wanyu/IteraTeR_v2 huggingface dataset:
Original:  The way we partition the graph is determined by solving a system of linear inequalities whose size is bounded by f^\prime(\varepsilon) depending only on \varepsilon.
Revised:  The way we partition the graph is determined by solving a system of linear inequalities whose size is bounded by a function depending only on \varepsilon.
Edit intent: clarity
Original:  Bydense we mean that |E(G)|\geq\alpha|V(G)|^2 for some fixed \alpha>0, and amaximal \varepsilon-near perfect matching is a maximal matching which covers at least (1-\varepsilon)|V(G)| vertices.
Revised:  By we mean that |E(G)|\geq\alpha|V(G)|^2 for some fixed \alpha>0, and amaximal \varepsilon-near perfect matching is a maximal matching which covers at least (1-\varepsilon)|V(G)| vertices.
Edit intent: fluency
Original:  Bydense we mean that |E(G)|\geq\alpha|V(G)|^2 for some fixed \alpha>0, and amaximal \varepsilon-near perfect matching is a maximal matching which covers at least (1-\varepsilon)|V(G)| vertices.
Revised:  Bydense maximal \varepsilon-near perfect matching is a maximal matching which covers at least (1-\varepsilon)|V(G)| vertices.
Edit intent: clarity
Original:  Bydense we mean that |E(G)|\geq\alpha|V(G)|^2 for some fixed \alpha>0, and amaximal \varepsilon-near perfect matching is a maximal matching which covers at least (1-\varepsilon)|V(G)| vertices.
Revised:  Bydense we mean that |E(G)|\geq\alpha|V(G)|^2 for some fixed \alpha>0, and aA maximal \varepsilon-near perfect matching is a maximal matching which covers at least (1-\varepsilon)|V(G)| vertices.
Edit intent: fluency
Original:  More precisely, we provide a deterministic algorithm that for a given (dense ) graph G of order n and a real number \varepsilon>0, returns either a conclusion that G has no \varepsilon-near perfect matching, or a positive real number m%DIFDELCMD < \leq1%%% /2, such that the logarithm of the number of maximal \varepsilon-near perfect matchings in G is at least mn\log n.
Revised:  More precisely, we provide a deterministic algorithm that for a given dense graph G of order n and a real number \varepsilon>0, returns either a conclusion that G has no \varepsilon-near perfect matching, or a positive real number m%DIFDELCMD < \leq1%%% /2, such that the logarithm of the number of maximal \varepsilon-near perfect matchings in G is at least mn\log n.
Edit intent: coherence
Original:  More precisely, we provide a deterministic algorithm that for a given (dense ) graph G of order n and a real number \varepsilon>0, returns either a conclusion that G has no \varepsilon-near perfect matching, or a positive real number m%DIFDELCMD < \leq1%%% /2, such that the logarithm of the number of maximal \varepsilon-near perfect matchings in G is at least mn\log n.
Revised:  More precisely, we provide a deterministic algorithm that for a given (dense ) graph G of order n and a real number \varepsilon>0, returns either a conclusion that G has no \varepsilon-near perfect matching, or a positive %DIFDELCMD < \leq1%%% /2, such that the logarithm of the number of maximal \varepsilon-near perfect matchings in G is at least mn\log n.
Edit intent: clarity
Original:  More precisely, we provide a deterministic algorithm that for a given (dense ) graph G of order n and a real number \varepsilon>0, returns either a conclusion that G has no \varepsilon-near perfect matching, or a positive real number m%DIFDELCMD < \leq1%%% /2, such that the logarithm of the number of maximal \varepsilon-near perfect matchings in G is at least mn\log n.
Revised:  More precisely, we provide a deterministic algorithm that for a given (dense ) graph G of order n and a real number \varepsilon>0, returns either a conclusion that G has no \varepsilon-near perfect matching, or a positive real number m%DIFDELCMD < \leq1%%% /2, such that the number of maximal \varepsilon-near perfect matchings in G is at least mn\log n.
Edit intent: clarity
Original:  In addition, we find that the roughness  , which measures de deviation from additivity of the fitness function, of the learning trajectories is always greater than the roughness estimated over the entire fitness landscape.
