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06f4017c29c9cef6c869571fd8744b5c22d4d56d | subsection | 31 | 69 | Polling Model and Notation | \sum _{y}\sigma (\pi , y^{(u+1)} = y) V\left(T\left(\pi ,y^{(u+1)}=y\right) \right) \ge \\
\sum _{y}\sum _{r}\Lambda (r)V\left(T(\pi ,y^{(u)} = r)\right)\sigma (\pi , y^{(u+1)} = y)\\
= \sum _{r}V\left(T\left(\pi ,y^{(u)}=r\right)\right) {\sigma \left(\pi ,y^{(u)} = r\right)}
\therefore C(\pi ,1) \le C(\pi ,u) ~\foral... | {
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7cb0150e5388df633ba9773b439986e7ce708c8a | subsection | 32 | 69 | Polling Model and Notation | Therefore Q^{j/K} \succeq _B Q^{(j+J)/K}.
For Theorem REF c, we have Q^{j/K} = Q^{j/K+1} \times Q^{j/K(K+1)} . Therefore Q^{j/K} \succeq _B Q^{j/K+1}.
EM Algorithm with Ultrametric Constraints
The parameters of the POMDP are computed using a sequence of observations obtained from level {N} in Fig.REF . Specifically, a... | {
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377a218c0aeffcf01946f2c4240bfb4b45151421 | subsection | 33 | 69 | Adaptive Polling and Blackwell Dominance | Armed with the POMDP formulation of the previous section, in this section we give sufficient conditions for Blackwell dominance of Theorem REF to hold for three polling mechanisms:
(i) Adaptive Intent Polling (Sec.REF ), (ii) Adaptive Expectation Polling (Sec.REF ), (iii) Adaptive Friendship Polling (Sec.REF ).The form... | {
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a79b732cf706b34a9b8cef6550695d4002cee888 | subsection | 34 | 69 | Adaptive Intent Polling | In adaptive intent polling, a node at level l is sampled with a probability \beta _l and is asked the following question:``\textit {What does it (a node at level) think the state is?}"
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3e6f8db6fc520aa2dfefcbfbbd2279dbecd16771 | subsection | 35 | 69 | Adaptive Intent Polling | The cost (REF ) expressed in terms of the belief state \pi captures the fact that a control with higher measurement cost should result in a smaller entropy (more reduction in uncertainty) cost and vice versa.
Main Result. Myopic Policies for Adaptive Intent Polling
Our main result on adaptive intent polling is Theorem ... | {
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2bdae4bf0e83efa859467e6f09e1d740d87dfcc5 | subsection | 36 | 69 | Adaptive Intent Polling | Proposition REF below provides a justification for the polynomial f_{U}(z) to be Hurwitz. If f_{U}(z) is Hurwitz, then a way to compute f_{g}(z) for g \in \lbrace U-1,\cdots ,2,1 \rbrace is by successive long-division of f_{U}(z) by linear or quadratic factors of f_{U}(z ).
Matrix polynomials and Blackwell Dominance
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2351a446e4d98c63b45713ae9eb909c84ec8ebd9 | subsection | 37 | 69 | Adaptive Intent Polling | Corollary 1.1
If the channel error probabilities (likelihoods) for the pollster satisfy f_u(B) \succeq _B f_{u+1}(B) ~\forall u \in \mathcal {U}, then
i.)
Shannon Capacity Ordering: \mathcal {C}^{(u)} \ge \mathcal {C}^{(u+1)}~\forall u \in \mathcal {U}.
ii.)
Rényi Divergence Ordering:
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682edef90e28865b4591854062a375c0a7ccae28 | subsection | 38 | 69 | Adaptive Intent Polling | \end{aligned}
In (REF ), \pi denotes the posterior distribution updated according to (\ref {eq:FIL}) and \bar{x} \in \lbrace e_1,e_2,\cdots , e_X\rbrace , where e_i is the unit indicator vector. In (REF ) using the law of iterated expectation, \eta _2(\pi ,u) can be expressed in terms of the belief \pi as follows :
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acbc00969d8864871c9e06612b5e9e128969ee8d | subsection | 39 | 69 | Adaptive Intent Polling | The proof of Theorem REF follows from Proposition REF below and Theorem REF . The expectation polling mechanism employed by the pollster determines how the opinions are gathered, and the opinions are distributed as O(u) for the pollster. Proposition REF below provides a justification for the opinion distribution B to b... | {
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bbbd2705dc726d4b199ec13baf39d96abdbd1ae2 | subsection | 40 | 69 | Adaptive Intent Polling | Corollary 2.1
If the channel error probabilities (likelihoods) for the pollster satisfy Q^{l_{u} / K} \succeq _B Q^{l_{v} / K} for any K>0, we have
i.)
Shannon Capacity Ordering: \mathcal {C}^{(l_u)} \ge \mathcal {C}^{(l_v)} for l_u>l_v.
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774b47da376a00c9524c5340dc9109b8470320f1 | subsection | 41 | 69 | Adaptive Intent Polling | Since more informative opinion fractions are costlier to obtain, from (REF ) below, S(u) \ge S(u+1)~\forall u \in \mathcal {U}.
ii.)
State-Estimation error:
The state-estimation error incurred in choosing action u is modelled as in (REF )
\begin{aligned}\eta _2(\pi ,u) = w_u\left(1 - \pi ^\prime \pi \right)
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443eac616aec68045433d1ef47ee15a5787b3633 | subsection | 42 | 69 | Adaptive Intent Polling | \times \cdots \times {n}^{(j)}_X!} & \prod _{h=1}^{X} (B_l)_{ih}^{{n}^{(j)}_h}.
Here \mathcal {N}_j and {n}^{(j)}_{i} indicate the total and the number in favor of x = i reported and B_l denotes the opinion distribution (REF ) at level l. The likelihood in (REF ) is the well known multinomial distribution.
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fc2706d9990f25ddf142584e5206d03d19ca98e3 | subsection | 43 | 69 | Adaptive Intent Polling | Proposition 3
For the channel error probabilities (likelihoods) O(u) \succeq _B O(u+1), we have
i.)
Shannon Capacity Ordering: \mathcal {C}^{(u)} \ge \mathcal {C}^{(u+1)} for u \in \mathcal {U}.
ii.)
