proof of Theorem | Write&K_{a,\sigma }:=K_a\cap g_\sigma ^{-1}(G_{a,\sigma }),\;K_{a,\tau }:=K_a\cap g_\sigma ^{-1}(G_{a,\tau });
\\ &\lambda _{a,\sigma }:=\lambda _a(\cdot |K_{a,\sigma }),\;\;\lambda _{a,\tau }:=\lambda _a(\cdot |K_{a,\tau }).Lemma 4.4
Assume that S_a=2. Let K_{a,\sigma },K_{a,\tau } be as defined in (REF ). There exis... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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4d2ca89e4afb27947af5512176592400699a397e | subsection | 31 | 42 | Proof of Theorem | Hence,\frac{\eta _rp_\tau }{p_\sigma }\le \frac{c_\sigma ^r}{c_\tau ^r}
\le \frac{\eta _r^{-1}p_\tau }{p_\sigma }.This, together with (REF ), implies that\eta _r(1-\overline{p}^3)\lambda _a(K_{a,\tau })\le \frac{c_\sigma ^r}{c_\tau ^r}
\le 2\eta _r^{-1}(1-\overline{p}^3)^{-1}\lambda _a(K_{a,\tau }).The remaining part o... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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acd25b35044c7ab273654448a33fd8ff4866dbe1 | subsection | 32 | 42 | Proof of Theorem | We have\lambda _{a,\sigma }(B(x,\epsilon ))&=&\frac{\lambda _a((B(x,\epsilon )\cap K_{a,\sigma }))}{\lambda _a(K_{a,\sigma })}
\\&=&\frac{\mu (g_\sigma ((B(x,\epsilon )\cap K_{a,\sigma })\cap G_a)}{\mu (G_a)\lambda _a(K_{a,\sigma })}\\&\le &\frac{\mu ((B(g_\sigma (x),c_\sigma \epsilon )\cap J_\sigma )}{\mu (G_{a,\sigma... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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7b37dde35e05702ec176951560e697663acdc00e | subsection | 33 | 42 | Proof of Theorem | Thus,\lambda _{a,\tau }(B(x,\epsilon ))=\widetilde{\lambda }_{a,\tau }(B( h_{\sigma ,\tau }(x),c_\sigma c_\tau ^{-1}\epsilon ))\le C_2(c_\sigma c_\tau ^{-1}\epsilon )^t.This completes the proof of the lemma.Our next lemma gives a lower estimate for e_{h,r}^r(\lambda _a)-e_{h+1,r}(\lambda _a). This estimate will be usef... | {
"cite_spans": [
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"doi": "10.48550/arxiv.1708.07657",
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"raw": "S. Zhu, Asymptotic local uniformity of the quantization error for Ahlfors-David probability measures. arXiv preprint arXiv:1708.07657 (2017).... | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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810a5c5df1b21f0763ae2d4ffb8d589349c16607 | subsection | 34 | 42 | Proof of Theorem | It follows that\lambda _a\bigg (K_{a,\sigma }\setminus \bigcup _{b\in \beta _h}B(b,\xi _{h,2})\bigg )\ge D_1-\frac{D_1}{2}=\frac{D_1}{2}.Since |K_{a,\sigma }|\le 1, we may find an integer l_h which depends on C_1,t and h such that K_{a,\sigma }\setminus \bigcup _{b\in \beta _h}B(b,\xi _{h,2}) may be covered by l_h clos... | {
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$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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e304ebf97c2c60e4c70a4e2ebe138267d6fbd620 | subsection | 35 | 42 | Proof of Theorem | Then we haveLemma 4.9
Assume that S_a=2 and let \lambda _a be as defined in (REF ).
Then there exists a number d_{H_a} depending on H_a and C_2 such that\min _{b\in \beta _a}\int _{P_{b}(\beta _a)}d(x,b)^rd\lambda _a(x)\ge d_{H_a}.For convenience, we simply write H for H_a. Note that \alpha _n\in C_{n,r}(\mu ). By , \... | {
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"so... | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
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4b8b42af9447b34097a12003e76b180ee848c403 | subsection | 36 | 42 | Proof of Theorem | In this case, we havee_{H-1,r}^r(\lambda _a)-e_{H,r}^r(\lambda _a)&\le & I(\gamma ,\lambda _a)-I(\beta _a,\lambda _a)\\&=&
\int _{P_b(\beta _a)}d(x,\gamma )^rd\lambda _a(x)-\int _{P_b(\beta _a)}d(x,\beta _a)^rd\lambda _a(x)\\&\le & \int _{P_b(\beta _a)}d(x,\gamma )^rd\lambda _a(x)\le 2A_{b,1}(\gamma ,\lambda _a).Using ... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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-0.02712247706949711,
... | |
3d42ed4c4fece983ef7a449d9583d7668e46ff98 | subsection | 37 | 42 | Proof of Theorem | In this case, by (REF ), we have\zeta _{H-1,r}\le e_{H-1,r}^r(\lambda _a)-e_{H,r}^r(\lambda _a)\le 2(c_\sigma ^{-1} c_\tau )^r\lambda _a(P_{b,2}(\beta _a)).This and Lemma REF lead to\lambda _a(P_{b,2}(\beta _a))\ge 2^{-1}(c_\sigma c_\tau ^{-1})^r\zeta _{H-1,r}\ge 2^{-1}D_2\lambda _a(K_{a,\tau })\zeta _{H-1,r}.It follow... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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843b9fa0293cdc2e19e175344fa5cbcec41cee08 | subsection | 38 | 42 | Proof of Theorem | Then we have\sup _{x\in \mathbb {R}^1}\widetilde{\lambda }_{a,\tau }(B(x,\epsilon ))\le C_2\epsilon ^t;\;\widetilde{\lambda }_{a,\tau }(h_{\sigma ,\tau }(P_{b,2}(\beta _a)))\ge 2^{-1}D_2\zeta _{H-1,r}.Thus, by Lemma REF , we deduce&&\int _{h_{\sigma ,\tau }(P_{b,2}(\beta _a))}d(x,h_{\sigma ,\tau }(b))^rd\widetilde{\lam... | {
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"doi": "10.48550/arxiv.1708.07657",
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$\mathbb{R}^1$ | [
"Sanguo Zhu"
