wanyu/IteraTeR_full_sent huggingface dataset:
Original:  Due to the global nature of the updatesof the Swendsen-Wang algorithm, the underlying Markov chain often mixes rapidly in the low temperature region (large \beta) where long-range correlations impair the convergence rate of local chains such as the Glauber dynamics. With few exceptions, tight bounds on the convergence rate of the Swendsen-Wang algorithm only hold for high temperature regions (small \beta) corresponding to the uniqueness or the decay of correlations region. We present tight bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree that extend to the non-uniqueness region . Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced by Martinelli et al. ( 2004)implies constant spectral gap of the Swendsen-Wang dynamics .
Revised:  Due to the global nature of its updates. We present optimal bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree that extend to the non-uniqueness region . Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced by Martinelli et al. ( 2004)implies constant spectral gap of the Swendsen-Wang dynamics .
Edit intent: clarity
Original:  We present tight bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree that extend to the non-uniqueness region . Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced by Martinelli et al. ( 2004)implies constant spectral gap of the Swendsen-Wang dynamics .
Revised:  We present tight bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree . Our bounds extend to the non-uniqueness region . Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced by Martinelli et al. ( 2004)implies constant spectral gap of the Swendsen-Wang dynamics .
Edit intent: clarity
Original:  We present tight bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree that extend to the non-uniqueness region . Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced by Martinelli et al. ( 2004)implies constant spectral gap of the Swendsen-Wang dynamics .
Revised:  We present tight bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree that extend to the non-uniqueness region and apply to all boundary conditions. We show that the spatial mixing conditions known as Variance Mixing and Entropy Mixing, introduced in the study of local Markov chains by Martinelli et al. ( 2004)implies constant spectral gap of the Swendsen-Wang dynamics .
Edit intent: meaning-changed
Original:  We present tight bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree that extend to the non-uniqueness region . Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced by Martinelli et al. ( 2004)implies constant spectral gap of the Swendsen-Wang dynamics .
Revised:  We present tight bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree that extend to the non-uniqueness region . Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced by Martinelli et al. ( 2003), imply \Omega(1) spectral gap and O(\log{n the Swendsen-Wang dynamics .
Edit intent: meaning-changed
Original:  We present tight bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree that extend to the non-uniqueness region . Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced by Martinelli et al. ( 2004)implies constant spectral gap of the Swendsen-Wang dynamics . This implies that the relaxation time (i. e., the inverse of the spectral gap) is O(1) for all boundary conditions in the uniqueness region or when \beta<\beta_1 where \beta_1 exceeds the uniqueness threshold for the Ising model  and for the q-state Potts model when q is small with respect to d.
Revised:  We present tight bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree that extend to the non-uniqueness region . Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced by Martinelli et al. ( 2004)implies constant spectral gap of the Swendsen-Wang dynamics on the d-ary tree. We also show that these bounds are asymptotically optimal. As a consequence, we establish \Theta(\log{n all boundary conditions in the uniqueness region or when \beta<\beta_1 where \beta_1 exceeds the uniqueness threshold for the Ising model and for the q-state Potts model when q is small with respect to d.
Edit intent: meaning-changed
Original:  This implies that the relaxation time (i. e., the inverse of the spectral gap) is O(1) for all boundary conditions in the uniqueness region or when \beta<\beta_1 where \beta_1 exceeds the uniqueness threshold for the Ising model  and for the q-state Potts model when q is small with respect to d.
Revised:  This implies that the relaxation time (i. e., the inverse of the spectral gap) is O(1) for all boundary conditions throughout the tree uniqueness region; in fact, our bounds hold beyond the uniqueness threshold for the Ising model and for the q-state Potts model when q is small with respect to d.
Edit intent: clarity
Original:  This implies that the relaxation time (i. e., the inverse of the spectral gap) is O(1) for all boundary conditions in the uniqueness region or when \beta<\beta_1 where \beta_1 exceeds the uniqueness threshold for the Ising model  and for the q-state Potts model when q is small with respect to d.