Revised:  In addition, we find that the roughness of the learning trajectories , which measures de deviation from additivity of the fitness function, of the learning trajectories is always greater than the roughness estimated over the entire fitness landscape.
Edit intent: clarity
Original:  In addition, we find that the roughness  , which measures de deviation from additivity of the fitness function, of the learning trajectories is always greater than the roughness estimated over the entire fitness landscape.
Revised:  In addition, we find that the roughness , which measures the deviation from additivity of the fitness function, of the learning trajectories is always greater than the roughness estimated over the entire fitness landscape.
Edit intent: fluency
Original:  In addition, we find that the roughness  , which measures de deviation from additivity of the fitness function, of the learning trajectories is always greater than the roughness estimated over the entire fitness landscape.
Revised:  In addition, we find that the roughness , which measures de deviation from additivity of the fitness function, is always greater than the roughness estimated over the entire fitness landscape.
Edit intent: coherence
Original:  The mean path divergence, however, is strongly affected by the search parameters - population size and imitation propensity - that obliterate the influence of the underlying landscape.
Revised:  The mean path divergence, however, is strongly affected by the search parameters -- population size and imitation propensity - that obliterate the influence of the underlying landscape.
Edit intent: fluency
Original:  The mean path divergence, however, is strongly affected by the search parameters - population size and imitation propensity - that obliterate the influence of the underlying landscape.
Revised:  The mean path divergence, however, is strongly affected by the search parameters - population size and imitation propensity -- that obliterate the influence of the underlying landscape.
Edit intent: fluency
Original:  The imitative search becomes more deterministic, in the sense that there are fewer distinct learning trajectories and those trajectories are more similar to each other, with increasing population size and imitation propensity.
Revised:  The learning trajectories become more deterministic, in the sense that there are fewer distinct learning trajectories and those trajectories are more similar to each other, with increasing population size and imitation propensity.
Edit intent: clarity
Original:  The imitative search becomes more deterministic, in the sense that there are fewer distinct learning trajectories and those trajectories are more similar to each other, with increasing population size and imitation propensity.
Revised:  The imitative search becomes more deterministic, in the sense that there are fewer distinct trajectories and those trajectories are more similar to each other, with increasing population size and imitation propensity.
Edit intent: clarity
Original: A family of problems that have been studied in the context of various streaming algorithms are generalizations of the fact that the expected maximum distance of a 4-wise independent random walk on a line over n steps is O(n).
Revised: Recently, many streaming algorithms have utilized generalizations of the fact that the expected maximum distance of a 4-wise independent random walk on a line over n steps is O(n).
Edit intent: clarity
Original: A family of problems that have been studied in the context of various streaming algorithms are generalizations of the fact that the expected maximum distance of a 4-wise independent random walk on a line over n steps is O(n).
Revised: A family of problems that have been studied in the context of various streaming algorithms are generalizations of the fact that the expected maximum distance of any 4-wise independent random walk on a line over n steps is O(n).
Edit intent: clarity
Original:  For small values of k, there exist k-wise independent random walks that can be stored in much less space than storing n random bits, so these properties are often useful for lowering space bounds.
Edit intent: coherence
Original:  In this paper, we show that  for all of these examples, 4-wise independence is required by demonstrating a pairwise independent random walk with steps uniform in%DIFDELCMD < \pm %%% 1 and expected maximum distance O (n \lg n) from the origin.
Revised:  In this paper, we show that 4-wise independence is required for all of these examples, 4-wise independence is required by demonstrating a pairwise independent random walk with steps uniform in%DIFDELCMD < \pm %%% 1 and expected maximum distance O (n \lg n) from the origin.
Edit intent: fluency
Original:  In this paper, we show that  for all of these examples, 4-wise independence is required by demonstrating a pairwise independent random walk with steps uniform in%DIFDELCMD < \pm %%% 1 and expected maximum distance O (n \lg n) from the origin.
Revised:  In this paper, we show that for all of these algorithms, by constructing a 3-wise independent random walk with steps uniform in%DIFDELCMD < \pm %%% 1 and expected maximum distance O (n \lg n) from the origin.