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4f90a05998b263b24a0ccc4029a306620803f59c | subsection | 39 | 42 | Proof of Theorem | One may see for more details. We still denote by d_H, the minimum of d_H and \widetilde{d}_H. Then Lemma REF holds true for both cases S_a=2 and S_a=1.For two \mathbb {R}-valued variables X,Y, we write X\lesssim Y (X\gtrsim Y) if there exists some constant D such that X\le DY (X\ge DY).Proof of Theorem REFNote that |{\... | {
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"s... | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
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d33f4750dd8ccaf6e1d41c154042344803974e41 | subsection | 40 | 42 | Proof of Theorem | Thus, by (REF ) and Lemmas REF , REF and Remark REF , we deduceI_a(\alpha _n,\mu )&=&\mu (G_a)\int _{P_a(\alpha _n)}d(x,a)^rd\lambda _a\circ g_\sigma ^{-1}(x)
\\&=&\mu (G_a)c_\sigma ^r\int _{g_\sigma ^{-1}(P_a(\alpha _n))}d(x,g_\sigma ^{-1}(a))^rd\lambda _a(x)\\&\gtrsim &\mathcal {E}(\sigma )\min _{1\le h\le 2M_3}\zeta... | {
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"so... | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
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40718d5f5a6e3162a7a78920b00150e2da4f2b90 | subsection | 41 | 42 | Proof of Theorem | Using (REF ), (REF ), Remark REF and Lemma REF , we deduce\Delta _{n,r}(\mu )&\ge &\int _{G_a}d(x,\alpha _n)^rd\mu (x)-\int _{G_a}d(x,\beta )^rd\mu (x)\\
&\ge &\int _{G_a}d(x,\gamma _\sigma )^rd\mu (x)-\int _{G_a}d(x,\Gamma _a)^rd\mu (x)\\&\gtrsim &
\mathcal {E}(\sigma )(e^r_{T_\sigma ,r}(\lambda _a)-e^r_{T_\sigma +1,r... | {
"cite_spans": []
} | 1802.03723 | Asymptotic uniformity of the quantization error for Moran measures on
$\mathbb{R}^1$ | [
"Sanguo Zhu"
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eba06c609550a86dc0e284eddd6e81e4df47261e | abstract | 0 | 117 | Abstract | This work constructs a discrete random variable that, when conditioned upon,
ensures information stability of quasi-images. Using this construction, a new
methodology is derived to obtain information theoretic necessary conditions
directly from operational requirements. In particular, this methodology is used
to derive... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
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] | 2,018 | en | Computer Science | [
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c81feb5f4ebb8a3973bfa6ae10360f8fb31e72c5 | subsection | 1 | 117 | Introduction | Consider arbitrary discrete random variables (DRVs)
(M,\mathbf {X},\mathbf {Y}), which form a Markov chain in that order, where
\mathbf {X} = (X_1,X_2,\dots ,X_n), \mathbf {Y} = (Y_1,Y_2,\dots ,Y_n), and\Pr \left( \mathbf {Y} = \mathbf {y} |\mathbf {X} = \mathbf {x}\right)
&= \prod _{i=1}^n \Pr \left( Y_i = y_i | X_i =... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1002/j.1538-7305.1948.tb01338.x",
"end": 740,
"openalex_id": "https://openalex.org/W1995875735",
"raw": "C. E. Shannon, “A mathematical theory of communication,” Bell system technical journal, vol. 27, no. 3, pp. 379–423, 1948.",
"... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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3e37e02dc08f6a1471c1af0e8054539b76df60bf | subsection | 2 | 117 | Introduction | To understand the importance of such a capability,
consider Fano's inequality , which states that
given DRVs M,\hat{M} and \epsilon \in (0,1), if
\Pr (M = \hat{M}) < \epsilon then\mathbb {H}(M|\hat{M}) \le \epsilon \log _{2}|\mathcal {M}| + \mathbb {H}(B_{\epsilon }),where B_{\epsilon } is a Bernoulli random variable w... | {
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"raw": "R. Fano, Class notes for transmission of information, course. 6.574. MIT, Cambridge, MA, 1952.",
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information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
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] | 2,018 | en | Computer Science | [
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2052ebcaf7281ca9618b91074373d74abdc43445 | subsection | 3 | 117 | Notation | Constants, random variables (RVs), and sets will be denoted by lower
case, upper case and script letters respectively. Function
\Pr (\cdot ) returns the probability of the event in the predicate. We
will always employ the corresponding script form of a letter to denote
the support set of any DRV. That is, if X is a DRV... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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c8a36351792db26739b091d2d07f5b262319864c | subsection | 4 | 117 | Notation | The set of all possible conditional
distributions of the form w(y|x), where y \in \mathcal {Y} and
x \in \mathcal {X}, is denoted \mathcal {P}(\mathcal {Y}|\mathcal {X}). For DRVs
\mathbf {Y} and \mathbf {X} if
p_{\mathbf {Y}|\mathbf {X}}(\mathbf {y}|\mathbf {x}) = \prod _{i=1}^n p_{Y_1|X_1}
(y_i|x_i), we will write
p_... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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85bfd6a59b1d66dc0aa52686351663a0a9593d1b | subsection | 5 | 117 | Notation | For any DRVs X,Y,Z, if X \gg Y then Y \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put (.5,.22){\circle {.3}} \end{}}X \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put (.5,.22){\circle {.3}} \end{}}Z.