Revised:  This implies that the relaxation time (i. e., the inverse of the spectral gap) is O(1) for all boundary conditions in the uniqueness region or when \beta<\beta_1 where \beta_1 exceeds the uniqueness threshold for the Ising model , and for the q-state Potts model when q is small with respect to d.
Edit intent: fluency
Original:  In past work on this and also the more general mixture-of-products problem (X_i independent conditional on U, but not necessarily iid), a barrier at m^{O(k^2) a sample of size \exp(k\log k) (with m=2k).
Revised:  We give an algorithm for identifying a k-mixture using samples of m=2k iid binary random variables using a sample of size \exp(k\log k) (with m=2k).
Edit intent: meaning-changed
Original:  In past work on this and also the more general mixture-of-products problem (X_i independent conditional on U, but not necessarily iid), a barrier at m^{O(k^2) a sample of size \exp(k\log k) (with m=2k). It is known that the sample complexity of any solution to the identification problem must be \exp(\Omega(k)) \left(}\right)\left(\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.
Revised:  In past work on this and also the more general mixture-of-products problem (X_i independent conditional on U, but not necessarily iid), a barrier at m^{O(k^2) a sample of size \left(}\right)\left(\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.
Edit intent: coherence
Original:  It is known that the sample complexity of any solution to the identification problem must be \exp(\Omega(k)) \left(}\right)\left(\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.
Revised:  It is known that the sample complexity of any solution to the identification problem must be \exp(\Omega(k)) \left(1/w_{\min}\right)\left(\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.
Edit intent: meaning-changed
Original:  It is known that the sample complexity of any solution to the identification problem must be \exp(\Omega(k)) \left(}\right)\left(\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.
Revised:  It is known that the sample complexity of any solution to the identification problem must be \exp(\Omega(k)) \left(}\right)^2 \cdot\left(\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.
Edit intent: meaning-changed
Original:  It is known that the sample complexity of any solution to the identification problem must be \exp(\Omega(k)) \left(}\right)\left(\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.
Revised:  It is known that the sample complexity of any solution to the identification problem must be \exp(\Omega(k)) \left(}\right)\left(1/\zeta\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.
Edit intent: meaning-changed
Original:  It is known that the sample complexity of any solution to the identification problem must be \exp(\Omega(k)) \left(}\right)\left(\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.
Revised:  It is known that the sample complexity of any solution to the identification problem must be \exp(\Omega(k)) \left(}\right)\left(\right)^{O(k) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.
Edit intent: meaning-changed
Original:  Here w_{\min} is the minimum probability of an outcome of U, and \zeta is the minimum separation between the distinct success probabilities of the X_is} . Stated in terms of the moment problem, it suffices to know the moments to additive accuracy \exp(-k\log k). Our run-time for the moment problem is only\cdot\zeta^{O(k)}.
Revised:  Here w_{\min} is the minimum probability of an outcome of U, and \zeta is the minimum separation between the distinct success probabilities of the X_is} . Stated in terms of the moment problem, it suffices to know the moments to additive accuracy w_{\min\cdot\zeta^{O(k)}.
Edit intent: clarity
Original:  It is known that the sample complexity of any solution to the identification problem must be at least exponential in k. Previous results demonstrated either worse sample complexity and worse } O(k ^{2+o(1) .
Revised:  It is known that the sample complexity of any solution to the identification problem must be at least exponential in k. Previous results demonstrated either worse sample complexity and worse } O(k ^c) runtime for some c substantially larger than 2, or similar sample complexity and much worse k^{O(k^2) .
Edit intent: meaning-changed
Original:  Our work is aimed at producing a (\alpha,\Delta)-kernel of D_P(\kappa), namely a set %DIFDELCMD < \calS %%%  such that (after a shift) it holds that (1-\alpha)D_P(\kappa)\subset %DIFDELCMD < \CH%%% (%DIFDELCMD < \calS%%%  ) \subset (1+\alpha)D_P(\kappa-\Delta).