Edit intent: clarity
Original:  In this paper, we show that  for all of these examples, 4-wise independence is required by demonstrating a pairwise independent random walk with steps uniform in%DIFDELCMD < \pm %%% 1 and expected maximum distance O (n \lg n) from the origin.
Revised:  In this paper, we show that for all of these examples, 4-wise independence is required by demonstrating a pairwise independent random walk with %DIFDELCMD < \pm %%% 1 and expected maximum distance O (n \lg n) from the origin.
Edit intent: clarity
Original:  In this paper, we show that  for all of these examples, 4-wise independence is required by demonstrating a pairwise independent random walk with steps uniform in%DIFDELCMD < \pm %%% 1 and expected maximum distance O (n \lg n) from the origin.
Revised:  In this paper, we show that for all of these examples, 4-wise independence is required by demonstrating a pairwise independent random walk with steps uniform in%DIFDELCMD < \pm %%% expected maximum distance O (n \lg n) from the origin.
Edit intent: coherence
Original:  We also show that this bound is tight for the first and second moment, i. e. the expected maximum square distance of a 2-wise independent random walk is always O(n \lg^2 n)
Revised:  We prove that this bound is tight for the first and second moment, i. e. the expected maximum square distance of a 2-wise independent random walk is always O(n \lg^2 n)
Edit intent: clarity
Original:  . Also, for any even k%DIFDELCMD < \ge %%% \dots \le \le  4, we show that the kth moment of the maximum distance of any k-wise independent random walk is O( n^{k/2 Kolmogorov's maximal inequality by showing an equivalent statement that requires only 4-wise independent random variables with bounded second moments, which also generalizes a result of 5 } .
Revised:  . Also, for any even k%DIFDELCMD < \ge %%% \dots \le i\le 4, we show that the kth moment of the maximum distance of any k-wise independent random walk is O( n^{k/2 Kolmogorov's maximal inequality by showing an equivalent statement that requires only 4-wise independent random variables with bounded second moments, which also generalizes a result of 5 } .
Edit intent: clarity
Original:  . Also, for any even k%DIFDELCMD < \ge %%% \dots \le \le  4, we show that the kth moment of the maximum distance of any k-wise independent random walk is O( n^{k/2 Kolmogorov's maximal inequality by showing an equivalent statement that requires only 4-wise independent random variables with bounded second moments, which also generalizes a result of 5 } .
Revised:  . Also, for any even k%DIFDELCMD < \ge %%% \dots \le \le 4, we show that the kth moment of the furthest distance traveled is O( n^{k/2 Kolmogorov's maximal inequality by showing an equivalent statement that requires only 4-wise independent random variables with bounded second moments, which also generalizes a result of 5 } .
Edit intent: clarity
Original:  . Also, for any even k%DIFDELCMD < \ge %%% \dots \le \le  4, we show that the kth moment of the maximum distance of any k-wise independent random walk is O( n^{k/2 Kolmogorov's maximal inequality by showing an equivalent statement that requires only 4-wise independent random variables with bounded second moments, which also generalizes a result of 5 } .
Revised:  . Also, for any even k%DIFDELCMD < \ge %%% \dots \le \le 4, we show that the kth moment of the maximum distance of any k-wise independent random walk is O( n^{k/2 Kolmogorov's maximal inequality that requires only 4-wise independent random variables with bounded second moments, which also generalizes a result of 5 } .
Edit intent: coherence
Original:  . Also, for any even k%DIFDELCMD < \ge %%% \dots \le \le  4, we show that the kth moment of the maximum distance of any k-wise independent random walk is O( n^{k/2 Kolmogorov's maximal inequality by showing an equivalent statement that requires only 4-wise independent random variables with bounded second moments, which also generalizes a result of 5 } .
Revised:  . Also, for any even k%DIFDELCMD < \ge %%% \dots \le \le 4, we show that the kth moment of the maximum distance of any k-wise independent random walk is O( n^{k/2 Kolmogorov's maximal inequality by showing an equivalent statement that requires only 4-wise independent random variables with bounded second moments, which also generalizes a result of 5 \l} .
Edit intent: coherence
Original: Memristors, resistors with memory whose outputs depend on the history of their inputs, have been used with success in neuromorphic architectures, particularly as synapses or non-volatile memories.