To simplify the statements of our results, we will adopt the standard
set notation when des... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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f2a53d49cd4e774d1bd0d301afffc59cc1bbd95e | subsection | 6 | 117 | Notation | In specific, the following quantities will receive heavy use:For DRVs U,X,Y,Z, and probability distributions
w, \hat{w}, \tilde{w} \in \mathcal {P}(\mathcal {Y}|\mathcal {X}) and
p \in \mathcal {P}(\mathcal {X}),\mathbb {H}_{u}(X|Z) &= - \sum _{(x,z) \in \mathcal {X} \times \mathcal {Z}}
p_{X,Z|U}(x,z|u) \log _{2}p_{X|... | {
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"doi": "10.1007/978-3-662-12066-8",
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"raw": "T. S. Han, Information-Spectrum Methods in Information Theory. Applications of mathematics, Springer, 2003.",
"source_ref_id": "bbbc6b9... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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ad895c65965751a537f2b634b6de90d39a2d4ecf | subsection | 7 | 117 | Notation | As a
compromise, we introduce the following order terminology which is
similar in spirit to Bachmann-Landau notation, but has a formal
definition which has to be context sensitive.Definition 1 For any \epsilon \in \mathbb {R}_+, we say f(\epsilon ) =
O(g(\epsilon )) if there exists a constant c \in \mathbb {R}_+ (that
... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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7c1c99beef6cced560ee821626a9d4c752692866 | subsection | 8 | 117 | Images and Quasi-images | The manipulation of images and quasi-images will play an important
role in establishing our theorems. Let us define these concepts. For
all discussions and results in this section, it is assumed that
(\emptyset , \mathbf {X}, \mathbf {Y}) is a regular collection of DRVs.Definition 3 () Let
p_{Y|X} \in \mathcal {P}(\mat... | {
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"source_ref_id": "d9... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
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a5363ca3c4657d4a1147e6b84edccf5be729d8d1 | subsection | 9 | 117 | Images and Quasi-images | Before pointing out the lemmas which will find use in this
paper, we refer readers to
,
and for an information theoretic context
of the blowing up lemma.Lemma 5
()
Given \mathcal {X}, \mathcal {Y}, \alpha \in (0,1), and \beta \in (0,1-\alpha ], there exists \tau _n : \mathbb {R}_+ \times \mathbb {R}_+ \rightarrow \... | {
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"source_ref_id": "d9... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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a3f716b81137912a3447ec9fbf5ffb314896fd5d | subsection | 10 | 117 | Images and Quasi-images | In specific the geometrical
interpretations of their work may lead to further insight which
allow for an improvement in the \tau _n term.In terms of applications Ahlswede used the blowing
up lemma to prove a local strong converse for maximal error codes over
a two-terminal DMC, showing that all bad codes have a good s... | {
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{
"arxiv_id": "",
"doi": "10.1007/bf00535683",
"end": 351,
"openalex_id": "https://openalex.org/W1995237169",
"raw": "R. Ahlswede and G. Dueck, “Every bad code has a good subcode: A local converse to the coding theorem,” Zeitschrift für Wahrscheinlichkeitstheorie un... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03826448693871498,
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0.01006197277456522,
-0.016599586233496666,
-0... | |
0694cf0b4328ea281972dc61b48e5bbc6b855e83 | subsection | 11 | 117 | Other works of interest | Here we wish to briefly highlight a few of the methods by which
information theoretic necessary conditions are generally obtained,
first and foremost being Fano's
inequality . Fano's inequality and generalizations
(for instance, Han and Verdú ), directly
provide information theoretic necessary conditions from probabili... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 175,
"openalex_id": "",
"raw": "R. Fano, Class notes for transmission of information, course. 6.574. MIT, Cambridge, MA, 1952.",
"source_ref_id": "17a5088a2e955026c48e0b5ce010c474226c9cc7",
"start": 0
},
{
... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0.04314517602324486,
-0.009497125633060932,... | |
a1ac3a490006b5f64dd11adebe49c34de83047b3 | subsection | 12 | 117 | Main results | Given a regular collection (M_{[1:l]},\mathbf {X},\mathbf {Y}_{\mathcal {W}}),
our primary goal is to “stabilize” \mathbf {Y}_{w}, when conditioned
on M_{j}, where j\in [1:l], in the sense that the entropy spectrum
of \mathbf {Y}_w|M_j is concentrated around a single frequency. More
precisely, we want\Pr \left( | h_{\m... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.012596449814736843... | |
d5eca0543094971ad2286534502cb34079afb21f | subsection | 13 | 117 | Main results | From
this exchange, we can easily create new necessary conditions for
different information theoretic problems, as we demonstrate in
Section .In order to construct the information stabilizing random variable,
first for a given regular collection (\emptyset ,\mathbf {X},\mathbf {Y}), we
find a subset \mathcal {A}^\dagge... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
ac461e5c2d2bdb62c590f66e3b24ada936386a5a | subsection | 14 | 117 | Main results | Thus directly building upon Theorem REF
we construct the following theorem.Theorem 11