Revised:  Our work is aimed at producing a (\alpha,\Delta)-kernel of D_P(\kappa), namely a set %DIFDELCMD < \calS %%% \mathcal{S such that (after a shift) it holds that (1-\alpha)D_P(\kappa)\subset %DIFDELCMD < \CH%%% (%DIFDELCMD < \calS%%% ) \subset (1+\alpha)D_P(\kappa-\Delta).
Edit intent: meaning-changed
Original:  Our work is aimed at producing a (\alpha,\Delta)-kernel of D_P(\kappa), namely a set %DIFDELCMD < \calS %%%  such that (after a shift) it holds that (1-\alpha)D_P(\kappa)\subset %DIFDELCMD < \CH%%% (%DIFDELCMD < \calS%%%  ) \subset (1+\alpha)D_P(\kappa-\Delta).
Revised:  Our work is aimed at producing a (\alpha,\Delta)-kernel of D_P(\kappa), namely a set %DIFDELCMD < \calS %%% such that (after a shift) it holds that (1-\alpha)D_P(\kappa)\subset %DIFDELCMD < \CH%%% %DIFDELCMD < \calS%%% ) \subset (1+\alpha)D_P(\kappa-\Delta).
Edit intent: fluency
Original:  Our work is aimed at producing a (\alpha,\Delta)-kernel of D_P(\kappa), namely a set %DIFDELCMD < \calS %%%  such that (after a shift) it holds that (1-\alpha)D_P(\kappa)\subset %DIFDELCMD < \CH%%% (%DIFDELCMD < \calS%%%  ) \subset (1+\alpha)D_P(\kappa-\Delta).
Revised:  Our work is aimed at producing a (\alpha,\Delta)-kernel of D_P(\kappa), namely a set %DIFDELCMD < \calS %%% such that (after a shift) it holds that (1-\alpha)D_P(\kappa)\subset %DIFDELCMD < \CH%%% (%DIFDELCMD < \calS%%% \mathsf{CH ) \subset (1+\alpha)D_P(\kappa-\Delta).
Edit intent: meaning-changed
Original: We present analytical solutions to two classes of generalized matrix eigenvalue problems ( GMEVPs) which naturally cover the matrix eigenvalue problems (MEVPs) with matrices such as the tridiagonal and pentadiagonal matrices.
Revised: It is well-known that the finite difference discretization of the Laplacian eigenvalue problem -\Delta u = \lambda u leads to a matrix eigenvalue problem (EVP) A x= \lambda x where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in Strang and MacNamara \mbox{%DIFAUXCMD strang2014functions analytical solutions to two classes of generalized matrix eigenvalue problems ( GMEVPs) which naturally cover the matrix eigenvalue problems (MEVPs) with matrices such as the tridiagonal and pentadiagonal matrices.
Edit intent: meaning-changed
Original: We present analytical solutions to two classes of generalized matrix eigenvalue problems ( GMEVPs) which naturally cover the matrix eigenvalue problems (MEVPs) with matrices such as the tridiagonal and pentadiagonal matrices.
Revised: We present analytical solutions to the generalized matrix eigenvalue problems ( GMEVPs) which naturally cover the matrix eigenvalue problems (MEVPs) with matrices such as the tridiagonal and pentadiagonal matrices.
Edit intent: clarity
Original: We present analytical solutions to two classes of generalized matrix eigenvalue problems ( GMEVPs) which naturally cover the matrix eigenvalue problems (MEVPs) with matrices such as the tridiagonal and pentadiagonal matrices. We refer to the matrices in the two classes as Toeplitz-regularized and  corner-overlapped block-diagonal matrices.
Revised: We present analytical solutions to two classes of generalized matrix eigenvalue problems ( GEVPs) A x= \lambda Bx which arise from the finite element method (FEM) and corner-overlapped block-diagonal matrices.
Edit intent: meaning-changed
Original:  We refer to the matrices in the two classes as Toeplitz-regularized and  corner-overlapped block-diagonal matrices. The first class generalizes the tridiagonal Toeplitz matrices to matrices with larger bandwidths where boundary entries are modified. For the second class, we decompose the problem into a lower two-block matrix problem where one of the blocks is a quadratic eigenvalue problem (QEVP) .