Revised: Memristors, resistors with memory whose outputs depend on the history of their inputs, have been used with success in neuromorphic architectures, particularly as synapses and non-volatile memories.
Edit intent: fluency
Original:  Therefore, in order to fill this gap, we introduce models for single and multilayer perceptrons based  on memristors.
Revised:  In the present work we introduce models for single and multilayer perceptrons based on memristors.
Edit intent: clarity
Original:  These showed that both of them perform well and in accordance with Minsky-Papert's theorem , which motivates the possibility of building memristor-based hardware for a physical neural network .
Revised:  These showed that both of them perform well and in accordance with Minsky-Papert's theorem , which motivates the possibility of building memristor-based paradigm, both in the classical and quantum learning realms .
Edit intent: clarity
Original:  The techniques we develop for analyzing our algorithm may be of use for other related problems.
Revised:  The techniques we develop for analyzing our algorithm maybe of use for other related problems.
Edit intent: coherence
Original: Non-Gaussian component analysis (NGCA) is a problem in multidimensional data analysis . Since its formulation in 2006, NGCA has attracted considerable attention in statistics and machine learning.
Revised: Non-Gaussian component analysis (NGCA) is a problem in multidimensional data analysis which, since its formulation in 2006, NGCA has attracted considerable attention in statistics and machine learning.
Edit intent: coherence
Original: Non-Gaussian component analysis (NGCA) is a problem in multidimensional data analysis . Since its formulation in 2006, NGCA has attracted considerable attention in statistics and machine learning.
Revised: Non-Gaussian component analysis (NGCA) is a problem in multidimensional data analysis . Since its formulation in 2006, has attracted considerable attention in statistics and machine learning.
Edit intent: fluency
Original:  NGCA is also related to dimensionality reduction and to other data analysis problems such as ICA.
Revised:  NGCA is also related to dimension reduction and to other data analysis problems such as ICA.
Edit intent: clarity
Original:  We show how to extend Fuzz to encompass more general relational properties of probabilistic programs, with our motivating example being the (\epsilon, \delta )-variant of differential privacy .
Revised:  We show how to extend Fuzz to capture more general relational properties of probabilistic programs, with our motivating example being the (\epsilon, \delta )-variant of differential privacy .
Edit intent: clarity
Original:  We show how to extend Fuzz to encompass more general relational properties of probabilistic programs, with our motivating example being the (\epsilon, \delta )-variant of differential privacy .
Revised:  We show how to extend Fuzz to encompass more general relational properties of probabilistic programs, with approximate, or (\epsilon, \delta )-variant of differential privacy .
Edit intent: clarity
Original:  We show how to extend Fuzz to encompass more general relational properties of probabilistic programs, with our motivating example being the (\epsilon, \delta )-variant of differential privacy .
Revised:  We show how to extend Fuzz to encompass more general relational properties of probabilistic programs, with our motivating example being the (\epsilon, )-variant of differential privacy .
Edit intent: clarity
Original:  We show how to extend Fuzz to encompass more general relational properties of probabilistic programs, with our motivating example being the (\epsilon, \delta )-variant of differential privacy .
Revised:  We show how to extend Fuzz to encompass more general relational properties of probabilistic programs, with our motivating example being the (\epsilon, \delta \delta)-differential privacy serving as a leading example .
Edit intent: clarity
Original:  First, we introduce the categorical notion of assignment on a monad to model composition properties of probabilistic divergences.
Revised:  First, we introduce the categorical notion of comonadic lifting of a monad to model composition properties of probabilistic divergences.
Edit intent: clarity
Original:  Then, we show how to express relational properties as sensitivity properties via an adjunction we call the path construction , reminiscent of Benton's linear and non-linear models of linear logic .
Revised:  Then, we show how to express relational properties in terms of sensitivity properties via an adjunction we call the path construction , reminiscent of Benton's linear and non-linear models of linear logic .
Edit intent: clarity
Original:  Then, we show how to express relational properties as sensitivity properties via an adjunction we call the path construction , reminiscent of Benton's linear and non-linear models of linear logic .