Rényi Divergence Ordering:
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2c56b4eb8a4ff9991ab323a7c87bc36b7651ad81 | subsection | 44 | 69 | Adaptive Intent Polling | Initialize: O(1) = \hat{O}(1)
For u \in \lbrace 1,2,\cdots ,U-1\rbrace , do:
R^*_{u+1} = \operatornamewithlimits{arg\,min}_{R \in \mathcal {M}}\Vert O(u+1) - \hat{O}(u) R \Vert
\hat{O}(u+1) = \hat{O}(u)R^*_{u+1}
end
Output: \hat{O}(u) for u \in \mathcal {U}.
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5d6239abb2bf33a021c526ed78aaf3e63600ec03 | subsection | 45 | 69 | Adaptive Intent Polling | \\
{\hspace{-17.07182pt}} \text{Mis-specified Policy:}&~J_{\mu ^*(\gamma )}(\pi ;\theta ) \le J_{\mu ^*(\theta )}(\pi ;\theta ) + 2G \Vert \gamma - \theta \Vert .
Here G = \max _{i \in \mathcal {X},u} \frac{C(e_i,u)}{1-\rho } and e_i denotes the indicator vector with a `1' in the i^{th} position, and
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e05f2bb79a0b47b33bd99eff36aea5f1c5ff41a3 | subsection | 46 | 69 | Adaptive Intent Polling | This implies that expectation polling is more informative than intent polling.
For l_1>1, there is no apparent comparison of ultrametric and polynomial channels. However, Algorithm REF can be used to design polling POMDPs for arbitrary l_1~\text{and}~f_2. Proposition REF and hence Theorem REF provides the performance b... | {
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cca7897c61cd3c92a87e9fc9e8a0d4604f684237 | subsection | 47 | 69 | Adaptive Intent Polling | We consider the following two measures for measuring the effectiveness of the myopic polling policy: (i) The percentage loss in optimality due to using the myopic policy \bar{\mu } instead of optimal policy \mu ^* is
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Dominance | [
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70885d78bdc91e24e5c134219968825a583e590b | subsection | 48 | 69 | Adaptive Intent Polling | Also, the critics and those who see the movie before its release will influence the future movie goers by sharing opinions on social media platforms. So production, media house, and critics are in \text{Level}~0 and the common movie goer is \text{Level}~1. We will use adaptive expectation polling (Sec.REF ) to poll the... | {
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Dominance | [
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6a8b39e50947325ac418b2c14b20f5d8285f71e1 | subsection | 49 | 69 | Adaptive Intent Polling | \end{aligned}
Note that the costs associated with the actions u=1 and u=2 in (REF ) assume the following structure: w_1 \le w_2 model the accuracy of the observations and S(1) \ge S(2) model the additional cost in expectation polling – nodes need to be compensated for exhausting their resources gathering information f... | {
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Dominance | [
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e52f07e266c869a7923a4ca1042a251d08b54f0c | subsection | 50 | 69 | Adaptive Intent Polling | The notion of Blackwell dominance was extended to the case of polynomial observation likelihoods (channels) described by matrix polynomials. Second, we presented an adaptive generalization of expectation polling to hierarchical social networks. The notion of Blackwell dominance was extended to the case of ultrametric o... | {
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} | 1810.00571 | Adaptive Polling in Hierarchical Social Networks using Blackwell
Dominance | [
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a6d4cb347ec19dd469aca6c54b65b682837ba45b | subsection | 51 | 69 | Adaptive Intent Polling | Observe that,
\begin{aligned}T\left(\pi ,y^{(u+1)}=y\right) =& \cfrac{O_{u+1}(y)P^\prime \pi }{\sigma \left(\pi ,y^{(u+1)} = y\right)}
=\sum _{r}\Lambda (r)T(\pi ,y^{(u)} = r)
\end{aligned}
where \Lambda (r) is a probability mass function w.r.t r and defined as
\Lambda (r) = \mathbb {P}\left(y^{(u+1)}= y|y^{(u)} = ... | {
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Dominance | [
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35ded951f31fef7fcbe2626fcffb647b4e7bd9db | subsection | 52 | 69 | Adaptive Intent Polling | We know that :
{\vspace{0.0pt}}O(u) \succeq _B O(u+1) \Rightarrow \\ D(O_i(u)|| O_j(u)) \ge D(O_i(u+1)|| O_j(u+1)),
for all i,j \in \mathcal {X}. From (REF ) and (REF ), the result follows.
Proof of Theorem REF:
It is given that p(z) \in \mathcal {P}_n and q(z) \in \mathcal {P}_m, with n>m. Clearly, f(Q) and g(Q) ... | {
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Dominance | [
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5310c38e6d94d80a9be65eb891455c4a8c1e7ebb | subsection | 53 | 69 | Adaptive Intent Polling | For all i,j,k \in \mathcal {X}, i \ne j \ne k:
B_{N+1}(i,j) &\ge B_{N+1}(i,k) + M (1 - \kappa ), \\
B_{N+1}(i,j) &\ge B_{N+1}(k,j) + M \kappa , \\
B_{N+1}(k,j) & \ge B_{N+1}(i,k) + M (1 - \kappa ), \\
B_{N+1}(i,k) & \ge B_{N+1}(k,j) +M \kappa , \\
\kappa & \ge 0, \\
-\kappa & \ge -1,
for some large positive value M. ... | {
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Dominance | [
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3d42ee2513b95c1190fe610f0a97bb18f5a75e36 | subsection | 54 | 69 | Approximate Blackwell Dominance, Performance Bounds, and Ordinal Sensitivity | So far we have discussed sufficient conditions for Blackwell dominance; when these conditions hold, the optimal adaptive polling policy is provably upper bounded
by a myopic policy.
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Dominance | [
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cee74efa42d99992a300951b3e2c7ad8173ef46d | subsection | 55 | 69 | Le Cam Deficiency | Given a collection of matrices, it is important to check whether there exists a Blackwell dominance relation, as Theorem REF can used to compute inexpensive policies. What if the pollster would like to choose between different polling mechanisms at each polling epoch to estimate the state?Now the control inputs u \in \... | {
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Dominance | [
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b7bd70b2852d4e7c83ae18a164db6fd87900f7e7 | subsection | 56 | 69 | Performance Bounds on Comparison of Polling POMDPs | Let \theta = (P,O(u),C,\rho ) denote the given adaptive polling POMDP model and \gamma = (P,\hat{O}(u),C,\rho ) denote the adaptive polling POMDP model having a Blackwell dominance relation between the observation distributions. Let J_{\mu ^*(\gamma )}(\pi ;\theta ) and J_{\mu ^*(\gamma )}(\pi ;\gamma ) be defined as i... | {
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2b28c5797f58c9bf0ef7abe77560c29691a9d13d | subsection | 57 | 69 | Performance Bounds on Comparison of Polling POMDPs | Algorithm REF and Theorem REF can be used to design polling POMDPs that have observation distributions that are not Blackwell comparable, for example, when the polling distributions in case of adaptive intent polling are not Hurwitz; in case of adaptive friendship polling; to name a few.Proposition REF below highlights... | {
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04a28caa77550908e56f945951b752276cc8196b | subsection | 58 | 69 | Ordering of Hierarchical Social Networks | So far we have discussed three types of polling mechanisms on a single hierarchical social network. In this section, we briefly discuss how to order hierarchical networks that differ in the opinion distributions B (defined in (REF )), according to the expected polling cost. Theorem REF below shows that some networks ar... | {
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3cac99cbba05d2144ba2f4b9bbfe1854c1fc9efa | subsection | 59 | 69 | Numerical Examples | The main results of this paper involve using Blackwell dominance to construct myopic policies that provably upper bound the optimal adaptive polling policy.