Revised:  We refer to the matrices in the two classes as Toeplitz-regularized and corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation .
Edit intent: meaning-changed
Original:  Analytical solutions are also obtained for these QEVPs. Moreover , we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for MEVPs) for GMEVPs and derive some trigonometric identities . Possible generalizations and applications to the design of better numerical methods for solving partial differential equations are discussed .
Revised:  Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly , we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for MEVPs) for GMEVPs and derive some trigonometric identities . Possible generalizations and applications to the design of better numerical methods for solving partial differential equations are discussed .
Edit intent: clarity
Original:  Moreover , we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for MEVPs) for GMEVPs and derive some trigonometric identities . Possible generalizations and applications to the design of better numerical methods for solving partial differential equations are discussed .
Revised:  Moreover , we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities . Possible generalizations and applications to the design of better numerical methods for solving partial differential equations are discussed .
Edit intent: meaning-changed
Original:  Moreover , we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for MEVPs) for GMEVPs and derive some trigonometric identities . Possible generalizations and applications to the design of better numerical methods for solving partial differential equations are discussed .
Revised:  Moreover , we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for MEVPs) for GMEVPs and derive some trigonometric identities .
Edit intent: coherence
Original: The most important feature of Wikipedia is the presence of hyperlinks in pages. Link placement is the product of people's collaboration, consequently Wikipedia naturally inherits human bias. Due to the high influence that links' disposition has on users' navigation sessions, one needs to verify that, given a controversial topic, the  hyperlinks' network does not expose users to only one side of the subject.
Revised: People eager to learn about a topic can access Wikipedia to form a preliminary opinion. Despite the solid revision process behind the encyclopedia's articles, the hyperlinks' network does not expose users to only one side of the subject.
Edit intent: clarity
Original:  Due to the high influence that links' disposition has on users' navigation sessions, one needs to verify that, given a controversial topic, the  hyperlinks' network does not expose users to only one side of the subject.
Revised:  Due to the high influence that links' disposition has on users' navigation sessions, one needs to verify that, given a controversial topic, the users' exploration process is still influenced by the hyperlinks' network does not expose users to only one side of the subject.
Edit intent: meaning-changed
Original:  Due to the high influence that links' disposition has on users' navigation sessions, one needs to verify that, given a controversial topic, the  hyperlinks' network does not expose users to only one side of the subject. A Wikipedia's topic-induced network that prevents users the discovery of different facets of an issue, suffers from structural bias.
Revised:  Due to the high influence that links' disposition has on users' navigation sessions, one needs to verify that, given a controversial topic, the hyperlinks' network . In this paper, we shed light on this overlooked phenomenon by investigating how articles describing complementary subjects of a topic interconnect, and thus may shape readers' exposure to diverging content. To quantify this, we introduce the exposure to diverse information, a metric that captures how users' exposure to multiple subjects of a topic varies click-after-click by leveraging navigation models. For the experiments, we collected six topic-induced network that prevents users the discovery of different facets of an issue, suffers from structural bias.
Edit intent: meaning-changed
Original:  A Wikipedia's topic-induced network that prevents users the discovery of different facets of an issue, suffers from structural bias. In thiswork, we define the static structural bias, which indicates if the strength of connections between pages of contrasting inclinations is the same, and the dynamic structural bias, which quantifies the network's level bias that users face over the course of their navigation sessions. Our measurements of structural bias on several controversial topics demonstrate its existence, revealing that users have low likelihood of reaching pages of opposing inclination from where they start, and that they navigate Wikipedia showing a behaviour much more biased than the expected from the baselines. Our findings advance the relevance of the problem and pave the way for developing systems that automatically measure and propose hyperlink locations that minimize the presence and effects of structural bias .