(Information stabilizing partitions)
For any regular collection (M_{[1:l]}, \mathbf {X}, \mathbf {Y}_{[1:k]}) and
real number
\alpha \in \left(\frac{\log _{2}n}{n}, \frac{1}{8 \ln 2}\right), we
have[leftmargin=*]
a DRV
V: \left\lbra... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0.05398300290107727,
... | |
fbf3c9fd05bbfd2a30999787e5baf2adc4fe349a | subsection | 15 | 117 | Main results | Still, the applicability of this methodology can be improved
by also stabilizing M_{[1:l]}.Theorem 12
For any DRVs M_{[1:l]}, positive integer \psi , and positive real
numbers \rho \in [1,\infty ), we have:[leftmargin=*]
a DRV
Q: \left\lbrace \begin{array}{c}|\mathcal {Q}| \le (\psi +1)^l\\ Q \ll M_{[1:l]}\end{array} ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
3900a344c4313ecabd6c5589aaf2f1e4683aa626 | subsection | 16 | 117 | Main results | Providing stability to M_j|\mathbf {Y}_i may be
instantly recognizable to the reader as stabilizing a message given an
observation.The need of our second augmentation theorem arises from the fact that Theorem REF cannot in and of itself simultaneously provide stable quasi images for all product distributions in \mathca... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03219969570636749,
0.026568563655018806,
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-0.002554229460656643,
-0.08234576135873795,
... | |
8ce9482b545fa9a1e349825b4366df35e7490bf0 | subsection | 17 | 117 | Main results | For the upcoming theorem, we begin
to adopt the notation outlined previously where
\mathbf {Y}_{\mathcal {P}(\mathcal {Y}|\mathcal {X})} \triangleq \bigotimes _{ w \in \mathcal {P}(\mathcal {Y}|\mathcal {X})} \mathbf {Y}_{w} and \mathbf {Y}_{w}|\mathbf {X} is
distributed w^n(\mathbf {y}|\mathbf {x}) for
w \in \mathcal ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.01939919777214527,
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0.018407106399536133,
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-0.00903566088527441,
-0.02352019026875496,
0.... | |
fb323648198130b6291e62da2b42722a6044ecef | subsection | 18 | 117 | Main results | But, it is clear that these Theorems are somewhat unwieldy. To simplify this procedure we will essentially combine Theorems REF , REF and REF into a single corollary which simultaneously stabilizes \mathbf {Y}_{w} |M_{i} and M_{i} for all w \in \mathcal {P}(\mathcal {Y}|\mathcal {X}) and i \in [1:l]. Because of the ten... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
0.010277262888848782,
0.010551934130489826,
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0.02603268064558506,
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-0.012581445276737213,
0.01403109636157751,
0.0... | |
bf5356b656ba4175ca71bbdf2a68d64468b08c80 | subsection | 19 | 117 | Main results | While such a trade off would be useful for scenarios such as ID coding, they would not be appropriate for the examples presented here.In order to simplify analysis we introduce the following definition.Definition 14 For any regular collection
(M_{[1:l]},\mathbf {X},\mathbf {Y}_{\mathcal {P}(\mathcal {Y}|\mathcal {X})})... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04901330545544624,
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0.011017312295734882,
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0.0342116542160511,
-0.002519714180380106,
-0.030213680118322372,
0.004020860884338617,
0.... | |
320defbb256378bebb954abcc8a4e93d6dfbc94f | subsection | 20 | 117 | Main results | In addition, if h(m_j|u) < n^2 - 2 n\delta , then
p(m_j|u) \approx p(m_j) \approx 2^{-\mathbb {H}_u(M_j) \pm n
\delta }. In that sense \mathcal {D}_{\scriptscriptstyle {(\text{stable})},(M_j)}(u,w;\delta ) and
\mathcal {D}_{\scriptscriptstyle {(\text{saturate})},(M_j)}(u,w;\delta ) consists of the probability terms
whi... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.07489264011383057,
0.0010671972995623946,
-0.015595003962516785,
-0.024872658774256706,
-0.005672644358128309,
0.0036126391496509314,
0.020050719380378723,
0.06353972107172012,
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0.01751767471432686,
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0.02610866166651249,
-0.012176920659840107,
-... | |
20a862ed60a8bb00fbc9a2c76c7d0fd471a728ad | subsection | 21 | 117 | Main results | For any
regular collection
(M_{[1:l]},\mathbf {X},\mathbf {Y}_{\mathcal {P}(\mathcal {Y}|\mathcal {X})}) and any DRV
T :\lbrace (T,M_{[1:l]}) \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put (.5,.22){\circle {.3}} \end{}}\mathbf {X} \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put (.5,.22){\... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03502423316240311,
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-0.009907769970595... | |
b610ae686b43a8d787d8ad686be0cfe754d945c3 | subsection | 22 | 117 | Main results | Furthermore if M_j is uniform over \mathcal {M}_j, then (REF ) holds.The proof of which is in Appendix REF . Note the error term is primarily due to the result holding simultaneously for all distributions in \mathcal {P}(\mathcal {Y}|\mathcal {X}). If this term is of importance in a potential application, and if only a... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.02601940743625164,
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-0.03061286173760891,
-0.01980832219123840... | |