In this section, the performance of this myopic upper bound is illustrated using numerical examples for adaptive polling. Let \Pi ^s_1 represent the set of belief ... | {
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653f3491ed54ff1a66465fc381b0453b7504544e | subsection | 60 | 69 | Numerical Examples | \end{aligned}In (REF ), the cumulative discounted cost is evaluated using 1000 Monte carlo simulations over a horizon of 100 time units. | {
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Dominance | [
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0f7aaaa188f0a1ec568f0f27015c33f0988b509e | subsection | 61 | 69 | Example 1: Market Research. Adaptive Expectation Polling with X=3, Y=3, U = 2 and N=1 | We describe how to estimate the revenue level a movie generates based on the response received on social media platform YouTube. The popularity in the movie is modeled as a 3 state Markov chain x taking values on the state-space \mathcal {X}=\lbrace \text{High}, \text{Medium}, \text{Low} \rbrace .
Prior to a movie's re... | {
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Dominance | [
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75f65bfc590b359d99881f9237c3536994121a8f | subsection | 62 | 69 | Example 1: Market Research. Adaptive Expectation Polling with X=3, Y=3, U = 2 and N=1 | The computed parameters for P, O(1) = O^{1/2}(2), and O(2) are as follows:\begin{aligned}\begin{pmatrix}
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0.0346 & 0.9433 & 0.0221\\
0.0065 & 0.0138 & 0.9797
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\begin{pmatrix}
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0.1809 & 0.1809 & 0.6382
\end{pmatrix}, \\
\begin{p... | {
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Dominance | [
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6cef616fe9b8e4775022b128b132ce37841349de | subsection | 63 | 69 | Example 2: Large Dimensional Example. Adaptive Intent Polling with X=20, Y=20, U = 5 and N = 9 | The Blackwell dominance structural result is particularly useful for large number of states and observation symbols since solving the POMDP (for the optimal policy) is intractable. RandomThe matrices are generated by stochastic simulation as follows: twenty (1 \times 20) probability vectors were simulated from the Diri... | {
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Dominance | [
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1a7971cebe0c8073c1e1c3882400199751010b9f | subsection | 64 | 69 | Conclusions | This paper considered the problem of adaptive (feedback control based) polling in hierarchical social networks, and the problem was formulated as a partially observed Markov decision process (POMDP). We presented three main results. First, we presented an adaptive generalization of intent polling to hierarchical social... | {
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} | 1810.00571 | Adaptive Polling in Hierarchical Social Networks using Blackwell
Dominance | [
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ae4aeee74ff93efe720985255c3783b4f439ad8c | subsection | 65 | 69 | Proofs | Proof of Theorem REF:Denote by y^{(u)} as the observations recorded when using action u. Then O(u+1) = O(u) R implies the following\mathbb {P}\left(y^{(u+1)}|x\right) = \sum _{y^{(u)}}\mathbb {P}\left(y^{(u+1)}|y^{(u)}\right)\mathbb {P}\left(y^{(u)}|x\right)For notational convenience, let T(\pi ,y,u) be written as T(\... | {
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Dominance | [
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3c7f45857cfa119ad9cbf90091c714f58cec8292 | subsection | 66 | 69 | Proofs | Observe that,\begin{aligned}T\left(\pi ,y^{(u+1)}=y\right) =& \cfrac{O_{u+1}(y)P^\prime \pi }{\sigma \left(\pi ,y^{(u+1)} = y\right)}
=\sum _{r}\Lambda (r)T(\pi ,y^{(u)} = r)
\end{aligned}where \Lambda (r) is a probability mass function w.r.t r and defined as\Lambda (r) = \mathbb {P}\left(y^{(u+1)}= y|y^{(u)} = r\right... | {
"cite_spans": []
} | 1810.00571 | Adaptive Polling in Hierarchical Social Networks using Blackwell
Dominance | [
"Sujay Bhatt",
"Vikram Krishnamurthy"
] | [
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fbe33d376b8bad615adabdd3e5fe3c8eb0fc908d | subsection | 67 | 69 | Proofs | From the definition of Rényi Divergence (REF ) we have :{\hspace{-14.22636pt}}D_\alpha (O_i(u+1)|| O_j(u+1)) \le \min \Big \lbrace (1-\alpha ) D(O_i(u+1)|| O_j(u+1)), \\ \alpha D(O_j(u+1)|| O_i(u+1)) \Big \rbrace .We know that :{\vspace{0.0pt}}O(u) \succeq _B O(u+1) \Rightarrow \\ D(O_i(u)|| O_j(u)) \ge D(O_i(u+1)|| O_... | {
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"doi": "10.1109/ciss.2012.6310920",
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Dominance | [
"Sujay Bhatt",
"Vikram Krishnamurthy"
] | [
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] | 2,018 | en | Computer Science | [
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a9a95143236445031f56921abb2f732a2d2058fd | subsection | 68 | 69 | EM Algorithm with Ultrametric Constraints | The parameters of the POMDP are computed using a sequence of observations obtained from level {N} in Fig.REF . Specifically, a modified version of the EM algorithm is used to compute the maximum likelihood estimate of the tuple \left(P, B_{N+1}\right), where B_{N+1} is restricted to the space of ultrametric stochastic... | {
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"raw": "A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” Journal of the Royal Statistical Society. Series B (methodological), pp. 1–38, 1977.",
... | 1810.00571 | Adaptive Polling in Hierarchical Social Networks using Blackwell
Dominance | [
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63f2b26139f4197b2dd13354503d4c04ff02e32f | abstract | 0 | 27 | Abstract | Smoothing (and decay) spacetime estimates are discussed for evolution groups
of self-adjoint operators in an abstract setting. The basic assumption is the
existence (and weak continuity) of the spectral density in a functional
setting. Spectral identities for the time evolution of such operators are
derived, enabling r... | {
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} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
] | [
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3f8d13cb28a7cf6a09630ad7b40cb95724f876f5 | subsection | 1 | 27 | Introduction | In this paper we present an abstract framework for global spacetime and smoothing estimates for evolution groups generated by self-adjoint operators. In particular, this approach leads to such estimates for various classes of pseudodifferential operators.Global spacetime estimates for solutions of partial differential ... | {
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67e43a5630d74f34c0f93e261b2b270d5bbcdfca | subsection | 2 | 27 | The basic setup and notation | Notation. The following notations are used throughout the paper.{\left\langle {x}\right\rangle }=(1+|x|^2)^\frac{1}{2}.