Revised:  A Wikipedia's topic-induced networks about polarizing topics and analyzed the extent to which their topologies induce readers to examine diverse content. More specifically, we take two sets of articles about opposing stances (e.g., guns control and guns right) and measure the probability that users move within or across the sets, by simulating their behavior via a Wikipedia-tailored model. Our findings show that the networks hinder users to symmetrically explore diverse content. Moreover, on average, the probability that the networks nudge users to remain in a knowledge bubble is up to an order of magnitude higher than that of exploring pages of contrasting subjects. Taken together, those findings return a new and intriguing picture of Wikipedia's network structural influence on polarizing issues' exploration .
Edit intent: meaning-changed
Original:  Bj \"{o}%DIFDELCMD < }%%% rklund , Husfeldt and Koivisto (SICOMP 2009) presented an O^\star(2^n) time algorithm for Bin Packing.
Revised:  \"{o}%DIFDELCMD < }%%% rklund , Husfeldt and Koivisto (SICOMP 2009) presented an O^\star(2^n) time algorithm for Bin Packing.
Edit intent: fluency
Original:  Bj \"{o}%DIFDELCMD < }%%% rklund , Husfeldt and Koivisto (SICOMP 2009) presented an O^\star(2^n) time algorithm for Bin Packing.
Revised:  Bj }%DIFDELCMD < }%%% rklund , Husfeldt and Koivisto (SICOMP 2009) presented an O^\star(2^n) time algorithm for Bin Packing.
Edit intent: coherence
Original:  Bj \"{o}%DIFDELCMD < }%%% rklund , Husfeldt and Koivisto (SICOMP 2009) presented an O^\star(2^n) time algorithm for Bin Packing.
Revised:  Bj \"{o}%DIFDELCMD < }%%% Bj\"orklund , Husfeldt and Koivisto (SICOMP 2009) presented an O^\star(2^n) time algorithm for Bin Packing.
Edit intent: meaning-changed
Original:  In this paper, we show that for every m \in \mathbb{N there exists a constant \sigma_m >0 such that an instance of Bin Packing with m bins can be solved in O(2^{(1-\sigma_m)n}) randomized time.
Revised:  In this paper, we show that for every m \in \mathbf{N there exists a constant \sigma_m >0 such that an instance of Bin Packing with m bins can be solved in O(2^{(1-\sigma_m)n}) randomized time.
Edit intent: meaning-changed
Original:  Before our work, such improved algorithms were not known even for m equals 4 A key step in our approach is the following new result in Littlewood-Offord theory on the additive combinatorics of subset sums: For every \delta >0 there exists an \varepsilon >0 such that if |\{ X\subseteq \{1,\ldots,n \} : w(X)=v \}| \leq 2^{(1-\varepsilon)n} for every v then |\{ w(X): X \subseteq \{1,\ldots,n\} \}|\leq 2^{\delta n}.
Revised:  Before our work, such improved algorithms were not known even for m equals 4. A key step in our approach is the following new result in Littlewood-Offord theory on the additive combinatorics of subset sums: For every \delta >0 there exists an \varepsilon >0 such that if |\{ X\subseteq \{1,\ldots,n \} : w(X)=v \}| \leq 2^{(1-\varepsilon)n} for every v then |\{ w(X): X \subseteq \{1,\ldots,n\} \}|\leq 2^{\delta n}.
Edit intent: fluency
Original:  Before our work, such improved algorithms were not known even for m equals 4 A key step in our approach is the following new result in Littlewood-Offord theory on the additive combinatorics of subset sums: For every \delta >0 there exists an \varepsilon >0 such that if |\{ X\subseteq \{1,\ldots,n \} : w(X)=v \}| \leq 2^{(1-\varepsilon)n} for every v then |\{ w(X): X \subseteq \{1,\ldots,n\} \}|\leq 2^{\delta n}.
Revised:  Before our work, such improved algorithms were not known even for m equals 4 A key step in our approach is the following new result in Littlewood-Offord theory on the additive combinatorics of subset sums: For every \delta >0 there exists an \varepsilon >0 such that if |\{ X\subseteq \{1,\ldots,n \} : w(X)=v \}| \geq 2^{(1-\varepsilon)n} for every v then |\{ w(X): X \subseteq \{1,\ldots,n\} \}|\leq 2^{\delta n}.
Edit intent: fluency