Revised:  Then, we show how to express relational properties as sensitivity properties via an adjunction we call the path construction .
Edit intent: clarity
Original:  Finally, we instantiate our semantics to model the terminating fragment of Fuzz , and extend the language with types carrying information about richer divergences between distributions.
Revised:  Finally, we instantiate our semantics to model the terminating fragment of Fuzz extended with types carrying information about richer divergences between distributions.
Edit intent: clarity
Original:  Finally, we instantiate our semantics to model the terminating fragment of Fuzz , and extend the language with types carrying information about richer divergences between distributions.
Revised:  Finally, we instantiate our semantics to model the terminating fragment of Fuzz , and extend the language with types carrying information about other divergences between distributions.
Edit intent: clarity
Original:  Our algorithm relies on a new, yet intuitive graph theoretic model of the optimization problem and performs binary search in a layered graph.
Edit intent: coherence
Original:  Moreover, we show that no deterministic online algorithm can achieve a competitive ratio smaller than  3.
Revised:  Moreover, we show that no deterministic online algorithm can achieve a competitive ratio smaller than ~ 3.
Edit intent: fluency
Original:  We develop a randomized online algorithm that is 2-competitive against an oblivious adversary and prove that 2 is a lower bound for the competitive ratio of randomized online algorithms  .
Revised:  In addition, we develop a randomized online algorithm that is 2-competitive against an oblivious adversary and prove that 2 is a lower bound for the competitive ratio of randomized online algorithms .
Edit intent: coherence
Original:  We develop a randomized online algorithm that is 2-competitive against an oblivious adversary and prove that 2 is a lower bound for the competitive ratio of randomized online algorithms  .
Revised:  We develop a randomized online algorithm that is 2-competitive against an oblivious adversary . Moreover, we prove that 2 is a lower bound for the competitive ratio of randomized online algorithms .
Edit intent: coherence
Original:  Finally, we address the continuous setting and give a lower bound of  2 on the best competitiveness of online algorithms.
Revised:  Finally, we address the continuous setting and give a lower bound of ~ 2 on the best competitiveness of online algorithms.
Edit intent: fluency
Original: Given a weighted graph G=(V,E) , and\to _+ and \pi:V}\to  U\subseteq V, the normalized cut value for U is defined as the sum of the weights of edges exiting U divided by the number of vertices in U.
Revised: Given a weighted graph G=(V,E) , and\to _+ and \pi:V}\to U\subseteq V, the normalized cut value for U is defined as the sum of the weights of edges exiting U divided by the weight of vertices in U.
Edit intent: clarity
Original:  The mean isoperimetry problem for weighted graphs {\it   is a generalization of the  sparsest cut problem in which, given a parameter k alongside a weighted graph G=(V, E), the objective is to find k disjoint nonempty subsets of V for which the average normalized cut value of the parts is as small as possible.
Revised:  The for weighted graphs {\it is a generalization of the sparsest cut problem in which, given a parameter k alongside a weighted graph G=(V, E), the objective is to find k disjoint nonempty subsets of V for which the average normalized cut value of the parts is as small as possible.
Edit intent: clarity
Original:  The mean isoperimetry problem for weighted graphs {\it   is a generalization of the  sparsest cut problem in which, given a parameter k alongside a weighted graph G=(V, E), the objective is to find k disjoint nonempty subsets of V for which the average normalized cut value of the parts is as small as possible.
Revised:  The mean isoperimetry problem {\it is a generalization of the sparsest cut problem in which, given a parameter k alongside a weighted graph G=(V, E), the objective is to find k disjoint nonempty subsets of V for which the average normalized cut value of the parts is as small as possible.
Edit intent: coherence
Original:  The mean isoperimetry problem for weighted graphs {\it   is a generalization of the  sparsest cut problem in which, given a parameter k alongside a weighted graph G=(V, E), the objective is to find k disjoint nonempty subsets of V for which the average normalized cut value of the parts is as small as possible.
Revised:  The mean isoperimetry problem for weighted graphs {\it is a generalization of the sparsest cut problem in which, given a parameter k , the objective is to find k disjoint nonempty subsets of V for which the average normalized cut value of the parts is as small as possible.
Edit intent: clarity