The Fourier transform in \mathbb {R}^n:
\mathcal {F}f(\xi )=\widehat{f}(\xi )=(2\pi )^{-\frac{n}{2}}\int _{\mathbb {R}^n}f(x)e^{-i\xi x}\,dx.
It is useful in multi-variable formulas to indicate by a... | {
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} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
] | [
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6a842434fec39c7b95c3f5f0924eb2cb13b7c524 | subsection | 3 | 27 | The basic setup and notation | The basic tool in our treatment is the hypothesis that the weak spectral derivativeA(\lambda )=\frac{d}{d\lambda }\Big (E(\lambda )P_{ac}(H)\Big ),\quad \lambda \in J,exists and
is bounded from \mathcal {X} into \mathcal {X}^\ast . Thus{\left\langle {A(\lambda )f,g}\right\rangle }=\frac{d}{d\lambda }\Big (E(\lambda )P_... | {
"cite_spans": []
} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
] | [
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a13d1bf8e3d28921952f0f862fb943941bd835d8 | subsection | 4 | 27 | Global smoothing estimates | The following assumption is fundamental in what follows.ASSUMPTION 3.1 Let J\subseteq \mathbb {R} be an open set . The operator-valued function A(\lambda ):\mathcal {X}\hookrightarrow \mathcal {X}^\ast , is weakly continuous on J.In particular, it is also locally bounded on J (in the uniform operator topology).The clas... | {
"cite_spans": []
} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
] | [
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5fbc4f2fbe4b05b9ac5d76ab4fdf6738e856a1ad | subsection | 5 | 27 | Global smoothing estimates | Otherwise, the set J should be replaced by J\setminus \mathcal {N}.To prove (REF ) we use the definition of A(\lambda ) and the spectral calculus to get(\sigma (H)e^{ita(H)}P_{ac}(H)E(J)\phi ,\psi )_{\mathcal {H}}=\int \limits _{J}e^{ita(\lambda )}\sigma (\lambda ){\left\langle {A(\lambda )\phi ,\psi }\right\rangle }d... | {
"cite_spans": []
} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
] | [
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4de8e84866253ee5102fd862bd7a9a52a74b376b | subsection | 6 | 27 | Global smoothing estimates | The method of proof used for Proposition REF in conjunction with the role of A(\lambda ) in the spectral calculus enable us to obtain a general spacetime estimate for any initial data \phi \in \mathcal {H}.THEOREM 3.4
Assume the conditions of Assumptions REF - REF and let \phi \in \mathcal {H}. Then\Big \Vert \sigma... | {
"cite_spans": []
} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
] | [
"math.SP",
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9bbca5f6f9ec2c2bf156bf29c3f2a9226fa73c79 | subsection | 7 | 27 | Global smoothing estimates | As in (REF ) we write, assuming first that \phi \in \mathcal {X},\aligned \Big \Vert \sigma (H)e^{ita(H)}P_{ac}(H)E(J)\phi \Big \Vert _{ L^2(\mathbb {R}_t,\mathcal {X}^\ast )}\hspace{100.0pt}\\
=\sup \limits _{\Vert g\Vert _{ L^2(\mathbb {R}_t,\mathcal {X})}=1} \sqrt{2\pi }\,\Big |\int _{J}\sigma (\lambda )
{\left\lan... | {
"cite_spans": []
} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
] | [
"math.SP",
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e7f09c7bfeb4da654b5120aaa05c7056b8c3c385 | subsection | 8 | 27 | Global smoothing estimates | However, this assumption can be dropped due to the density of \mathcal {X} in \mathcal {H}.Remark 3.5 Note that in the above statements we could “absorb” E(J) into \sigma (H), but we preferred to emphasize the localization aspect of the estimates (in “energy” space) and to leave \sigma (\lambda ) as a continuous functi... | {
"cite_spans": []
} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
] | [
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a9aa6faa943edec45edb15b6f5e398ce392adabe | subsection | 9 | 27 | Global smoothing estimates | Then for all \phi \in L^2(\mathbb {R}^n_x),
\aligned \Big \Vert {\left\langle {x}\right\rangle }^{-s}\sigma (H)e^{ita(H)}P_{ac}(H)E(J)\phi \Big \Vert _{L^2(\mathbb {R}_t\times \mathbb {R}^n_x)}\hspace{100.0pt}\\ \le \sqrt{ 2\pi }\,\,\sup \limits _{\lambda \in J}\Big [\frac{|\sigma (\lambda )|}{|a^{\prime }(\lambda )|^... | {
"cite_spans": []
} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
] | [
"math.SP",
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b0e5d535d3713450a0daea73af2f3f8a39b252d0 | subsection | 10 | 27 | Global smoothing estimates | Then (REF ) can be rewritten as\Vert \mathfrak {H}_J\Vert \le \sqrt{2\pi }\,\,\sup \limits _{\lambda \in {J\setminus \mathcal {N}}}{\left[{\frac{|\sigma (\lambda )|}{|a^{\prime }(\lambda )|^\frac{1}{2}}
\Vert A(\lambda )\Vert _{B(\mathcal {X},\mathcal {X}^\ast )}^\frac{1}{2}}\right]}.In fact, we now show that the oper... | {
"cite_spans": []
} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
] | [
"math.SP",
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4fbbad4de826c0011fa488209af2a12165513606 | subsection | 11 | 27 | Global smoothing estimates | Using Equation (REF ) we therefore write, for \phi \in \Gamma ,\aligned \Big \Vert \sigma (H)e^{ita(H)}P_{ac}(H)E(J)\phi \Big \Vert _{ L^2(\mathbb {R}_t,\mathcal {X}^\ast )}\hspace{100.0pt}\\
=\sup \limits _{\Vert g\Vert _{ L^2(\mathbb {R}_t,\mathcal {X})}=1}\Big |\int _\mathbb {R}\int _{J_\lambda }\sigma (\lambda )
e... | {
"cite_spans": []
} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
] | [
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4cd89603407cb8f67e501921019d0be35e0b6181 | subsection | 12 | 27 | Global smoothing estimates | In view of (REF ) there exists \psi \in \mathcal {X}
such that \Vert \psi \Vert _{\mathcal {X}}=1 and{\left\langle {A(\lambda _0)\psi ,\psi }\right\rangle }\,\,\,\ge \Vert A(\lambda _0)\Vert _{B(\mathcal {X},\mathcal {X}^\ast )}-\delta .Let h>0 so that D_h=(\lambda _0-\frac{h}{2},\lambda _0+\frac{h}{2})\subseteq J, an... | {
"cite_spans": []
} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
] | [
"math.SP",
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"math.FA"
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490f22f7a193c9fb6d2a0ce112f36fb2fb5deacd | subsection | 13 | 27 | Global smoothing estimates | In the case of the Laplacian H=-\Delta such a result was obtained in . | {
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"doi": "10.1016/0022-1236(92)90100-w",
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"raw": "B. Simon, Best constants in some operator smoothness estimates, J. Funct. Anal.107 (1992), 66–71.",
"source_ref_id": "6509b98ba4c47d0... | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
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58c024d119e76397cd86b64339a360d611eeb375 | subsection | 14 | 27 | Comparison principles | As an application of the spectral identities established in Section ,
we present here applications of spectral comparison