634a5955f9ddb2a6e6c03dd68ec5f40e7cf5037e | subsection | 23 | 117 | Applications | In this section we will highlight a new methodology by which to obtain information theoretic necessary conditions. First we will apply this new methodology to a classical problem to highlight how it works, and how it differs from conventional approaches. In doing so we will provide extra commentary at each step in orde... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
0.006077051628381014,
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-0.025620514526963234,
... | |
e4f8839d3eeed95d3fa9a0e4e47efa5325a516d7 | subsection | 24 | 117 | One way communications over a DMC | Here we consider a classical problem in information theory, channel coding over a DMC p_{Y|X}. In this model a source wants to send a message M, which will be chosen at random according to some arbitrary distribution over \mathcal {M}, to the destination. Connecting the source and destination is a DMC characterized by ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0340656079351902,
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0.0032585158478468657,
... | |
e054f21236c1337a0a01caac3bbdc513a2b12b81 | subsection | 25 | 117 | Fano's inequality | Without a uniform distribution over M, Fano's inequality can only (essentially) provide\mathbb {H}(M) < \mathbb {I}(\Phi (\mathbf {Y}) ; M) + \Pr \left( \Phi (\mathbf {Y}) \ne M \right) \log _{2}|\mathcal {M}|.Now, if it were the case that M was uniform over \mathcal {M}, then (REF ) reduces to\log _{2}|\mathcal {M}| <... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05351397767663002,
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-0.0031749133486300707,
-0.0229280237108469,... | |
547fb4880dddd3e6b9beed0a96a4b4ef464349ba | subsection | 26 | 117 | Fano's inequality | To see this consider a case where any potential decoder is given the side information that determines whether M =0 or M \ne 0. When the decoder is informed that M=0, then clearly the probability of error of the decoder can be eliminated. On the other hand when M\ne 0, then the number of potential messages greatly excee... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.026492265984416008,
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0.010842600837349892,
... | |
2d78be6bbd1500afd18d0e0032f3ebdeb7f6531d | subsection | 27 | 117 | Information stable partitions | Now we move onto our methodology, which even without the assumption that M is information stable, nor that \Pr \left( \Phi (\mathbf {Y}) \ne M \right) \rightarrow 0 as a function of n, yields\Pr \left( n^{-1} h(M) > \max _{p(x)} \mathbb {I}(Y;X) + \zeta _n \right) < \delta + 2^{-n\zeta _n},for some \zeta _n: \zeta _n \... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-662-12066-8",
"end": 531,
"openalex_id": "https://openalex.org/W4251984466",
"raw": "T. S. Han, Information-Spectrum Methods in Information Theory. Applications of mathematics, Springer, 2003.",
"source_ref_id": "bbbc6b9... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03947873041033745,
-0.028831064701080322,
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0.029303956776857376,
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0.013477379456162453,
-0.04301777854561806,
0.011578190140426159,
-0.01445366907864809,
0... | |
e6bce8a58625f6373e19d2097a98fe1c837a4d72 | subsection | 28 | 117 | Information stable partitions | Here the
operational requirement (Equation (REF )) can be written
as\Pr \left( \Phi (\mathbf {Y}) = M \right) = \sum _{\mathbf {y},m} p_{\Phi |\mathbf {Y}}(m|\mathbf {y}) p_{\mathbf {Y},M}(\mathbf {y},m) > 1 - \delta .Next, because ((M,\emptyset ),\mathbf {X},\mathbf {Y}) constitute a regular
collection of DRVs, there ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.044138796627521515,
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0.021977823227643967,
-0.038888540118932724,
0.0013697984395548701,
0.010088431648910046,
... | |
2f56d80e4f4e25e1604ed818c60882ef888b3933 | subsection | 29 | 117 | Information stable partitions | The set
\mathcal {D}_{\scriptscriptstyle {(\text{saturate})},\emptyset }(U,p_{Y|X};\nu _n) is not considered because
the random variable \emptyset is trivially uniform by
convention. Introducing U into the LHS of (REF ) via
the law of total probability yields\Pr \left( \Phi (\mathbf {Y}) = M \right)
& =
\sum _{u} \sum ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0781107172369957,
0.012364987283945084,
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0.006537195760756731,
-0.019176790490746498,
0.011487767100334167,
0.05452493950724602,
0.031076470389962196,
0.0036805097479373217,
-0.03374626860022545,
0.02607250213623047,
-0.005987979471683502,
-0.... | |
8832c1419c57fb42d0d9507c770fd875f95b7f0b | subsection | 30 | 117 | Information stable partitions | It is provided
mainly to show convergence.&\hspace{-10.0pt}
\sum _{u \in \mathcal {\tilde{U}}_{\scriptscriptstyle {(\text{stable})}}}
\sum _{(\mathbf {y},m) \in \mathcal {D}_+(u,p_{Y|X};\nu _n) } \hspace{-20.0pt}
p_{\Phi |\mathbf {Y}}(m|\mathbf {y})
p_{\mathbf {Y},M,U}(\mathbf {y},m,u)
\\
&\le \Pr \left( \Phi (\mathbf ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0... | |
c24052002d1e459593a10734f9c03d50908a21f8 | subsection | 31 | 117 | Information stable partitions | That is,\begin{array}{r l l l}
2^{-\mathbb {H}_u(\mathbf {Y}|M) - n \nu _n} &\le &p(\mathbf {y}|m,u) &\le 2^{-\mathbb {H}_u(\mathbf {Y}|M) + n \nu _n} \\
2^{-\mathbb {H}_u(\mathbf {Y}) - n \nu _n} &\le &p(\mathbf {y}|u) &\le 2^{-\mathbb {H}_u(\mathbf {Y}) + n \nu _n} \\
2^{-\mathbb {H}_u(\mathbf {M}) - n \nu _n} &\le &... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