principles for self-adjoint operators. Given two such operators, these principles enable us to carry smoothing estimates for one of them to the other, provided their spectral densities can be “comp... | {
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"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
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2c08f8256b19b6729209a715b810aea76da8dea6 | subsection | 15 | 27 | Comparison principles | Suppose that we have\frac{|\sigma (\lambda )|}{{\left|{{a}^{\prime }(\lambda )}\right|}^{1/2}}
{\left|{
{\left\langle { A(\lambda )\phi ,\,\psi }\right\rangle }}\right|}
\ge \frac{|\widetilde{\sigma }(\lambda )|}{{\left|{\widetilde{a}^{\prime }(\lambda )}\right|}^{1/2}}
{\left|{
{\left\langle {
\widetilde{A}(\lambda )\... | {
"cite_spans": []
} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
"Mitsuru Sugimoto"
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... | |
800ebf943e2b67c1480f10b88dcb2c68646026b1 | subsection | 16 | 27 | Comparison principles | The absence of such a scalar product in the second estimate means that it is “space-global comparison” .The following theorem is a direct consequence of Theorem REF (and (REF )).THEOREM 4.4 (uniform comparison)
Assume the conditions of Assumption REF .Suppose that for almost all \lambda \in J and for every unit vec... | {
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"Matania Ben-Artzi",
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f40353a2c665f59086b53588d34fcce10715475c | subsection | 17 | 27 | Comparison principles | Suppose also that there is a constant C_0>0 such that{\left\Vert {\sigma (H)e^{itH}P_{ac}(H)E(J)\phi }\right\Vert }_{ L^2(\mathbb {R}_t,\mathcal {X}^\ast )}
\le C_0 \Vert \phi \Vert _{\mathcal {H}}for all \phi \in \mathcal {H}.Then we have the estimate{\left\Vert {|a^{\prime }(H)|^{1/2}\sigma (H)e^{ita(H)}P_{ac}(H)E(J)... | {
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} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
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d3b3e90b0b01405e793f1e94c791434cc28aa721 | subsection | 18 | 27 | Comparing unperturbed and perturbed operators | Comparing unperturbed and perturbed operatorsThe comparison principles presented above provide a very effective way of dealing with global spacetime estimates for perturbations, as we shall now see.Turning back to the setup in Section
we impose on the self-adjoint operator H an assumption which is stronger than Assum... | {
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ad0fe7fae7a9989b5832e56d5efc97faaa9e3a0d | subsection | 19 | 27 | Comparing unperturbed and perturbed operators | \end{array}\right.}Remark 4.8
The compactness hypothesis on V can be replaced by the assumption that the operators VR^\pm (\lambda ):\mathcal {X}\hookrightarrow \mathcal {X} are compact for any \lambda \in J.Under these assumptions, the operator \widetilde{H}=H+V is essentially self-adjoint on \mathcal {D} and we us... | {
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} | 1807.11611 | Spectral identities and smoothing estimates for evolution operators | [
"Matania Ben-Artzi",
"Michael Ruzhansky",
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c391e95e59adadb60083d9fdb6e96b0be6fdd60d | subsection | 20 | 27 | Comparing unperturbed and perturbed operators | Under the conditions of the Proposition, we can invoke the resolvent equation to obtain the Limiting Absorption Principle for \widetilde{H} :{\widetilde{R}}^\pm (\lambda )=\lim \limits _{\epsilon \downarrow 0}{\widetilde{R}}(\lambda \pm i\epsilon ))=R^\pm (\lambda )[I+VR^\pm (\lambda )]^{-1},\quad \lambda \in J.The exi... | {
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"Matania Ben-Artzi",
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4fd3a55921c7f00084756b9ef63f61bb3a8bca68 | subsection | 21 | 27 | Applications to operators of mathematical physics | We shall now give a few examples that illustrate the scope of the abstract results when applied to a variety of operators that are frequently studied in mathematical physics. | {
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a69db099354fd6c689eb4535ef8d60f6e9022ad0 | subsection | 22 | 27 | The fractional Laplacian | Consider the operator H=-\Delta in \mathcal {H}=L^2(\mathbb {R}^n), n\ge 3. It is absolutely continuous in J=(0,\infty ) and the condition of Assumption REF is satisfied with \mathcal {X}=L^2_s(\mathbb {R}^n),\,s>\frac{1}{2} .Furthermore, it was shown in that for all \phi \in L^2(\mathbb {R}^n),\int _{\mathbb {R}}\... | {
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ae92bc2c6393697d61554e425b838295161d7788 | subsection | 23 | 27 | The Stark Hamiltonian | Consider the self-adjoint operator in \mathcal {H}=L^2(\mathbb {R}^n), n\ge 1,H=-\Delta -x_1,where x=(x_1,x_2,\ldots ,x_n)\in \mathbb {R}^n.This operator is the well-known “Stark Hamiltonian”, describing the motion of a quantum-mechanical charged particle in a uniform electric field (all physical constants scaled to un... | {
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24e088cda548f9a0f86e6bf611dcfdb030cbe47f | subsection | 24 | 27 | The Schrödinger operator with potential | Consider the operator \widetilde{H}=H+V in L^2(\mathbb {R}^n),\,n\ge 3, where H=-\Delta . This operator can be studied in terms of Theorem REF .Employing the notation of Subsection REF we let \mathcal {X}=L^2_{s}(\mathbb {R}^n), so that \mathcal {X}^\ast =L^2_{-s}(\mathbb {R}^n) and \mathcal {X}^\ast _H=H^2_{-s}(\mat... | {
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053463948bbdee7acc0e7701b435fc176c95a2a0 | subsection | 25 | 27 | The Schrödinger operator with potential | Then there exists a constant C_{\delta ,s} so that\int _{\mathbb {R}}\int _{\mathbb {R}^n}{\left\langle {x}\right\rangle }^{-2s}|(I+\widetilde{H})^\frac{1}{4} {\widetilde{E}}(J_\delta )e^{it{\widetilde{H}}}\phi (x)|^2\,dx\,dt\le C_{\delta ,s}\Vert \phi \Vert _{L^2(\mathbb {R}^n)}^2,for all \phi \in L^2(\mathbb {R}^n).T... | {
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b8504f25b3fd204726e73dd5a63a16c2ce1875b8 | subsection | 26 | 27 | The Schrödinger operator with potential | For example, we can take V to be the pseudodifferential operator
V={\left\langle {x}\right\rangle }^{-s}(I-\Delta )^\beta {\left\langle {x}\right\rangle }^{-s},
for any \beta <\frac{1}{2}.We refer the reader to , , for related spacetime estimates for the Schrödinger operator, using very different methods. | {
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def140cad4d780ad9cc072267636353ae272fa3c | abstract | 0 | 31 | Abstract | Blind image deblurring plays a very important role in many vision and
multimedia applications. Most existing works tend to introduce complex priors
to estimate the sharp image structures for blur kernel estimation. However, it
has been verified that directly optimizing these models is challenging and easy
to fall into ... | {
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Deblurring and Beyond | [
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"Xin Fan",
"Zhongxuan Luo"
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8b0ffdaadeb95dc7b542afcfe75639f5ac97756c | subsection | 1 | 31 | Introduction | Blind image deblurring is a fundamental component in many multimedia and computer vision applications. This problem involves the estimation of latent sharp image and blur kernel from a blurry observation. The most commonly used formulation for the blurry phenomenon can be given as follows:\mathbf {y}=\mathbf {u}\otimes... | {
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"Risheng Liu",
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9889328695609f7bfad084646f0f604ec9f1563d | subsection | 2 | 31 | Introduction | Fig. REF shows the performance of GCM on various applications.Our contributions can be summarized as follows:We establish two fundamental propagative modules (i.e., Generator and Corrector) to respectively learn latent image structures from training data and investigate principled mathematical rules for image propagati... | {
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} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
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89905c92e870968bdfb39c1910a6c5d6fb795f1d | subsection | 3 | 31 | Related Works | In this section, we briefly review some related works on the prior models and the existing inference strategies for blind image deblurring.
Specifically, the most commonly used deblurring formulation is the following regularized variational minimization model\min \limits _{\mathbf {u}}\Psi (\mathbf {u})=f(\mathbf {u}) ... | {
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Deblurring and Beyond | [
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c4919d078941e5a405660d45711e82a790fb6e47 | subsection | 4 | 31 | Related Works | (REF ) can also be utilized to formulate other computer vision and multimedia tasks.
For example, by defining f with physical rules of different problems and enforcing other task-related priors for \phi , a variety of applications, such as image interpolation and edge-preserved smoothing, can all be formulated by Eq. (... | {
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c1f49a001ec5c2bbbf675a587a312dc5ba8063ac | subsection | 5 | 31 | The Collaborative Modules | In this section, we propose a collaborative framework to learn Generation and Correction Modules (GCM) for latent image propagation. The strict theoretical analysis on GCM is also established at the end of this section. | {
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86f6bdf40595bf9b317eda022482c7bd5005f649 | subsection | 6 | 31 | Generator with Fidelity Warm Start | Inspired by the success of deep networks in visual processing areas, we would like to first establish our Generator as a parameterized network architecture (denoted as \mathcal {N}), i.e., at t-th stage, we consider\tilde{\mathbf {u}}^{t+1}=\mathcal {N}^t(\mathbf {u}_0^{t+1};{\omega }^t),where {\omega }^t is the learna... | {
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5764e77da4700913bdae5b922bdd3c3970d4df49 | subsection | 7 | 31 | Corrector by Proximal Prior Descent | Since generating the latent image structure is a highly ill-posed problem,
only performing Generator may not guarantee the exact recovery of our desired optimal solution. Moreover, no prior knowledge is enforced into the current scheme, thus it is natural to introduce another module to incorporate our prior assumptions... | {
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Deblurring and Beyond | [
"Risheng Liu",
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"Shichao Cheng",
"Xin Fan",
"Zhongxuan Luo"
] | [
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67400d2ae4b393436152ca1a809aca5282aaea38 | subsection | 8 | 31 | GCM with Theoretical Guarantee | We first summarize the complete GCM framework in Alg. REF and express the pipeline of GCM in Fig. REF . Please notice that due to the CNN-based Generator, GCM is indeed not a standard optimization scheme. Thus existing convergence analysis is not available for the propagations generated by our GCM. But fortunately, we ... | {
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125b24cfd47aac04bb3476b0294556be03443c2b | subsection | 9 | 31 | Applications | Now we demonstrate how to apply our GCM to address blind image deblurring. Thanks to the flexibility of our Generation and Correction modules, GCM indeed can also be applied to address other related multimedia applications. | {
"cite_spans": []
} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
"Risheng Liu",
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316d2a561ebf3c50d87283a01a66038b5e15f470 | subsection | 10 | 31 | Blind Image Deblurring | As discussed in Sec. , generating latent image with rich salient edges and sharp structures often plays important role in blind image deblurring. Therefore, we would like to train our Generator in image gradient domain as follows. We first calculate the warm start with the fidelity f(\mathbf {u}) = \Vert \mathbf {u}\ot... | {
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Deblurring and Beyond | [
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ee860baadc1c5907a651a601718034ecbcab3e25 | subsection | 11 | 31 | Byproduct Applications | As nontrivial byproducts, we would also like to demonstrate how to apply GCM to other image related vision and multimedia applications.Image Interpolation: The purpose of this task is to remove corrections (e.g., text, blocking or mask) from the partially invisible observation. When using GCM to deal with this problem,... | {
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Deblurring and Beyond | [
"Risheng Liu",
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5fc4f0e8c5e27daed03266dd76b00a93997663b2 | subsection | 12 | 31 | Discussions | Here we would like to discuss and highlight some important aspects of GCM. | {
"cite_spans": []
} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
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1676df9bc18ea7af7d57fcb7b7305532082f454b | subsection | 13 | 31 | Theoretically Convergent Ensemble Framework for Image Modeling | Indeed, our GCM can be viewed as a general framework to integrate deep network architectures (i.e., Generator) and physical principles (i.e., Corrector) to address not only blind image deblurring, but also other related vision and multimedia tasks. The main advantage against existing heuristic ensemble strategies (e.g.... | {
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Deblurring and Beyond | [
"Risheng Liu",
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"Zhongxuan Luo"