a9819cc8b28a4dcf653a09029f7a9df782e4b670 | subsection | 32 | 117 | Information stable partitions | But also note that&\Pr \left( \begin{array}{c} \mathbb {H}_{U}(M) \le \mathbb {I}_{U}(\mathbf {Y};M) +3n \nu _n + n\varepsilon _n \\
U \in \mathcal {\tilde{U}}_{\scriptscriptstyle {(\text{stable})}} \end{array} \right) \\
&\le O( 2^{-n\varepsilon _n}) \\
&\hspace{5.0pt} + \Pr \left( \begin{array}{c} h(M) \le \mathbb {I... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.01... | |
be0318f456084bd9e93f89c36e49630b6ed162fe | subsection | 33 | 117 | Information stable partitions | O(2^{-n\varepsilon _n}),directly follows from Equation (REF ), which bounds the sum over all terms not relating to a u \in \mathcal {\tilde{U}}. Given that u \in \mathcal {\tilde{U}}, we bound the sum of all terms for which u \notin \mathcal {\tilde{U}}\setminus \mathcal {\tilde{U}}_{\scriptscriptstyle {(\text{stable})... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0.... | |
673b89a4c0885d501e6cc5c69ee5fa87de2a9bae | subsection | 34 | 117 | Information stable partitions | In specific,&\sum _{{(\mathbf {y},m,u): \\ u \in \mathcal {\tilde{U}} \setminus \mathcal {\tilde{U}}_{\scriptscriptstyle {(\text{stable})}} }} p_{\Phi |\mathbf {Y}}(m|\mathbf {y}) p_{\mathbf {Y},M,U}(\mathbf {y},m,u) \\
&\le O(2^{-n\varepsilon _n}) \hspace{-1.0pt} + \hspace{-48.0pt} \sum _{{(\mathbf {y},m,u): \\ u \in ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
9a2bf5146f47a8caae715a6a23d573b5bc40921c | subsection | 35 | 117 | Information stable partitions | Thus all terms other than those with u \in \mathcal {\tilde{U}}_{\scriptscriptstyle {(\text{stable})}} do not contribute to the sum, and for the remaining terms it follows that&\sum _{{(\mathbf {y},m,u): \\ u \in \mathcal {\tilde{U}}_{\scriptscriptstyle {(\text{stable})}} }} p_{\Phi |\mathbf {Y}}(m|\mathbf {y}) p_{\mat... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-... | |
b3b852e399d393219d1299d3d29cef53808c2ebb | subsection | 36 | 117 | Body | In the wiretap channel, a source wants to reliably send a message M chosen uniformly from \left\lbrace 1,\dots ,2^{nr} \right\rbrace , to a given destination while ensuring a certain level of secrecy from an eavesdropper. The source is connected to the destination through a DMC p_{Y|X} \in \mathcal {P}(\mathcal {Y}|\ma... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1002/j.1538-7305.1975.tb02040.x",
"end": 867,
"openalex_id": "https://openalex.org/W2043769961",
"raw": "A. D. Wyner, “The wire-tap channel,” The Bell System Technical Journal, vol. 54, pp. 1355–1387, Oct 1975.",
"source_ref_id": "... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0.05275149270892143,
... | |
6ed74e1b15c85a9d5dc3cf3f30195fdfbd119f43 | subsection | 37 | 117 | Body | As a first point of order, note that for any (\delta , \ell )-code subject to weak information leakage, where \delta \rightarrow 0 and \ell \rightarrow 0 as a function of n, using Fano's inequality it is possible to obtainnr &\le \mathbb {I}(\mathbf {Y};M) + n\delta _1 \\
n\ell &> \mathbb {I}(\mathbf {Z};M) - n \delta ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/tit.1978.1055892",
"end": 1125,
"openalex_id": "https://openalex.org/W2144007657",
"raw": "I. Csiszár and J. Körner, “Broadcast channels with confidential messages,” IEEE Transactions on Information Theory, vol. 24, pp. 339–348, May... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.08360083401203156,
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-0.05852058157324791,
0.... | |
c8783bfbc19e1cef5b0aed3de2497f14db56c988 | subsection | 38 | 117 | Body | For our purposes, we will directly assume Equation (REF ) as an implication of (REF ) and ().Observe that ((M,\emptyset ),\mathbf {X},\mathbf {Y}_{\mathcal {P}(\mathcal {Y}|\mathcal {X})}) is a regular collection, and therefore there exists[leftmargin=*]
a DRV U: \left\lbrace \begin{array}{c} (U,M) \operatorname{ \begi... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03534582257270813,
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0.015368413180112839,
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0.0049485680647194386,
0.005318661220371723,
-... | |
2fd8966287f36f5fbd77a7b546fa1c93e2793f64 | subsection | 39 | 117 | Body | Proof these conditions are sufficient can be found in our earlier work .First, let us consider the weak information leakage.Theorem 16r \le c\left( \frac{\ell }{1-\delta } \right) + O(-\sqrt{\varepsilon _n} \log _{2}\varepsilon _n)for any (\delta ,\ell ) code subject to weak information leakage.First, repeating the der... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/isit.2017.8006627",
"end": 72,
"openalex_id": "https://openalex.org/W2963563362",
"raw": "E. Graves and T. F. Wong, “Wiretap channel capacity: Secrecy criteria, strong converse, and phase change,” in Int. Sym. Info. Theory, pp. 744–... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.049082137644290924,
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-0.0423974432051181... | |
05660be07d339e6f7e9be0aee154640a319b2915 | subsection | 40 | 117 | Body | In fact, starting from Equation (REF ) and using basic information inequalities we haven\ell &> \mathbb {I}(\mathbf {Z};M) \ge - \log _{2}|\mathcal {U}| + \sum _{u} \mathbb {I}(\mathbf {Z};M|u)p_{U}(u) \\
&\ge - \log _{2}|\mathcal {U}| + \sum _{u \in \mathcal {U}^{+} } \mathbb {I}(\mathbf {Z};M|u)p_{U}(u)\\
&\ge - 2\lo... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