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8aa89dd8ed8324ab8e37b474ee0eb7d26c5b2cd0 | subsection | 14 | 31 | Analogy to Adversarial Learning Methodology | Since there exist two cascaded modules in our GCM (i.e., Generator and Corrector), which is similar to that in the popular adversarial learning methods (i.e., Generator and Discriminator in Generative Adversarial Network, GAN for short), we would like to provide some brief comparisons between these two learning methodo... | {
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} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
"Shichao Cheng",
"Xin Fan",
"Zhongxuan Luo"
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8945bf454a075258db9c57237cfbc7ebbfd043e1 | subsection | 15 | 31 | Experimental Results | We first conduct experiments to verify the mechanism of GCM. Then a range of results are demonstrated to evaluate GCM on blind image deblurring. Finally, we show the performance of GCM on other related applications, such as image interpolation and edge-preserved smoothing. | {
"cite_spans": []
} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
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"Xin Fan",
"Zhongxuan Luo"
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bbea91b939d0e427c2f1dbd8dfa7430d0ad98641 | subsection | 16 | 31 | Experimental Setup | To provide fair comparisons on the blind image deblurring problem, we adopt as the final non-blind deconvolution process for all the compared methods.
We also execute these approaches with their default parameter settings. For the training data of GCM, we randomly select 800 natural images from the training set of th... | {
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Deblurring and Beyond | [
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b57e56a0861b3a6e2806484ed0e49e47c1f95249 | subsection | 17 | 31 | Model Verification | To verify the efficiency of our proposed collaborative learning strategies, we first compare the deblurring performance of GCM with different settings on an example image.
In Fig. REF , we denote the naive cascade of the designed Generator and Corrector as “Generator" and “Corrector”, respectively. While “GCM” denotes ... | {
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} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
"Risheng Liu",
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a2b4b6690a4177d5dae0ceffeb3ca12a3125befe | subsection | 18 | 31 | Synthetic Dataset | We first consider synthetic test data and compare GCM with several state-of-the-art blind deblurring methods (e.g., , , , , ) on widely used Levin et al.'s dataset , including 32 blurry images with size 255\times 255, which are produced by 4 clear images and 8 blur kernels.
Quantitative scores (e.g., PSNR, SSIM, Error ... | {
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Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
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"Xin Fan",
"Zhongxuan Luo"
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0e9bd7f88ebbd8e007f0aa4bb76918f339b368f8 | subsection | 19 | 31 | Real Blurry Images | We also evaluate the compared methods on real-world blurry images (collected by ) in Fig. REF . We can see on the top row that GCM can recover more details (e.g., the tail of bird with legible), compared with other methods. On the bottom row, it is also easy to observe that the numbers in license plate have been succes... | {
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Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
"Shichao Cheng",
"Xin Fan",
"Zhongxuan Luo"
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0ff73fd24b846f4a4af7846493b0f3742721f275 | subsection | 20 | 31 | Detection by Deblurring | It is known that motion blurs caused by the shaking of capture device often reduce the performance of detection algorithm. See Fig. REF (b) for an example. Thus a natural strategy to evaluate the effectiveness of deblurring algorithms is just to perform object detection on the restored images.
In this experiment, we ad... | {
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Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
"Shichao Cheng",
"Xin Fan",
"Zhongxuan Luo"
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9ff9b723446d10691f3d52b6596b87daa11ab838 | subsection | 21 | 31 | Other Applications | To verify the flexibility of our collaborative modules, we also express the performance of GCM on other applications, including image interpolation and edge-preserved smoothing. | {
"cite_spans": []
} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
"Shichao Cheng",
"Xin Fan",
"Zhongxuan Luo"
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613b2c2737724e5859118351f12dad73920b2b9b | subsection | 22 | 31 | Image Interpolation | The purpose of image interpolation is to recover an image in which some pixels are lost or deteriorated.
To evaluate the performance of our method in this task, we compare GCM with other state-of-the-art image interpolation methods , , , on both text and random missing pixels masks. The test images are randomly chosen ... | {
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Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
"Shichao Cheng",
"Xin Fan",
"Zhongxuan Luo"
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fe7fac7a0b0c5da5bc3687eae42a73742e6d7375 | subsection | 23 | 31 | Edge-Preserved Smoothing | Edge-preserved image smoothing is a fundamental tool for image editing and processing, such as pencil sketch rendering and cartoon artifact removal . Here we compare our method with state-of-the-art image smoothing approaches, including
the classic BLF , WLS and recently proposed L_0 , RTV . Fig. REF illustrates the ... | {
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Deblurring and Beyond | [
"Risheng Liu",
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"Zhongxuan Luo"
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7005f447dc3200b7f44ca55d866eb60d832857d7 | subsection | 24 | 31 | Conclusions | This paper proposed GCM, a collaborative learning framework to estimate the latent image structures. By
integrating the learnable-architecture-based Generator and the model-driven Corrector in a principled manner, we obtained a convergent image propagation, which can promote kernel estimation for blind image deblurring... | {
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} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
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87bb5708b5b112b3382c980797f096601c1c203d | subsection | 25 | 31 | Proof of Theorem | To present our proof in a clear manner, here we reorganize the results in Theorem REF as the following two successive theorems, referring to “non-increasing properties of the objectives” (i.e., Theorem REF ) and the “critical point convergence” (i.e., Theorem REF ), respectively.