ee9be7edfbce2bbafa4e0b19081025c790f140d9 | subsection | 41 | 117 | Body | Which of these two codes will be used to transmit the information will then be selected at random prior to transmission.This result stands in contrast to the more modern metric which exhibits an “all or nothing” dichotomy of the region.Theorem 17r - O(-\sqrt{\varepsilon _n} \log _{2}\varepsilon _n) &< {\left\lbrace \be... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0... | |
f927282b54e20213cb9e1dcf1eba62bdcfb4ba79 | subsection | 42 | 117 | Converse for error exponents: keyed authentication | For this next example we consider a communication model recently employed by Lai et al. , andTo be more precise, this model is a special case of the model found in . and Gungor and Koksal . Here the source and destination must now maintain reliable communications in the presence of an interloper who has the ability to ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/tit.2008.2009842",
"end": 189,
"openalex_id": "https://openalex.org/W2150264238",
"raw": "L. Lai, H. El Gamal, and H. V. Poor, “Authentication over noisy channels,” Trans. Info. Theory, vol. 55, no. 2, pp. 906–916, 2009.",
"so... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
3235a63c818555098220f453278b7d56ec812414 | subsection | 43 | 117 | Converse for error exponents: keyed authentication | With this the probability of intrusion given intercession (that is, the false authentication probability) can be written as2^{- \beta } &\triangleq \hspace{-8.0pt} \sup _{\psi \in \mathcal {P}(\mathcal {Y}^n|\mathcal {Y}^n)} \sum _{{\mathbf {y},\mathbf {y}_{w_i}, k: \\ \mathbf {y} \in \mathcal {S}(k)} } \hspace{-3.0pt}... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/3-540-39568-7_32",
"end": 1058,
"openalex_id": "https://openalex.org/W1743965195",
"raw": "G. J. Simmons, “Authentication theory/coding theory.,” in Advances in Cryptology, Proceedings of CRYPTO '84, Santa Barbara, California, USA, ... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.... | |
0827855ef3ea39b990723c6440b25853a0a80408 | subsection | 44 | 117 | Converse for error exponents: keyed authentication | Therefore there exists:[leftmargin=*]
a DRV U: \left\lbrace \begin{array}{c} U \gg T \\ \log _{2}|\mathcal {U}| = O(\log _{2}n - n \varepsilon _n \log _{2}\varepsilon _n) \\ (U,M_{[1:l]}) \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put (.5,.22){\circle {.3}} \end{}}\mathbf {X} \operatorname{ \begin{}(1,1... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0016655785730108619,
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... | |
153b42ac6b492de94fa76e248fe5dee4c17142d3 | subsection | 45 | 117 | Converse for error exponents: keyed authentication | Then\beta &\le \hspace{-25.0pt} \inf _{{ (u,w) \in \mathcal {\tilde{U}} \times \mathcal {P}(\mathcal {Y}|\mathcal {X}) : \\ \Pr \left( \mathbf {Y}_w \in \mathcal {S}(K) |u \right) > 17 \cdot 2^{-n\varepsilon _n} } } \hspace{-28.0pt} \mathbb {I}(K;\mathbf {Y}_w|u)-h(u) + O(-n\sqrt{\varepsilon _n} \log _{2}\varepsilon _n... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.009256518445909023,
-0.022954333573579788,... | |
ba98f4a16080f9befc3dda7bc732a67b493f5057 | subsection | 46 | 117 | Converse for error exponents: keyed authentication | This is compounded by the fact that \Pr \left( \mathbf {Y}_w \in \mathcal {S}(K) \right) > 17\cdot 2^{-n\varepsilon _n} does not imply \mathbb {I}(K;\mathbf {Y}_w) \approx \mathbb {H}(K) since the sets \mathcal {S}(k) are not necessarily disjoint for different values of k. This is unfortunate since choosing a code more... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.030848339200019836,
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0.0... | |
db51c607922e0733f27325fc6061d79f60fb9ac0 | subsection | 47 | 117 | Converse for error exponents: keyed authentication | Doing so, we may derive Equation (REF ) as follows:&2^{-\beta } \\
&\ge \sum _{{\mathbf {y},\mathbf {y}_{w_i},k: \\ \mathbf {y} \in \mathcal {S}(k)}} p_{\mathbf {Y}_w|u}(\mathbf {y}|u) p(\mathbf {y}_{w_i},k,u) \\
&= \sum _{{\mathbf {y},\mathbf {x},k :\\ \mathbf {y} \in \mathcal {S}(k)}} p_{\mathbf {Y}_w,K,\mathbf {X},U... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.01896362379193306,
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0... | |
b282eac18b3a8e613edf4b36ecad43f554b839db | subsection | 48 | 117 | Converse for error exponents: keyed authentication | First understand that if \mathcal {S}(k) are not pairwise disjoint (i.e., \mathcal {S}(k) \cap \mathcal {S}(k^{\prime }) \ne \emptyset for some k \in \mathcal {K} and k^{\prime } \in \mathcal {K}\setminus \left\lbrace k \right\rbrace ), then the RHS of Equation (REF ) is\ge \sum _{{\mathbf {y} ,\mathbf {y}_{w_i},\mathb... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.016739577054977417,
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-0.003587869694456458,
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... | |
0e8dc505bf7a159aaef224d873718e9b210d4293 | subsection | 49 | 117 | Converse for error exponents: keyed authentication | Furthermore since all summands are positive, we may restrict the summation to only consider \mathbf {y}_{w_i} such that p_{\mathbf {y}_{w_i}|\mathbf {x}} = w, hence giving2^{-\beta }
&\ge \hspace{-10.0pt} \sum _{{\mathbf {y}_{w_i},\mathbf {x},k : \\ p_{\mathbf {y}_{w_i}|\mathbf {x}} = w }} \hspace{-10.0pt} p_{K|\mathbf... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1561/9781933019543",
"end": 2396,
"openalex_id": "https://openalex.org/W2055309977",
"raw": "I. Csiszár, P. C. Shields, et al., “Information theory and statistics: A tutorial,” Foundations and Trends® in Communications and Information Th... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.054824165999889374,
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0.02... | |
d830341c7b1c25b75fde9b6040191685a3b91df4 | subsection | 50 | 117 | Proof of Theorem | To prove Theorem REF , we construct here the subset
\mathcal {A}^\dagger \subseteq \mathcal {X}^n with non-negligible probability
for which the quasi-image of
\mathbf {X}\,|\left\lbrace \mathbf {X} \in \mathcal {A}^\dagger \right\rbrace by
p_{Y|X} \in \mathcal {P}(\mathcal {Y}|\mathcal {X}) is stable. Our construction ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-662-12066-8",
"end": 365,
"openalex_id": "https://openalex.org/W4251984466",
"raw": "T. S. Han, Information-Spectrum Methods in Information Theory. Applications of mathematics, Springer, 2003.",
"source_ref_id": "bbbc6b9... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.017727067694067955,
-0.04253275692462921,
0... | |
c5a7c3de42950917c4a664f68cdaba66c3510064 | subsection | 51 | 117 | Proof of Theorem | Furthermore s^* \ne t, since
p_{\mathbf {Y}}(\mathbf {y}) \le \left| \mathcal {Y} \right|^{-2n} for each
\mathbf {y} \in \mathcal {S}_{\mathbf {Y}}(t), and thus
p_{\mathbf {Y}}(\mathcal {S}_{\mathbf {Y}}(t)) \le \left| \mathcal {Y} \right|^{-n} < n^{-1}
\log _{2}n. The theorem follows by setting\mathcal {A}^\dagger = \... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.016695499420166016,
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0.022021576762199402,
0.029743626713752747,
0.013009974732995033,
0.03683997690677643,
-0.040319476276636124,
0.... | |
20366fda9422dfd6a48d09a9ecc82ae0a47387a2 | subsection | 52 | 117 | Proof of Theorem | Assume for the moment that\Pr \left( h(\mathbf {Y}|U) > s^* \lambda + n \tilde{\delta }| U=u \right)
&\le 2\cdot 2^{-n\alpha }
\\
\Pr \left( h(\mathbf {Y}|U) \le s^-\lambda -h(U) |U=u \right)
&< 2^{-n\alpha }.Clearly, applying the union bound
with (REF ), () gives\Pr \left( |h(\mathbf {Y}|U) - s^* \lambda | < n \delta... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.04697292298078537,
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0.02928558737039566,
0.011590622365474701,
0.03... | |
0819606f979cd33d55f0c2441d7e8d7bdbb8e158 | subsection | 53 | 117 | Proof of Theorem | Equation (REF ) now
directly follows from () and
Lemma REF since the support set of
\mathbf {X}|\left\lbrace U=u \right\rbrace is a subset of \mathcal {A}^+.On the other hand () can be derived as follows&\hspace*{-10.0pt} \Pr \left( h(\mathbf {Y}|U) \le s^- \lambda - h(U) |U=u \right)\\
&\le \Pr \left( h(\mathbf {Y})\l... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.007202377542853355,
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0.05639827996492386,
-0.02884002961218357,
0.037... | |
3b79caf6563d785b89438f51f693fa34639ded0f | subsection | 54 | 117 | Proof of Theorem | To do so,
we start by noting that if \mathcal {A}^- \ne \emptyset ,&\log _{2}g^n_{Y|X}(\mathcal {A}^-,1-2^{-n\alpha })
\\ &\hspace{10.0pt}
\le \log _{2}\bar{g}^n_{Y|X}(\mathbf {X},\eta _{s^-} ) + n \tau _{n}(2^{-n\alpha },2^{-n\alpha }) \\
&\hspace{10.0pt}
< s^- \lambda + \lambda + \log _{2}(t+1) + n \tau _{n}(2^{-n\al... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.03299928084015846,
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0.029961880296468735,
-0.04701099917292595,
0.0... | |
79c7d03fb5b9035cabd4bc8602734d12c612754d | subsection | 55 | 117 | Proof of Theorem | A lower bound on the probability of
\mathcal {S}_{\mathbf {Y}}(s^*)\setminus \mathcal {B}^- can be constructed as
follows:&\hspace{-20.0pt}p_{\mathbf {Y}}(\mathcal {S}_{\mathbf {Y}}(s^*) \setminus \mathcal {B}^-) \\
&= p_{\mathbf {Y}}(\mathcal {S}_{\mathbf {Y}}(s^*) ) -
p_{\mathbf {Y}}(\mathcal {S}_{\mathbf {Y}}(s^*) \... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.028193116188049316,
0.027338780462741852,
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0.033624257892370224,
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0.06840797513723373,
-0.011098751798272133,
0... | |
b59b53fcf835b9512dce8fc0a1f1fd267a1ecc90 | subsection | 56 | 117 | Proof of Theorem | But this implies a lower bound on the
probability of \mathcal {A}^\dagger since now&\hspace*{-10.0pt} \frac{\log _{2}n}{n} - 2^{-n\alpha } \\
&\le p_{\mathbf {Y}} (\mathcal {S}_{\mathbf {Y}}(s^*) \setminus \mathcal {B}^-) \\
&= \sum _{\mathbf {x} \in \mathcal {A}^- } p^n_{Y|X}(\mathcal {S}_{\mathbf {Y}}(s^*) \setminus ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.025233525782823563,
0.008009431883692741,
-0.03844527527689934,
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0.012365037575364113,
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0.07335114479064941,
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... | |
6235028cd2476daa535ef6507b2256c54f606d62 | subsection | 57 | 117 | Proof of Theorem | Solving () for
p_{\mathbf {X}}(\mathcal {A}^\dagger ) and simplifying, we havep_{\mathbf {X}}(\mathcal {A}^\dagger )
& \ge \frac{\log _{2}n}{n} - 3 \cdot 2^{-n\alpha } \\
&\ge \frac{\log _{2}n}{n} - \frac{3}{n} = \frac{1}{n} \log _{2}\frac{n}{8}since \alpha \ge n^{-1} \log _{2}n.Throughout this section we will once aga... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.002903867745772004,
0.014927634969353676,
-0.02341410331428051,
0.... |
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