Moreover, our theoretical analysis is b... | {
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"raw": "R Tyrrell Rockafellar and Roger J-B Wets. 2009. Variational analysis. Vol. 317. Springer Science & Business Media.",
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"start":... | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
"Shichao Cheng",
"Xin Fan",
"Zhongxuan Luo"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
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32dff8f896cff9cf9138dd4af78065015fcf09bf | subsection | 26 | 31 | Non-increasing Properties of the Objectives | Theorem 2
If \mu ^{t} < 1/L,
both \lbrace \mathbf {u}^{t}\rbrace and \lbrace \mathbf {v}^{t}\rbrace are the sequence generated by GCM, we have the objectives \lbrace \Psi (\mathbf {u}^{t})\rbrace is a non-increasing sequence and satisfied the following relationship:\Psi (\mathbf {u}^{t+1}) \le \Psi (\mathbf {v}^{t+1})... | {
"cite_spans": []
} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
"Shichao Cheng",
"Xin Fan",
"Zhongxuan Luo"
] | [
"cs.CV"
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f6579290986b4482b881928f2bea8af3d2048915 | subsection | 27 | 31 | Non-increasing Properties of the Objectives | (REF ) and (REF ), we have\Psi (\mathbf {u}^{t+1}) \le \Psi (\mathbf {v}^{t+1}) - \left( \frac{1}{2\mu ^t} - \frac{L}{2} \right) \Vert \mathbf {u}^{t+1} - \mathbf {v}^{t+1}\Vert ^2.Setting \mu ^t < \frac{1}{L}, we have \Psi (\mathbf {u}^{t+1}) \le \Psi (\mathbf {v}^{t+1}).
So far, we get the following relationship of o... | {
"cite_spans": []
} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
"Shichao Cheng",
"Xin Fan",
"Zhongxuan Luo"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
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8523329afc6144502c5086fb980b43d894a23870 | subsection | 28 | 31 | Critical Point Convergence | Theorem 3
If \lbrace \mathbf {u}^t\rbrace be the image sequence generated by GCM, we have any accumulation point of \lbrace \mathbf {u}^t\rbrace is the critical point of \Psi (i.e., it satisfies the first-order necessary optimal condition of Eq. (REF )).In this proof, we first verify the existence of accumulation poin... | {
"cite_spans": []
} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
"Shichao Cheng",
"Xin Fan",
"Zhongxuan Luo"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
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2009235e5f9f295c8c1c61598de56ce0d73e1322 | subsection | 29 | 31 | Critical Point Convergence | \end{array}Then sum over t to obtain\min \limits _{t}\left\lbrace \frac{1}{2\mu ^t} - \frac{L}{2} \right\rbrace \sum _{t=0}^{\infty } \Vert \mathbf {u}^{t+1} - \mathbf {v}^{t+1}\Vert ^2 \le \Psi (\mathbf {u}^{0}) - \Psi ^{*} < \infty ,which implies \Vert \mathbf {u}^{t+1} - \mathbf {v}^{t+1}\Vert \rightarrow 0 when t \... | {
"cite_spans": []
} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
"Shichao Cheng",
"Xin Fan",
"Zhongxuan Luo"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
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b5312bdc40594992bccac98b784ee613e0dcd355 | subsection | 30 | 31 | Critical Point Convergence | Note that Lipschitz smooth of f implies f is continuity, which yields \lim \limits _{j\rightarrow \infty } f(\mathbf {u}^{t_j}) = f(\mathbf {u}^{*}). Thus we conclude\lim \limits _{j\rightarrow \infty } \Psi (\mathbf {u}^{t_j}) = \Psi (\mathbf {u}^{*}).Recall that \lim \limits _{j\rightarrow \infty } \Psi (\mathbf {u}^... | {
"cite_spans": []
} | 1807.11706 | Learning Collaborative Generation Correction Modules for Blind Image
Deblurring and Beyond | [
"Risheng Liu",
"Yi He",
"Shichao Cheng",
"Xin Fan",
"Zhongxuan Luo"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
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952afb81bded3d5c88271906e5b213b70b8cc54d | abstract | 0 | 16 | Abstract | Given two graphs $G$ and $H$, it is said that $G$ percolates in $H$-bootstrap
process if one could join all the nonadjacent pairs of vertices of $G$ in some
order such that a new copy of $H$ is created at each step. Balogh, Bollob\'as
and Morris in 2012 investigated the threshold of $H$-bootstrap percolation in
the Erd... | {
"cite_spans": []
} | 1806.10425 | On $K_{2,t}$-bootstrap percolation | [
"M. R. Bidgoli",
"A. Mohammadian",
"B. Tayfeh-Rezaie"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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c8d69586d5df6dc169067e7760983fd15d44fcd4 | subsection | 1 | 16 | Introduction | Bootstrap percolation on graphs has been extensively investigated in several diverse fields such as combinatorics, probability theory, statistical physics and social sciences. Many different models of bootstrap percolation have been defined and studied in the literature including the r-neighbor bootstrap percolation an... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1002/rsa.20458",
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"raw": "J. Balogh, B. Bollobás and R. Morris, Graph bootstrap percolation, Random Structures Algorithms 41 (2012), 413–440.",
"source_ref_id": "04af684... | 1806.10425 | On $K_{2,t}$-bootstrap percolation | [
"M. R. Bidgoli",
"A. Mohammadian",
"B. Tayfeh-Rezaie"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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af675609aed1c27845c3129f8f0a231d7758f8f6 | subsection | 2 | 16 | Introduction | A function { is a {\sl threshold} for a sequence {E}_n of events in {G}(n, p) if
{\lim }_{n\rightarrow \infty }{P}({E}_n)=
\left\lbrace \begin{array}{ll}
0, & \mbox{ if } p\ll {\mbox{;}\\
\vspace{-2.84526pt}\\
1, & \mbox{ if } p\gg {\mbox{.}
}
We say that {E}_n holds {\sl with high probability} if \lim _{n\rightarrow \... | {
"cite_spans": []
} | 1806.10425 | On $K_{2,t}$-bootstrap percolation | [
"M. R. Bidgoli",
"A. Mohammadian",
"B. Tayfeh-Rezaie"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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93f42164a204dde15bf07285fa0e1060422c4ff0 | subsection | 3 | 16 | Introduction | The degree of a vertex v\in V(G), denoted by \deg _G(v), is defined as |N_G(v)|. A graph G is a complete split graph if one can partition V(G) into an independent set I and a clique C such that each vertex in I is adjacent to each vertex in C. | {
"cite_spans": []
} | 1806.10425 | On $K_{2,t}$-bootstrap percolation | [
"M. R. Bidgoli",
"A. Mohammadian",
"B. Tayfeh-Rezaie"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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